tintarev k., fieseler k. h. - concentration compactness(2007)(264)

CONCENTRATION COMPACTNESS

functional-analytic grounds and applications

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CONCENTRATION COMPACTNESS

functional-analytic grounds and applications

KyriI Tintarev Karl-Heinz Fieseler

Uppsala University, Sweden

Imper ia l Col lege Press

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CONCENTRATION COMPACTNESS Functional-Analytic Grounds and Applications

Copyright Q 2007 by Imperial College Press

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ISBN-13 978-1-86094-666-0 ISBN-I 0 1-86094-666-6 ISBN-13 978-1-86094-667-7 (pbk) ISBN-I0 1-86094-667-4 (pbk)

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Preface

The subject of this book, concentration compactness, is a method for establishing convergence, in functional spaces, of sequences that are not a priori located in a compact set. This situation occurs, in particular, in variational problems with functionals that are invariant under some non- compact group of operators, and therefore have non-compact level sets. The concentration compactness argument considers possible "dislocated" limits of the sequence, that is, limits under sequences of the "gauge" operators. The proof of convergence then can be based on elimination of the dislocated limits. Since a concentration compactness argument using blow- up sequences appeared in the paper of J. Sacks and K. Uhlenbeck [lo31 on harmonic maps and in the paper of H. BrCzis and L. Nirenberg [24] on a semilinear elliptic equation with a critical nonlinearity, the term "concentration", rooted in the use of unbounded sequences of dilations, has become a common designation for all convergence arguments involving dislocated limits, whatever group of transformations is involved. This was the term adopted in the celebrated series of four papers 1861, 1871, [88] and 1891 of P.-L. Lions, which laid the broad foundations of the method and outlined a wide scope of its applications. The book presents a function-analytic formulation of the concentration compactness, inspired by the connection between weak convergence under sequences of Euclidean shifts and convergence in LP made in the paper [80] of E. Lieb, the celebrated improve- ment of the Fatou Lemma, known today as BrCzis-Lieb lemma, [22], the use of the BrCzis-Lieb lemma in P.-L. Lions' subadditivity reasoning, and the "multi-bump" expansions of M.Struwe [lll], H.BrCzis and J.-M. Coron [25], P.-L. Lions [go] and numerous later works. The function-analytic theory of concentration compactness follows the spirit of the work of P.- L. Lions in one important respect: it gives attention to convergence of

v

vi Concentration Compactness

arbitrary sequences before studying properties of sequences that originate in specific problems. The functional-analytic framework for concentration compactness is the dislocation space H, D, where H is a separable Hilbert space and D is a fixed group of unitary operators on H, satisfying certain compactness-related properties. The purpose of endowing a Hilbert space with a group D is to define an enhancement of the weak convergence: we say that a sequence uk converges to zero D-weakly if for every sequence gk E D, gkuk - 0. A refinement of the Banach-Alaoglou theorem (weak compactness of the unit balls) then can be stated in terms of such convergence: any bounded sequence has a convergent subsequence that, after subtraction of all dislocated weak limits (terms of the form gkw, gk E Dl w E H), converges to zero D-weakly. If D is the group of all unitary operators, the D-weak convergence becomes convergence in norm, but the group is too large for the above decomposition to hold. On the other hand, the convergence result of Lieb ([80]) states that weak convergence in H1(RN) enhanced by the group of Euclidean shifts yields convergence in measure (which implies, together with the Sobolev imbedding, convergence in the correspondent space Lp(RN)).

We have selected the contents for the book in order to give an access- ible, rather than technical, presentation of the concentration compactness. We have opted to present the topic in Hilbert space, rather than Banach space, and included three chapters with background material: Chapter 1 - a compilation of theorems from functional analysis, Chapter 2 - a com- pendium on Sobolev spaces with focus on H1(R) and unbounded sets, and Chapters 7-8 on differentiable manifolds and Lie groups. The reader is expected to be familiar with basics of point-set topology, metric spaces and measure theory. The presentation of Sobolev spaces in Chapter 2 implicitly emphasizes the role of the conformal group of Euclidean space, an approach which is later generalized in the concentration compactness argument for a conformal group of a manifold in the treatment of subelliptic Sobolev spaces in Chapter 9. The functional-analytic grounds of the concentration compactness are presented in Chapter 3, followed by applications in Chapters 4, 5 and 6 to functions on Euclidean domains. Chapter 9 is an introduction of subelliptic Sobolev spaces on Lie groups, followed by some analogs of problems considered in the preceding chapters that involve subelliptic operators and "magnetic" Laplace-Beltrami operators on manifolds. Chapter 10 surveys several additional applications. The authors will use a follow-up web page www. math. uu. se /~ t i n ta rev / cc . html to provide additional materials, problems, corrections etc.

Preface vii

The authors acknowledge, with their unreserved gratitude, the role of Karen Uhlenbeck in initiating the theme of this book by her inspiring remarks on the role of transformation groups in analysis during her 1996 visit to Sweden. This led to discussions of the functional-analytic formal- ization of concentration compactness between one of the authors (K.T.) and Ian Schindler that yielded the core statement of this book, Theorem 3.1. The authors thank the head of CEREMATH (Univ.Toulouse I), Jacqueline Fleckinger, for financial support and the warm hospitality throughout the years. They acknowledge with appreciation the editorial involvement of Maria Esteban which brought forth the publication of the core theorem in [106].

The authors acknowledge with enthusiasm the crucial role of R.Schoen who encouraged writing a book on the subject. The first author would like to thank several mathematical departments for offering him visiting positions in 2003-2005 that allowed the work on the manuscript: University of California, Irvine; Technion - Haifa Institute of Technology; University of Queensland; University of Toulouse 1; University of Cyprus (with partial support from the University of Crete); Hebrew University at Jerusalem, and in particular the financial support of the Lady Davis Fellowship Trust and of the Ethel Raybould Fellowship; they also acknowledge a partial use of funds from the Swedish Research Council. The authors would like to extend their gratitude to their home department at Uppsala for allowing, for the final three months of writing, to reschedule part of their teaching to the following semester.

The authors thank J. Chabrowski for careful reading and commenting on a main portion of the manuscript, and V. Benci, D.-M. Cao, G. Ce- rami, H. Brkzis, E. Hebey, E. Lieb, V. Maz'ya, Y. Pinchover, M. Schechter and I. Schindler for their comments and remarks during the work on the manuscript. Their special gratitude is to Hildegard Fieseler and Sonia Pratt - Tintarev for their warm support and patience.

Karl-Heinz Fieseler, Kyril Tintarev

Contents

Preface v

1 . Functional spaces and convergence 1

. . . . . . . . 1.1 Definitions and examples of functional spaces 1 . . . . 1.2 Holder inequality . Young inequality for convolutions 4

. . . . . . . . . . . . . . . . . . . . . 1.3 ArzelbAscoli theorem 7 . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Hilbert space 8

. . . . . . . . . . . . . . . . . . . . . . . 1.5 Weak convergence 13 . . . . . . . . . . . . . . . 1.6 Linear operators in Hilbert space 16

. . . . . . . . . . . . . . . . . . . . 1.7 Differentiable functionals 20 1.8 Continuous and differentiable functionals in LP-spaces . . . 23

2 . Sobolev spaces 29

. . . . . . . 2.1 Weak derivatives . Definition of Sobolev spaces 29 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Chain rule 32

2.3 Coordinate transformations . . . . . . . . . . . . . Trace domains and extension domains 34

2.4 Friedrichs inequality . . . . . . . . . . . . . . . . . . . . . . 37 2.5 Compactness lemma . . . . . . . . . . . . . . . . . . . . . . 39

. . . . . . . . . . . . . . . . . . . . . . . 2.6 Poincarkinequality 41 2.7 Space V ~ ~ ~ ( I W ~ ) . Sobolev, Hardy and Nash inequalities . . . 43 2.8 Sobolev imbeddings . . . . . . . . . . . . . . . . . . . . . . 47

. . . . . . . . . . . . . . . . . . . . . 2.9 Trace on the boundary 51 2.10 Differentiable functionals in Sobolev spaces . . . . . . . . . 56 2.11 Sobolev spaces of higher order . . . . . . . . . . . . . . . . . 57

ix

x Concentration Compactness

3 . Weak convergence decomposition 59

3.1 D-weak convergence and dislocation spaces . . . . . . . . . 60 3.2 D-weak convergence in l 2 with shifts . . . . . . . . . . . . . 61 3.3 Weak convergence decomposition . . . . . . . . . . . . . . . 62

. . . . . 3.4 Uniqueness in the weak convergence decomposition 68 . 3.5 D-flask subspaces D-weak compactness . . . . . . . . . . . 69

3.6 D-weak convergence with shift operators in RN . . . . . . . 70 . . . . . . . . . . . . . . . . . . . 3.7 Constrained minimization 75

3.8 Compactness in the presence of symmetries . . . . . . . . . 77 3.9 The concentration compactness argument . . . . . . . . . . 79 3.10 Bibliographic remarks . . . . . . . . . . . . . . . . . . . . . 80

4 . Concentration compactness with Euclidean shifts 83

4.1 Flask sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.2 Existence of Sobolev minimizers on flask domains . . . . . . 89

. . . . 4.3 Rellich sets and compactness of Sobolev imbeddings 90 4.4 Concentration compactness with symmetry . . . . . . . . . 91 4.5 Concentration compactness and the Friedrichs

inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.6 Solvability in non-flask domains . . . . . . . . . . . . . . . . 95 4.7 Convergence by penalty a t infinity . . . . . . . . . . . . . . 98 4.8 Minimizers with finite symmetry . . . . . . . . . . . . . . . 100 4.9 Positive non-extremal solutions . . . . . . . . . . . . . . . . 102 4.10 Bibliographic remarks . . . . . . . . . . . . . . . . . . . . . 107

5 . Concentration compactness with dilations 109

5.1 Semilinear elliptic equations with the critical exponent . . . 109 5.2 Oscillatory critical nonlinearity and the minimizer in the

Sobolev inequality . . . . . . . . . . . . . . . . . . . . . . . 116 5.3 The Brkzis-Nirenberg problem . . . . . . . . . . . . . . . . 121 5.4 Minimizer for the critical trace inequality . . . . . . . . . . 126 5.5 A singular subcritical problem . . . . . . . . . . . . . . . . . 131 5.6 Minimizer for the Hardy-Sobolev-Maz'ya inequality . . . . 136 5.7 Bibliographic remarks . . . . . . . . . . . . . . . . . . . . . 137

6 . Minimax problems 141

6.1 The mountain pass theorem . . . . . . . . . . . . . . . . . . 142 6.2 Functionals for the semilinear elliptic problems . . . . . . . 145

Contents xi

6.3 Critical points of the mountain pass type . . . . . . . . . . 149 6.4 Mountain pass problems with the critical exponent . . . . . 155 6.5 Critical problem with punitive asymptotic values . . . . . . 157 6.6 Bibliographic remarks . . . . . . . . . . . . . . . . . . . . . 159

7 . Differentiable manifolds 161

7.1 Differentiable manifolds . . . . . . . . . . . . . . . . . . . . 161 7.2 Tangent vectors and vector fields . . . . . . . . . . . . . . . 164 7.3 Cotangent vectors and 1-forms . . . . . . . . . . . . . . . . 170 7.4 Tensor fields of degree 2 . . . . . . . . . . . . . . . . . . . . 172 7.5 Differential forms . . . . . . . . . . . . . . . . . . . . . . . . 175

8 . Riemannian manifolds and Lie groups 181

8.1 Riemannian manifolds . . . . . . . . . . . . . . . . . . . . . 181 8.2 Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 8.3 The exponential map . . . . . . . . . . . . . . . . . . . . . . 193 8.4 Lie group actions . . . . . . . . . . . . . . . . . . . . . . . . 197 8.5 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 8.6 Bibliographic remarks . . . . . . . . . . . . . . . . . . . . . 201

9 . Sobolev spaces on manifolds and subelliptic problems 203

9.1 Sobolev inequality on periodic manifolds . . . . . . . . . . . 203 9.2 “Magnetic” Sobolev space . . . . . . . . . . . . . . . . . . . 205 9.3 Magnetic shifts and D-convergence . . . . . . . . . . . . . . 206

Carnotgroups . . . . . . . . . . . . . . . . . . . . . . . . . . 210 9.4 Subelliptic mollifiers and Sobolev spaces on

9.5 Compactness of subelliptic Sobolev imbeddings . . . . . . . 217 9.6 Subelliptic Friedrichs and Poincark inequalities . . . . . . . 218 9.7 Subelliptic Sobolev inequality . . . . . . . . . . . . . . . . . 221 9.8 Concentration compactness on Carnot groups

due to shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 9.9 Concentration compactness on Carnot groups

due to dilations . . . . . . . . . . . . . . . . . . . . . . . . . 224 9.10 Bibliographic remarks . . . . . . . . . . . . . . . . . . . . . 227

10 . Further applications 23 1

10.1 Dilations on the sphere and Yamabe problem . . . . . . . . 231 10.2 Global compactness in spaces H”(RN) and D”. 2 ( R N ) . . 232

xii Concentration Compactness

10.3 Minimizer in the Nash inequality . . . . . . . . . . . . . . . 10.4 A minimization problem with nonlocal term . . . . . . . . . 10.5 Concentration compactness with topological charge . . . . . 10.6 Bibliographic remarks . . . . . . . . . . . . . . . . . . . . .

Appendix A Covering lemma

Appendix B Rearrangement inequalities

Appendix C Maximum principle

Bibliography

236 237 240 244

247

249

251

253

Index 26 1

Chapter 1

Functional spaces and convergence

In this chapter we include some background material concerning LP-spaces and the weak convergence in Hilbert space. The reader is expected to know basics of linear algebra, measure theory and metric spaces. The few statements that we give with only partial proofs can be found in many textbooks on functional analysis, in particular in [124].

1.1 Definitions and examples of functional spaces

Definition 1.1 A norm on a real vector space V is a function 1 1 : V -+

[0, co) satisfying the following conditions:

( I ) [lull = 0 e=. v = 0 ( 2 ) Homogeneity: IIXvII = I X I . llvll 'v'X E IR,v E V . (3) Triangle inequality: Ilv + wll < \lull + Ilwll VV,W E V

A normed vector space is a real vector space V together with a norm 1 1 . 11. Remark 1.1 Any normed space defines a natural metric d : V x V 4 IR , namely d ( v , w ) := Ilv - wll; in particular we can define the open ball

B T ( v ) := {w E V ; d(v , W ) = ( ( v - w J J < r )

with radius r > 0 around a vector v E V as well as convergent sequences: A sequence of vectors v,, n E N, converges to a vector v E V , written as limn,, v, = v or simply v , + v if and only if for any E > 0 there are only finitely many v , lying outside the open ball B, (v) , or, equivalently, if llv, - v11 -+ 0 in R.

Definition 1.2 Let V be a normed vector space. A sequence v,, n E N, is called a Cauchy sequence, if for any E > 0 there is a number no E N, such

a

2 Concentration Compactness

that llun - un,II < E for all n 2 no. A normed vector space V is called complete or a Banach space if any

Cauchy sequence in V is convergent.

It is immediate from the definition that any convergent sequence is a Cauchy sequence. The converse is in general not true.

Example 1.1

(1) Take V := RN and for x := (xl, ..., XN) define 11x11 := Jxf + ... + x$, the Euclidean norm . We will follow the convention to denote the norm in RN as I . I rather than 1 1 . 11. Other possible, metrically equivalent choices are 1x1 := max{lxll, ..., IxNl), the maximum norm, or 1x1 :=

lxll + ... + lxNl, but, unless specified otherwise, the notation of the norm in EXN will always refer to the Euclidean norm. The space RN is complete.

(2) Let X be a topological space (e.g. a set in RN). The normed space C(X) is the vector space of all real valued bounded continuous functions on X equipped with the norm

Ilf l l := SUP If (XI(. x E X

The space C(X) is complete. (3) Let R C IRN be a Lebesgue measurable set, p E [ l ,m) and let

w : R + (0, oo) be a measurable function. Then the set IP(R, w) consisting of all Lebesgue-measurable functions f : R + R such that

1 f(x)IPw(x)dx < oo constitutes a vector space, containing as a subspace the set N(R) of all measurable functions f : R -+ R with f (R \ 0) c R being of Lebesgue measure 0. We denote LP(R, w) the factor space IP(S1, w)/N(R), and, following the usual convention, shall not distinguish in our notation between a function f E IP(R, w) and its residue class f + N(R) E LP(R, w). The space LP(R, w) is equipped with the following norm (the "p-norm")

1

if I := I l f llP := (l if (x)l~w(x)dx)

If w = 1, one writes LP(R) instead of LP(R,w). The space IW(R) consists of all a.e. bounded Lebesgue-measurable functions with the norm

sf I 1 := Ilf llW := , ~ ~ o x , " , p A I f (')I

Chapter 1 Functional spaces 3

and the space Loo (R) is, as above, the factor space IP(R)/N(R). The spaces LP(R, w), as well as Lm(R), are complete, and therefore, Banach spaces.

(4) Let R c IRN be an open set. The space Cm(R) consists of functions that are continuous and bounded in R, and whose derivatives up to the order m are continuous and bounded in 52, and it is equipped with the norm

Here the notation DOu refers to u, Dku, k E N, is the n-tuple, n = Nk, of all partial derivatives of order k and 1 . I is any (fixed) norm on I tNk. In these notations C(R) may be also referred to as CO(R). The spaces Cm(R) are complete (an elementary proof is based on completeness of R and uniform convergence). The subspace of Cm (R), m = 0,1, . . . that consists of functions whose support supp u = {x E R, u(x) # 0) is compact in R will be denoted Cr (R) .

(5) When X and Y are two normed vector spaces contained in a vector space V, one considers their intersection X n Y as a normed vector space equipped with the norm 1) . JJx + 1) JJr It is immediate that if X and Y are complete, so is X n Y.

We will also consider some vector spaces of functions without an assigned norm.

(1) Let X be a metric space. If u is continuous on X we shall write u E

Cloc(X). If X is compact, then CI,,(X) = C(X). (2) When R c IRN is an open set we define the vector space Clot( R ) asthe

space of functions continuous in R that have derivatives up to order m < co that are all continuous in 52. In contrast to that notation, the vector space of functions that have continuous derivatives in R of any order is called simply Cw(R) . The subspace of Cm(R) of functions with compact support in R is denoted Cp(R) .

(3) When R c RN is an open set and p E [I, co), the vector space Lyoc (R) is the space of functions (or more precisely, classes of equivalent functions modulo a.e.) whose restriction to any set K whose closure is compact in R (we write K g 0 ) is in LP(K).


1.2 Holder inequality. Young inequality for convolutions

Proposition 1.1 Holder inequality. Let R c EXN be a measurable set, let p 2 1 and let q E (1, +m] satisfy $ + = 1. For every f E LP(R),g E Lq(R), their product is an integrable function, i.e. f g E L1(R), and

Proof. Assume without loss of generality that 1 1 flip > 0 and llgllq > 0. 1. Case p = 1. By the definition of the Lm-norm, for every E > 0 there is a set A, with lAE 1 = 0 such that lg(x) 1 5 11g11 + E for a11 x E R \ A,. Then

which, since E is arbitrary, proves (1.1). 2. Case p > 1. Since the function t H $ + - t , t > 0, attains its minimal value 0 at t = 1, the substitution t = + yields the inequality

b p -

From here follows, since I f lp, lglQ E L1 ( 0 ) that 1 f gl E L1 ( a ) . Moreover, substituting a = and b = and integrating over R, we have

which immediately implies (1.1).

Iteration of (1.1) yields the following

Corollary 1.1 Let 52 c RN be a measurable set, let pi > 1, i = 1,. . . , m satisfy EL1 A = 1 (allowing values pi = m with = 0). If fi E LPi(R), i = 1 , . . . , m, then their product is an integrable function, i.e. IIzl fi E L1(R), and

Let V, W be normed vector spaces. A linear map T : V -, W is continuous if and only if it is bounded on the closed unit ball Bl(0) c V, i.e. if and only if

llTll := SUP{IIT(V)II; v E Bl(0)) < .

(1.1)

ew ext


We denote L(V, W) the set of all continuous linear maps from V to W; endowed with the above norm it becomes a normed vector space. In the case W = R we obtain the dual space V* := L(V, R) of the normed vector space V. From the Holder inequality one derives easily that Tg : LP(R) -+

R, f H Tg(f) := Ja f (x)g(x)dx with g E Lq(R) defines a continuous linear form Tg E LP(R)* satisfying llTgll 5 llgllq. With other words, the linear map T : L4(R) + LP(R)*,g H Tg, is continuous and J JTJJ < 1. In fact it is known that \IT11 = 1 and that it establishes an isometric isomorphism between L4(R) and LP(R)*.

Using the well known fact that for any open bounded set R C RN, C(R) is a dense subspace of LP(R), p E [I, co), we show that even the smaller space C r ( R ) is dense in LP(R) for any open set R c RN:

Lemma 1.1 Let R c RN be an open set and let p E [l, m). The space C r ( R ) is dense in LP(R).

Proof. 1. Assume first that R is bounded. Let Rd = {x E 0, d(x, RN \R) > 61, 6 > 0. Take any function u E Co(R6). Let p E C r (B1 (O), [O, 11) \ (0) be such that J p = 1 (if not, divide p by Jp), let pt(x) = t-Np(t-lx), 0 < t < 6 , and set

It is easy to see that ut E C r ( R ) . Moreover,

Ilut - ullm = sup / lu(x + ty) - u(x)Ip(y)dy zEaa Bl(0)

5 sup lu(x+z)-u(x)I. ~ E f i d ~ I z l < t

Since u is continuous and defined on a compact set, it is uniformly continuous. Therefore llut - ulloo, and thus, by the Holder inequality, llut - ullp, can be made arbitrarily small by choice of t .

2. The assertion of the lemma follows now once we verify that Co(R) is dense in LP(R). Let u E L*(R). Since JulP is integrable in R, for every E > 0 there is R > 0 such that ( (u - x ~ ~ ( ~ ) u ( ( ~ < E . Thus without loss of generality we may assume that R is bounded. From the integrability of (u(P in R follows that Ja,n, (u(P + 0 as b -+ 0. Then, from the density of C(R) in LP(R), using multiplication by a "cut-off' function from Cr(s2,) that equals 1 on Raa, the density of Co(R) in LP(R) is immediate.


Definition 1.3 The convolution is the following map cr(IRN) x c,-(rwN) -, (7,-(RN):

Due to Lemma 1.1 and the statement that follows the operation of convolution can be extended by continuity as a map from LQ x Lr to L3 with s = s(q, r ) defined below.

Proposition 1.2 (Young inequality for convolutions) Let u E

L Q ( I R ~ ) , v E L " ( R ~ ) and 11 E L 3 ( P N ) , where q, r , s 2 1 and $ + f + $ = 2. Then

Proof. Assume without loss of generality that u , v and 11 are nonnegative. Let

Noting that $ + $ + 5 = 1, we have due to (1.2)

IIN $(x) (u * v ) (x )dx l< Ilallrt I I ~ I I ~ / I I C I I ~ ~ ,

Let us estimate Ilallrt.

Similarly b 5 u and C ~ I $ 2 Il$:IIvII;. Substitution of these estimates into (1.5) gives (1.4).

Corollary 1.2 Let u E LQ(RN) , v E L ' ( I R ~ ) , where q,r > 1. Then 1 u * v € Lp(IRN), where , + l = ; + f and


Proof. If p > 1, then (1.6) follows from (1.5). If p = 1, then with necessity q = r = 1 and (1.6) follows immediately from the identity

The best constant in the inequality (1.4) resp. (1.6) is generally less than one. For the proof with the best constant see [82], Section 4.2.

1.3 Arzeld-Ascoli theorem

Definition 1.4 Let K C RN be a compact subset. A subset M C C(K) is called equicontinuous if for every E > 0 there is a 6 > 0 such that for all f E M and x, y E K we have Ix - yI < S ==+ I f (x) - f (y)I < E. It is called pointwise bounded if for any x E M the set {f (x), f E M ) is bounded.

Theorem 1.1 (Arzelh-Ascoli) Let K c RN be a compact set. A subset M c C(K) is relatively compact if and only if it is pointwise bounded and equicontinuous.

Proof. We recall first that a subset M of a metric space is relatively compact if and only if every sequence of points in M has a convergent subsequence. (L-X Since a sequence f, E C(K) with l l f n l l -+ co does not have a convergent subsequence, any relatively compact set M is bounded and in particular pointwise bounded. Assume it is not equicontinuous. Then we can find an E > 0 and sequences f, E M , x,, y, c K such that IX,- y,J -+ 0 and I fn(xn) - fn(yn)l > E for all n E N. We may replace the given sequences by convergent subsequences, i.e. assume that x, -+ xo E K, y, -+ yo E K and fn -+ f E C(K). But then necessarily yo = xo and 0 = I f(xo) - f (yo) 1 = limn,, I fn(xn) - fn(yn) 1 > E, a contradiction. "+": First of all note that there is a dense countable subset D c K: For m E N choose a finite set Dm C K such that the open balls with radius cover K, then take D := Uz==, Dm. Now it suffices to find a subsequence gk = fn,, such that gk(x) converges for every x E D: Take E > 0. Choose m E N, such that )x - yl < 1 f (x) - f (y)J < for all f E M and x, y E K, furthermore an index Ice E N with Igk(x) - gk, (x)J < $ for all k 2 Ico and x E Dm. Then consider an arbitrary point y E K and choose


x E Dm with lx - yI < 5 . Then

So lgk - gko I < E for k > kg. Altogether we have seen that gk E C(K) is a Cauchy sequence in the Banach space C(K) , hence convergent.

It remains to construct the subsequence gk. By induction we define for every m E N subsequences g p , k E N, which converge on Dm. Take g: := fk . Assume g p has been found. That sequence admits a subsequence g z k ) converging on all points of the finite set D,n+l\Dm, since the sequences

gp(x) with x E Dm+1 \ Dm are bounded. Take gp+l := gzk) . Eventually

we take the diagonal sequence (gk) with gk := gE.

1.4 Hilbert space

On RN the Euclidean norm plays a particular role due to the fact that it is induced by an inner product. In the following, assuming familiarity with finite-dimensional inner product spaces, we outline the infinite-dimensional analog of Euclidean space.

Definition 1.5 An inner product on a vector space H is a symmetric bilinear map

which is positive definite, i.e. (v,v) > 0 for all v E H with equality iff v = 0. A vector space H together with such an inner product is also called a pre-Hilbert space. It becomes a normed vector space together with the norm

Two vectors v, w E H are called orthogonal, written as v I w if (v, w) = 0. A complete pre-Hilbert space is called Hilbert space.

The proof of the triangle inequality for 1 1 - 1 1 uses another important inequality, the Cauchy-Schwarz inequality :


the proof of which is the same as in the finite dimensional case. An important consequence is the fact that (., .) is a continuous function on H x H due to the fact that 11 . 11 is continuous on H and the estimate

holds.

Remark 1.2 The inner product can be reconstructed from the associated norm:

Indeed a norm ( 1 . 1 1 o n a vector space H is induced by an inner product (., .) iff i t satisfies the parallelogram law

for all v, w E H , i.e., the s u m of the squares of the lengths of the diagonals of a parallelogram equals the sum of the squares of the lengths of its four sides.

Example 1.2

(1) The space L 2 ( R ) is a Hilbert space with the inner product

(2) The space

is a Hilbert space with the inner product

One of the most important geometric features of a Hilbert space is that given a closed convex set K c H , for every point v E H there is a unique point in K closest to v:


Proposition 1.3 Let K C H be a closed convex subset of the Hilbert space H , i.e. with any points u , w E K we have [u, W ] c K for the line segment [u, w ] := { t u + ( 1 - t ) w ; 0 5 t 5 1 ) with end points u, W . Then for any v E H there is a unique vector P ( v ) E K , such that

Ilv - P(v)II = dist(v, K ) := inf{llv - 2111 : u E K ) .

In fact, the map P : H -+ K is a projection onto K , i.e. PIK = idK resp. p2 = P .

Proof. T h e existence and uniqueness o f the vector P ( v ) follows from t h e fact that any sequence (u,) c K wi th llu, - vll tending t o d := dist(v, K ) is a Cauchy sequence and hence has a limit in t h e Hilbert space H . For this we may assume v = 0. T h e n we have

for all n, m E N . T h e right hand side converges t o 4d2, while t h e second t e rm o f the left hand side satisfies

because o f 2-'(un + urn) E K . Hence the first t e rm is arbitrarily small for sufficiently big indices n, m.

Proposition 1.4 Let U c H be a closed subspace of the Hilbert space H . Then for any v E H there is a unique vector P ( v ) E U , such that

I I v - P(v)II = dist(v, U ) := inf{llv - ull : u E U ) .

In fact, P : H a H is a continuous linear projection onto U , i.e. P E

L ( H , H ) and P satisfies P ( H ) = U and PIu = idu. Furthermore,

is the orthogonal complement of U . In particular

Proof. W e may apply Proposition 1.3 with K = U . I t provides the projection map P : H --+ U . Indeed

P-'(o) = U' := { w E H ; ( w , ~ ) = 0 V u E U ) . (1.10)

I f v E U L , then Ilv - u1I2 = 1 1 ~ 1 1 ~ + l l ~ 1 1 ~ for u E U , SO u = 0 is the vector in U closest t o v . O n t h e other hand, i f P ( v ) = 0 and u E U , then the


differentiable function (lv - tu1I2 = 1 1 ~ 1 1 ~ - 2t(u, v) + t211u112 with t E R attains at t = 0 its minimum, hence (u, v) = 0. Since P ( u + v) = u + P(v) for u E U = P(H), we get

for any v E H resp. U + U' = H . But (., .) being positive definite we have U n U L = {0}, such that H = U $ U L and any vector w E H has a unique decomposition w = u + u'. From this we obtain readily that P is a continuous, (indeed contractive) linear operator with P2 = P .

Remark 1.3 The above fact that any closed subspace U C H admits a closed complementary subspace W characterizes Hilbert spaces: Any Ba- nach space V with that property is isomorphic to a Hilbert space H , i.e. there is a linear isomorphism F : V -+ H such that both F and F-' are continuous.

An important consequence of Proposition 1.4 is

Theorem 1.2 (Riesz Representat ion Theorem) The map

with Tw : H -+ R given by Tw(v) := (v, w) is an isometric isomorphism, i. e. T is an isomorphism of vector spaces preserving lengths: l(Tw(l = (1~11.

Proof. The map T is obviously linear; the Cauchy-Schwarz inequality gives ITw (v) 1 5 ((v(( . ((w(( = ((w(( . llvll, whence (ITw (1 5 ( ( ~ ( 1 , while setting v := w yields (ITw 1 1 > ( ( w 11. In particular, T is injective. Now let cp : H -+ R be a continuous linear functional. Denote P : H -t H the orthogonal projection onto U := ker(cp). Then there is a unique vector u E ker(P) =

ker(cp)' satisfying p(u) = 1, and we have cp =Tw, where w := l l ~ l l - ~ u . So T is also surjective.

In order to do explicit calculations in a finite dimensional vector space one needs bases. For Hilbert spaces there is a corresponding notion:

Definition 1.6 Let H be a pre-Hilbert space. A sequence of vectors en, n 2 1, is called orthonormal if (en, em) = dnm. It is called an orthonormal basis if the subspace generated by the vectors el, e2, ... is dense in H .


Proposition 1.5 Let (en)nEN be an orthonormal basis in the pre-Hilbert space H . Then, for any v E H we have

v = C ( v , e n ) e n := lim C ( v , e n ) e n m-oo n= 1 n=l

and the following equality (known as Parseval equality)

holds true.

Proof. For v E Ho := span(e1, ez, ...) we write v = Cr=l Akek and take the inner product of both sides with en in order to see An = ( v , en). In the general case we have to approximate by vectors in Ho: An arbitrary vector v E H can be written

where m E N is arbitrary and vm I e l , ..., em, as is easily seen. We have to show that vm -, 0. Let E > 0. Take any vector w E Ho with Ilv - w1I2 < E

and choose mo E N with w = C ~ ~ l ( w , e n ) e n . That equation holds also with any m 2 mo instead of mo, in particular

with other words lim,,, vm = 0 resp. v = lim,,, CT=l(v , en)en. Fi- nally Parseval's equality follows from the fact that the norm is a continuous function.

We conclude that any vector v E H has a unique representation

where the coefficients An := ( v , en) satisfy C r = l A: < oo, and we can look at X = (A1, X 2 , . . . ) as a sort of coordinate sequence for the vector v E H . Conversely, whenever Cr==l A: < oo, the sequence of partial sums ( C y = l Xnen)mEN is a Cauchy sequence in H , and if H is a (complete)


Hilbert space, the sequence converges. Hence, if H is a Hilbert space admitting an orthonormal basis (en)nEN, then it can be identified with the vector space C2 by

in analogy to the isomorphism Rn E V, n = dim V, for a finite dimensional vector space V. Furthermore note that H has an orthonormal basis if and only if there is a countable dense subset. Indeed, denote Ho the span of that set. If dim Ho < oo, we have H = Ho, otherwise Ho is a dense subspace with a countable basis (V,),~N Using the (finite dimensional for every step) Gram-Schmidt orthogonalization procedure we can construct by induction from that basis of Ho an orthonormal basis (en)nGw for H .

Definition 1.7 A normed vector space is called separable if it admits a countable dense subset.

Example 1.3

(1) Both L2(R) and C2 are separable. For C2 that is immediate, while the rational linear combinations of characteristic functions of open boxes contained in R with rational vertices form a countable dense subset of L2 (R) .

(2) Any closed subspace U c H of a separable Hilbert space is itself separable, in particular admits an orthonormal basis: The image P(X) of a dense countable subset X c H under the orthogonal projection P : H + H onto U is countable and dense in U .

1.5 Weak convergence

Definition 1.8 A sequence (V,),~W C V in a normed vector space V is said to converge weakly to a vector v E V, written as w lim,,,~, = v or simply v, --\ v, if for all cp E V* we have limn,, cp(v,) = cp(v).

Note that if v, -+ v in V then for every cp E V*,

which implies v, --\ v, while the converse in general is false. For example, if en is an orthonormal basis in an infinite-dimensional Hilbert space, then, by Parseval's equality, for cp = T, (cf. Theorem 1.2) we obtain En lcp(en)I2 =


C , ( W , ~ , ) ~ = l l ~ 1 1 ~ < m and thus cp(en) -+ 0 although llenll = 1. The relation cp E V* + cp(un) -+ 0 defines therefore a weaker convergence.

Remark 1.4 The weak limit of a sequence i n a Hilbert space is unique, since any u E H , u # 0 , is separated from 0 by a linear functional ( . ,u) . The same is true for general normed vector spaces, where continuous linear functionals separate points as a consequence o j the Hahn-Banach theorem.

In what follows we consider weak convergence only for Hilbert spaces.

Proposition 1.6 If un --\ u i n a Hilbert space H , then

and

JJuJJ I liminf JJunJJ. (1.14)

Proof. From the linearity of the scalar product follows

Since the right hand side converges to zero by definition of weak convergence, (1.13) follows. Relation (1.14) is immediate.

Theorem 1.3 Uniform Boundedness Principle: A sequence u,, n E N, i n a Hilbert space H is bounded if and only if for any w E H the sequence ( w , un) is bounded.

Proof. "=+": This is an immediate consequence of the Cauchy-Schwarz inequality I(w,un)I I llwll . IJunlJ. "t" : For an unbounded sequence un, n E N, we construct a vector w E H, such that the sequence ( w , u,) is unbounded.

We may assume limn,, llunll = m or even llunll = 4n resp. u, = 4,vn with vectors v, of length 1. Namely, for any n E N there is a kn such that lluknII 2 dn, then replace un with the sequence 6 , := 4n l l~knI l -1~k , . We define

where ak = f 1 is inductively chosen: We take a1 := 1, and given a l , ..., u k - 1 we let

Chapter 1 Functional spaces

using the convention sign(0) = 1. Now

--

( w , u,) = 4, C 0 i 3 - ~ (vi , u,)

and thus

Corollary 1.3 Every weakly convergent sequence in a Hilbert space is bounded.

Proof. Let u , --\ u in H . Then for every v E H , (u,, u ) = ( u , - u , v ) + ( u , v ) is bounded. Therefore the sequence u,, n E N, is bounded by Theorem 1.3.

Corollary 1.4 Let u , - u and v , + v be sequences in the Hilbert space H . Then limn,, (u,, v,) = (u, u ) .

Proof. By Corollary 1.3 the sequence u,, n € N is bounded, i.e. there is M > 0 such that IIun(l 5 M for all n E N. Thus

An orthonormal basis in an infinite-dimensional Hilbert space is a bounded sequence that has no convergent subsequence: the distance between any two different terms in the sequence is Ile, - e,ll = Jlle,112 + llen112 = JZ. An analog of the Bolzano-Weierstrass theorem for Hilbert spaces cannot therefore assert convergence in norm, but it holds true in the sense of weak convergence.

Theorem 1.4 Banach-Alaoglu Theorem. Let v,, n E N, be a bounded sequence in the separable Hilbert space H . Then there is a subsequence u,,, k € N, which converges weakly to some v E H .

15


For the proof of the theorem we need the following

Lemma 1.2 Let X c H be a dense subset of the Hilbert space H and let v, E H , n E W, be a bounded sequence. I f the limit limn,,(w,v,) exists for all w E X , then vn - v for some vector v E H . If (w, v,) -t 0 for all w E X, then v = 0.

Proof. We show first that the sequence (w, v,), n E N, converges for all w E H and then determine its weak limit. Choose M 2 1 with llvnll < M for all n E N. Let E > 0. Choose wo E X with Ilw - wo(( < & < 5 . Now take no E N, such that I (wo, v,) - (wo, v,,) 1 < 5 for all n 2 no. Then

So the sequence (w, v,), n E N, is a Cauchy sequence and hence converges. It is obvious that cp(w) := lim,,,(w, v,) defines a linear functional; in fact since [(w, vn)l < IIvnll . llwll < M 11wll for a11 n E N it is even bounded: (cp(w)( < M((w(1 for all w E H. According to the Riesz representation theorem, Theorem 1.2, we have cp(.) = (.,v) for some v E H I in other words, v, - v. Finally, since cp is continuous and X dense, cpJx = 0 implies cp = 0 resp. v = 0.

Proof of Banach-Alaoglu's theorem. Let wk, k E N, be a dense sequence in H. If for all k E N the sequence (wk,vn), n E N, already converges, we may apply the previous lemma with X := {wk; k € N). So it remains to show that starting with the sequence vn, n E N, we can pass to a subsequence satisfying our assumption: We define for j E N successively sequences v i , n E N, such that the sequence (wk,vi), n E N, converges for 1 5 k 5 j . Take v: := v,. Assume vi , n E N, is constructed. Then, the sequence (wk+], vi) , n E N, being bounded, let ( w ~ + ~ , vie) , e E N, be a convergent subsequence and set v;+' := vil. Finally the diagonal sequence v:, n E W, has the desired property.

1.6 Linear operators in Hilbert space

In this section we consider the vector space C(H) := C(H, H ) of all bounded linear operators. A multiplication of operators in C(H) is defined by


(AB)v := A(Bv), v E H, and since IlABll < IlAll . llBlll it is a continuous operation. The adjoint operator A* of a bounded linear operator A E C(H) is defined as follows.

Definition 1.9 Let H be a Hilbert space and T : H -+ H* the Riesz representation isomorphism. Then the adjoint operator A* of a bounded linear operator is defined as

In order to be able to do calculations with the adjoint operator, note that A* is uniquely determined by the relation

that holds for all v, w E H, since any operator B E C(H) is uniquely determined by the values (v, Bw) = (Bw, v), e.g. for a separable Hilbert space H with an orthonormal basis { e n I n E ~ one has B(w) = X,(Bw, e,)e,. That leads to the following equalities

(A*)* = A, (A + B)* = A* + B*, (/!A)* = XA*, (1.15)

(AB)* = B*A*, IIA*ll = IlAll.

An operator A-' : H -, H is called an inverse of A if A-'A = AA-' = id. Since id = (AA-I)* = (APIA)* = (A-')*A* = A*(A-I)*, we conclude that if A has an inverse, then A* has an inverse given by

The space C(H) of bounded linear operators carries not only a natural structure of a normed vector space, but is in fact a Banach space. The corresponding notion of convergence for a sequence of operators A, G C(H) is that of uniform convergence on bounded subsets resp. on the closed unit ball B1(0) c H . That is a quite strong requirement, in practice pointwise convergence - strong or weak - has a much better chance to be realized:

Definition 1.10 We say that a sequence A, E C(H) converges strongly resp. weakly to the operator A, if for all v E H we have A,v -+ Av resp. A,v Av. In case of weak convergence we shall use the notation A, - A.

Lemma 1.3 Let X be a dense subset of H and let A, be a bounded sequence of bounded linear operators on H. If for every u , v E X, (A,u,v) -+ 0, then A, - 0. If for every u E X , A,u + 0, then A, converges to 0 strongly.


Proof. For the first part we have to show ( A n y v ) -+ 0 for all u , v E H . Fix first u E X . Then the sequence Anu, n E N, is bounded and so we may apply Lemma 1.2 and obtain, that we have convergence to 0 for any v E H. Now apply the same argument once more to (u , A z ( v ) ) in order to see that we may take u to be any vector in H. Here we have used, that the sequence A; is also bounded. For the second part, let u E H , let E > 0 and let u, E X , Ilu, - u11 5 E . Then

limsup llAnull 5 limsup IIAn(u - u,))II + 1imsl-l~ IIAnuEII n-tw n+m 12-00

5 sup ((A,IJE + limsup JJAnucIJ = SUP IIAnII&. n n+m n

Since E is arbitrary, A,U -+ 0.

Lemma 1.4 If A E L ( H ) and un ---\ 0 then Au, - 0. If A, E L ( H ) and An - 0 then A; - 0.

Proof. Let w E H. Then

(Auk, W ) = (uk, A* W ) -+ 0 ,

which proves the first assertion, since any linear functional on a Hilbert space is of the type cp = T, = ( . ,w) with some w E H. The second assertion follows from

Definition 1.11 A bounded operator A E L ( H ) is called self-adjoint if A* = A, it is called isometric if A*A = id and it is called unitary if it has an inverse and A* = A-l.

Note that a unitary operator is always an isometry. The converse is not true: the isometry A E L(12), (AX) = (0, XI, X p , . . . ) is not surjective.

Lemma 1.5 If an operator A E C ( H ) is self-adjoint and there is a X > 0 such that for all u E H

then A h,as an inverse and IIA-lll 5 A-l.

Proof. From (1.17) follows that A is injective. Let us show that A is surjective. Note first that the range AH of A is closed. Indeed, if Au, -+ w E H , then by (1.17) un is a bounded sequence, and therefore


by Theorem 1.4 has a subsequence unk - u with some u E H . Then by Lemma 1.4 w = Au E A H and, consequently, A H is closed. Let v E (AH)'. Then, since A2v E AH and taking into account A = A*, we have 0 = (A2v, v) = (Av, A*v) = (Av, Av) 2 Xll~11~, which implies that (AH)I = (0), i.e. that A is surjective. Therefore, A-l : H 4 H is defined. Substituting u = A-'v, v E H into (1.17), we get IIA-l~I( 5 X-lllvlll which proves the lemma.

Definition 1.12 One says that a normed vector space X is continuously imbedded into a Banach space Y if there is a bounded linear injection T : X + Y. The operator T is called imbedding operator . One says that the imbedding is compact if the set {Tu : u E X, llullx 5 1) is relatively compact in Y.

Remark 1.5 Note that for an imbedding operator T : X -+ T(X) is a linear isomorphism, but in general not a homeomorphism!

Proposition 1.7 Let Y be a Banach space and let y E Y. If for every q E Y*, q(y) = 0, then y = 0.

Proof. This statement is an equivalent form of the Hahn-Banach theorem (see [124] or any other comprehensive textbook on functional analysis). We give here a proof in two specific cases. When Y is a Hilbert space H , any linear functional has the form y H T,(y) = (y, v) with some v E H , and (y, v) = 0 for any v implies (y, y) = 0 and thus y = 0. When Y = Lp(R), p E [I, m), due to the Holder inequality, the functional u H Sn gu, g E Lq with + $ = 1, is continuous. Since g = ~ l y l p - ~ is in Lq, gy = lylP E L1 and

implies y = 0.

Proposition 1.8 Assume that a Hilbert space H is compactly imbedded into a Banach space Y and let T be the embedding operator. If u E H and uk E H , k E N, uk - u in H , then IlTuk - Tull + 0.

Proof. Assume without loss of generality that u = 0. Note first that any convergent subsequence of Tuk converges to 0. Indeed, assume that Tukj + w . Then for every L E Y*, LTukj + Lw. However, since ILTuJ 5 IILlly* IITIIY+HIIUII for d l u E H , L T is a continuous functional on H and since ukj - 0 on H , LTukj -+ 0 and consequently, Lw = 0 for every L E Y*. By Proposition 1.7, w = 0. Assume that Tuk does not converge to


0. Then there exists an E > 0 and a subsequence ukJ such that IITukj I l l y 2 E .

Since uk is weakly convergent, uk is bounded (Corollary 1.3). Then, since the imbedding is compact, the sequence Tukj has a convergent subsequence, which by the first step of the proof converges to 0, a contradiction.

1.7 Differentiable functionals

Definition 1.13 Let V be a vector space and let G : V --t R and u, v E L. If the derivative

dG(u + tv) DvG(u) := dt It- to

exists, it is called the derivative of G at the point u in the direction v. If D,(u) is defined for all v E V and the map G1(u) : v ++ D,G(u) is linear, G1(u) is called the linear derivative of G at u.

Problem 1.1 Show that if G : V -t R has a point of minimum at uo E V and if G has a linear derivative at uo, then G1(uo) = 0.

Problem 1.2 (Lagrange multipliers rule) Let F, G : V --+ R. Show that if the infimum inf,Ev,F(u)so G(u) is attained at some uo E V, and F and G have linear derivatives at uo, then F1(uo) and G1(uO) are linearly dependent.

Sketch of the proof: Step 1. Reduce the problem to finite dimensional subspaces of V containing uo, taking into account that the coefficients in linear combinations of F1(uo) and G1(uo) can be considered in the intersection spaces and thus are independent of the subspace. Note that it suffices to consider Ff(uo) # 0 and G1(uO) # 0. Step 2. There exists a vector v such that F1(uo)v < 0 and G1(uO)v < 0. Let ut = uo - tv. Then, by linear differentiability, F(ut) < 0 and G(ut) < G(uo) for all t > 0 sufficiently small, which contradicts the definition of the infimum.

Proposition 1.9 Let V be a normed vector space. Let U c V be an open set and let G : U -+ R. If D,G(u) exists for all u E U, v E V and the map (u,v) E U x V H D,G(u) is continuous, then for every u E U the map G1(u) : V -4 R, v H D,G(u) is linear.

Note that under the assumptions of the proposition Gf(u) E V* for every u E U .

Proof. It is immediate from the definition that DX,G(U) = XD,G(u) whenever X 2 0. Hence it suffices to prove that D,+,G(u) = D,G(u) +

Chapter 1 Functional spaces

D,G(u). Indeed,

D,+,G(u) = lim t-I (G(u + tv + tw) - G(u) ) t-io

= lim ~ - ' ( G ( u + tv + tw) - G(u + tw) + G(u + tw) - G(u) ) t+O

= lim t - l ( ~ ( u + tv + tw) - G(u + t w ) ) + D,G(u) t+O

1

= lim Jo D,G(U + stv + tw)ds + D , G ( ~ ) t-0

The last step in the chain of equalities is due to the estimate

where the right hand side converges to zero by continuity of D,G(u).

11' D,G(u + stv + tw) - D,G(u)ds

Problem 1.3 Show that if G is as in the proposition above, then for every u E U,

5 sup ID,G(u+stv+tw)-D,G(u)I, sE[O,lI

The linear derivative G f ( u ) satisfying (1.19) is called Frechet derivative. It is easy to see that (1.19) fails if one replaces G'(u) with any other T E

V*, so the Frbchet derivative is uniquely defined by (1.19). One says that G E C1(U), where U c V is an open set, if Gf : U -+ V* exists and is continuous.

Corollary 1.5 Let L be a dense subspace of a normed vector space V, let G : L H R, and assume that D,G(u) exists for all u , v E L. If the map (u, v ) H D,G(u) has a continuous extension T : V x V -+ R, then G has a continuous eztension G to V, D,G(U) = T ( u , v ) for u , v E V, G is Fke'chet differentiable at every point and e ( u ) v = T ( u , v ) .

Proof. Define ~ ( u ) := G(0) + T(su,u)ds, u E V, noting that G = G on L and that G is continuous in V. It is easy to see that D,G(U) = T ( u , v ) for u,v E L. By Proposition 1.9, the map T ( u , - ) , u E L, is linear on L, and therefore, by continuity of T and density of L, T ( u , .) E V* for u E V. We leave to the reader to prove, using the density of L and continuity of G

21


and T that the identity D,G(U) = T(u, v) extends to u, v E V and that G satisfies (1.19).

The following statement concerns the question whether a given differential expression is a "variational" one (that is, locally equals G1(u) for some functional G).

Proposition 1.10 Let V be a vector space, let v be a set of linear maps V + R and let T : V + v be such that for every v E V, the map u H T(u)v has a linear derivative, continuous on every finite-dimensional subspace of V. There exzsts a functional G : V + R with the linear derivative GI = T on V if and only if for every v, w E V,

Proof. Necessity. Let G : V + R have the linear derivative GI = T . Set, for fixed u, v, w E V, g(s, t) = G(u + sv + tw). From the definition of directional derivative follows that d,g(s, t ) = D,G(u+sv+tw) = T(u+sv+tw)v, and therefore dtd,g(s, t) = DwT(u + sv + t w ) ~ , and d,dtg(s, t) = D,T(u + sv + tw)v. By the continuity assumption on DwT the second derivatives are continuous and, consequently, dtdsg(O, 0) = d,dtg(O, 0), which yields DwTv = D,Tw. Sufficiency. Assume that the map T has the required continuity properties and satisfies (1.20). Define G(u) = J; T(su)uds. Then, using (1.20), we obtain

For example, if V = C r (IRN) and


with an infinitely differentiable function @, then the criterion (1.20) reads

for every u ,v ,w E C r ( R N ) , which is a condition on a. In particular, when the map T is linear, that is, @u = Ck ak(x) .k Dku, then the correspondent equation is "variational" on spaces of smooth functions with compact support if and only if @ is formally self-adjoint, that is C~ a k ( x ) .k D ~ U = C , ( - I ) ~ D ~ .k (ak(x)u) .

Problem 1.4 Let V = CA(RN). Show that there is no functional G E C1(V) with differentiable G'v, v E V , such that G'(u)v =

J, az,u(x)v(x)dx.

Let H be a Hilbert space. If G : H -+ R has a linear derivative at u and G'(u) is a bounded linear functional, then by Theorem 1.2 there exists a w, E H such that G1(u)v = ( w , , ~ ) . The vector w, is called the gradient V G ( u ) of the functional G at the point u.

Problem 1.5 Show that if G(u) = llu112, then V G ( u ) = 2u.

1.8 Continuous and differentiable functionals in LP-spaces

Lemma 1.6 Let R be a measurable subset of RN and 1 < p 5 q < oo. Ifuk 4 u i n LP(R) n Lq(R), then there exists a subsequence ukj of u k and v E LP(R) n Lq(R) such that for almost every x,

Proof, Consider a renumbered subsequence uk such that uk(x) 4 u(x ) a.e. in S1 and such that

Then it is easy to see that V ( X ) := lul ( x ) 1 + I ~ k + i ( x ) - ~k (x)I is in E LP(R) n Lq(R) and satisfies (1.21).

Lemma 1.7 Let 1 5 p _< q < oo. Let F E Cl,,(R x R ) and assume that for every E > 0 there is a C, < oo and a p, such that p 5 p, 5 q and


Let 52 C IRN be a measurable set. Then the functional

is continuous in Lp(R) n LQ(R), and moreover, if uk E Lp(R) n LQ(R) is a bounded sequence and uk -, u in LT(R) for some r E [p,q], then G(uk) -, G(u).

Proof. Let M > 0 be such that

Observe that uk -+ u in ~ ~ ' ( 0 ) for all r' E (p, q). Indeed, if r' < r , by the Holder inequality

If r' > r , the same conclusion can be made from a similar interpolation between Lr and LQ.

Let 6 > 0 and E > 0 be such that EM < 614. By Lemma 1.6, there is a 0 5 v(x) E LPE such that on a renumbered

subsequence luk(x) l,1u(x) I < v(x) almost everywhere in R. Due to (1.24), we have

Chapter 1 F!unctional spaces 25

The dominated convergence theorem implies that the integral in the last line converges to 0 since IF(x, uk) l , IF(x, u)l 5 2C,v(x)p~ E L1 on the set of integration. Since 6 is arbitrary, G(uk) + G(u) on the selected subsequence. If there is another renumbered subsequence where IG(uk) - G(u)l 2 E with some E > 0, then, by the argument above, it will still have a renumbered subsequence where G(uk) -+ G(u) , a contradiction.

Lemma 1.8 Let R C RN be a measurable set, let f E CI,,(R x R), F(x , t ) = f ( x , s)ds and let G(u) = J F(x , u(z))dx. Then for every u , v E

CF'(Q> 1

Proof. By the definition of directional derivative,

The expression under the integral converges to F'(u) and is bounded by

the constant ~ u P , ~ ~ ~ ~ ~ ~ u ~ ~ ~ ~l f ( x 1 + S V ( X ) ) I BY the dominated convergence theorem the integral converges to the right hand side of (1.25).

Lemma 1.9 Let R C RN be a measurable set, let 1 5 p 5 q 5 oo, and let f E C ( R x R) be a function such that for every E > 0 there is a C, < co and a p, such that p 5 p, 5 q and

Then for every r E [p, q], the mapping

restricted to any bounded set of LP(R) n Lq(R) x Lp(R) n LQ(S1) is continuous in L' (R ) x (Lp(R) n Lq(R)).

Proof. Let uk -+ u in Lr(R) be a bounded sequence in LP(R) n Lq(R) and let vk + v in LP(R) n Lq(R). Consider the inequality

Let us estimate the first integral in (1.28). By (1.26) and the Holder in-


equality,

In particular, the last term converges to zero due to the Holder interpolation

IIwllr L I I W I I : I I W I I ; - ' , r E [p,q], with an appropriate 8 = O(~ ,q , r ) . It remains now to estimate the second integral in (1.28).

Let M > 0 be such that

(Such M exists for the first integral due to the Holder inequality J lullr-'lql 5 I1wII:-lIIuIIT and the bounds on the respective norms for uk, u and v. The second integral is then bounded due to the Fatou lemma.)

Let b > 0 and E > 0 be such that EM 5 614. By the argument in the proof of Lemma 1.7, uk -+ u in LPe(R), since p, E [p, q]. By Lemma 1.6, there is a 0 L w(x) E LPE such that on a renumbered subsequence Juk(x)J, Ju(x)J 1 w(x) a.e. Then

Lebesgue dominated convergence theorem implies that the integral in the last line converges to 0 since If(x,uk)llvl, If(x,u)IIvI 5 2CEw(x)Ps-'1~1 E

L1 on the set of integration. Since S is arbitrary, we have, for a renumbered subsequence of uk, T(uk, vk) --+ T(u, v). If there is another renumbered subsequence where IT(uk,vk) - T(u,v)l 2 E with some E > 0, then, by


the argument above, it will still have a renumbered subsequence where T(uk, vk) -+ T(u, v), a contradiction.

Problem 1.6 Prove that the functional G(u) = Ja F(x , u), where Cl c iRN is a measurable set and F E Cl,,(R x R) satisfies I F (x , s) 1 < C(lslP + Islq), 1 < p < q I ca, is continuous in LP(R) n LQ(R).

Problem 1.7 Prove the following statement. Let R c E l N be a measurable set, let 1 < p < q _< ca, and let f E Cl,,(R x R) be a function such that

(f (x, s)l I C ( ( S ( ~ - ~ + (~lq-l) , %7 E W, x E a. (1.30)

Then the mapping (1.27) is continuous.

Chapter 2

Sobolev spaces

Let R c RN be an open set. Expressions Sa Vu . Vv and (VU . Vv + uv) define inner products on C,"(fl), but like in the case of the scalar product in L2, the completion space of CF(R) with respect to these products is not trivial. Of course, the second of the inner products guarantees that it will be a subspace of L2, but even then the question rises: what is the meaning of Vu in the formula for the scalar product for a general u in the completion?

The exposition below is subject to several restrictions. Of all Sobolev spaces wk+ we focus narrowly on the case k = 1, m = 2, while a brief survey of the general case is given in the last section. We do not explicitly consider domains with a boundary that is not C1, instead of it speaking about "extension domains", "trace domains" or "Friedrichs domains". We do not use the notion of capacity for characterization of domains, and thus, do not present necessary and sufficient conditions for classes of domains mentioned in this chapter that are established in terms of capacity. On the other hand, the presentation has a strong emphasis on unbounded domains. For more details about Sobolev spaces we refer the reader to the books of Adams 131, Maz'ya [93] and Ziemer 11251 or to advanced textbooks of Evans [52] and Lieb and Loss [82].

2.1 Weak derivatives. Definition of Sobolev spaces

Definition 2.1 Let 52 be an open set in ItN, N E N, and let u E L:,,(R). If there is a function v E L;,,(R, RN) such that for every cp E C r ( R ) ,


then v is called a weak derivative of u, denoted as Du.

Obviously, the weak derivative is defined by (2.1) uniquely and, whenever u E Ckc(R), Du = Vu. Unless it is needed to emphasize that existence of usual derivative is not assumed, notation Vu is used also for the weak derivative. When there is no locally integrable function to satisfy (2.1), one still may regard the functional e, : cp H - Ja uVcp as a weak derivative of u (derivative in the sense of distributions), but this book considers only locally integrable weak derivatives defined above.

Lemma 2.1 Let R C RN be an open connected set and let u E L:,,(Z;Z). If Du = 0, then the function u is constant.

Proof. Let U G R be an open connected set and let E = d(U, E l N \ a ) . if cp E C r ( U ) , then for every y E B,(O) and t E [ O , l ] , one has cp(. + ty) E

c,- ( a ) 7

d ~ ( x + ty) = y . Vp(x + ty), dt

and therefore

Then

which implies u(. - y) = u a.e. in U. Indeed, by the density of C,"(U) in L2(U) - Lemma 1.1 - we can choose cpk E C r ( U ) such that cpk -+

(u(. - y) - U) I u in L ~ ( U ) so that JU(u(. - y) - u)' = 0. Let ~ ( x ) = 1 u(x - y)dy, x E U, with c > 0 sufficiently small.

l B E l 4!1<€ Note that f i is a continuous function and that integration of the chain of equalities above gives ti = u a.e. in U. Therefore DE = Du = 0 and, by continuity, G(X - y) = ~ ( x ) for every x E U, lyl < E . Since U is connected,

Chapter 2 Sobolev spaces 31

ii is constant on U and thus u is constant a.e. in U. Since E is arbitrary and R is connected, u equals the same constant a.e. in R.

The Hilbert space HA (R) for an open set R c RN is defined as a completion of Ci (0 ) with respect to the scalar product

The scalar product (2.2) corresponds to the norm

and since I I u I I ~ ~ ~ ~ ~ 2 I l ~ l l ~ 2 ~ ~ , , the space HA(R) is continuously imbedded into L2(R). This allows us to regard H i (R) as a subspace of L2(R). Since C r (R) is dense in L2 (R) (Lemma 1. I), H i (0 ) is a dense subspace of L2 (0). When R1 c R2 are two open sets in RN, we regard the space Cr(R1) as a subspace of C?(R2), extending functions in C,"(RI) to 522 \ 01 by zero. This also allows to regard H;(R1) as a closed subspace of HA(R2). If uk E C,"(R), uk 4 u in Hd (R), then from (2.3) follows that Vuk is a Cauchy sequence in L2(R,RN) and so it has a limit u in L~(O,IW~). Moreover, u = Du in the sense of Definition 2.1: since

and the left and the right hand side are continuous functionals in L2 evaluated on Vuh resp. uk, passing to the limit we get 1 vv = - 1 uVp.

Problem 2.1 Show that

(a) u H DU is a bounded linear map Hi(R) + L' (Q, IW~) ; (b) D(cpu) = uDcp + cpDu for cp E C1(R); (c) if U E RN is an open set, then D(u 0 $) = [(Du) o $]V$ for 4 E

C1 (U -, 0 ) .

The space ~ ' ( 0 ) is defined as a vector space of functions u E L2(R), such that Du E L'(R), equipped with the scalar product (2.2). To show that it is a complete Hilbert space, note that every Cauchy sequence uk in H1(R) converges in L2(R) to some u E L2 and that Duk is a Cauchy sequence in L2(R, R ~ ) , and thus it converges to some v E L2(R, RN). It is easy to see that the limit values of both sides in (2.4) considered for uk yield v = Du, which implies that uk 4 u in H1(R). The space HA (0 ) is naturally injected


into H 1 ( R ) . From inequality (2.51), that we prove later in this chapter, it is immediate that H i (0) # H 1 ( R ) as long as the complement of R is large enough, in particular if it contains (locally) a hypersurface.

Problem 2.2

(a) Prove that H ; ( I W ~ ) = H ~ ( I W ~ ) by showing that u j E N H i ( B j ( o ) ) is dense in H ' ( I W ~ ) : for any u E H ' ( I W ~ ) consider uj = x(j- ' .)u with appropriate x E C F ( B l ( 0 ) ) and show that u j E H 1 ( R N ) and ~ ~ u - u ~ ~ ~ ~ ~ 4 0.

(b) Let R c RN be an open bounded set. Show that C 1 ( R ) is continuously imbedded into H1 ( R ) .

Lemma 2.2 Let R c RN be an open set and let x E C 1 ( R ) , u E HA (a) . Then xu E Hi (0) and the operator H i (0) 4 H; ( R ) , u H x u , is continuous.

Proof. If u E C F ( R ) , then

The relation extends to all u E H; ( R ) by the density of C F ( R ) in H t ( R ) .

2.2 Chain rule

Proposition 2.1 Let R c IRN be an open set and assume that a function $ E Ck,(R) has a bounded derivative and satisfies $(0) = 0. Then the map T : u w 1C, o U , u E COW (a) , extends to a map H;(R) -+ H i ( R ) , D($ o u ) = $'(u) Du, and

where A4 = supR ) $ ' I

Proof. Let uk E C r ( R ) , uk + u in HA(0). Note that I1C,(u)l 5 Mlul and thus $(u ) E L2(R) and $ ' (u)Vu E L2(R) . Then


and, taking into account that I$'(uk) - $ J ' ( u ) ~ ~ ~ v u ~ ~ 5 4 ~ ~ 1 V u 1 ~ , by the Lebesgue convergence theorem

and

Jn IV$(ur) - $'(u)vul2 = I$'(.k)vur - $'(u)vu12

In other words, $(uk) + $(u) and V$(uk) -+ $'(u)Du in L2(R), so that $(uk) is a Cauchy sequence in H,'(R), $(u) is its limit and D($ o u) =

$'(u) Du. This and the earlier observed inequality I$(u) 1 < M 1.1~1 imply (2.6).

Let sign(t) = tlltl for t # 0 and set sign(0) = 0. Let u+ (x) := max{u(x), 0) and u- := (-u)+.

Proposition 2.2 Let 52 c KtN be an open set. If u E HA(R) then

(a) 14, u+, 21- E H i (% (b) Dlu( = sign(u)Du, (c) Du+(x) = Du(x) whenever u(x) > 0 and Du+(x) = 0 otherwise, (d) -DL (x) = Du(x) whenever u(x) < 0 and Du- (x) = 0 otherwise, (e) Du(x) = 0 for almost every x E u-'(0).

Note that (e) is meaningful only if the set u-'(0) has positive measure and that due to (e) the relation (b) is independent of the value assigned to sign (0).

Proof. Let $k(t) = d p - k-l, k E N, and let Tk(u) = $k o U.

Since I$; 1 5 1, from Proposition 2.1 follows that qk o u E Hi (0) and 0 ~ 1 1 ~ 1 < IIuIIHi. By the Lebesgue convergence theorem, $k o u 4 lul

and D($k o u) -' sign(u)Du in L2(R). Thus gk o u is a Cauchy sequence in HA (0) whose limit is lul with Dlul = sign(u)Du. Repeating the argument above with u replaced by u+, one has $k o

U+ + U+ and D($k o u+) + x ~ , ( , ) , ~ ~ D u in L2(R), which verifies that $k 0 u+ is a Cauchy sequence in HA(R) whose limit is u+ with


Du+ = s ign (u )Du~~ , ( , ) ,~ )~ , . The argument for u- is completely analogous. It remains to prove (e). Let x E uW1(0). Then

by (c) and (d).

2.3 Coordinate transformations. Trace domains and extension domains

Definition 2.2 Let R C RN be an open set. One says that dR E C1 if for every point x E dR there is a map cp E C1((-1, I ) ~ , R ~ ) such that cp((-1, x (0 , l ) ) = cp((-1, I ) ~ ) n 0 , and cp : (-1, l)N-l x (0, l )) -+

p((-1, I ) ~ ) n R is a bijection with a C1-inverse.

Lemma 2.3 Let U, V C RN be two open sets and let $k E C1(U -+ V), k = 0,1, . . . , be bijective with a C1-inverse and assume that $k 4 11, in C1(u, V). Then the sequence of operators u E H ~ ( v ) H u o Gk E Ht(U) is bounded and u o qk -+ u o ?I) for eve y u E Hi (V).

Proof. Consider the change of variables x H y = ?lk(x) in the integral (2.3) for u E C,OO(V) and notice that by the Cauchy inequality in RN, I I u o $ ~ l l & ~ ( ~ ) I C I I U ~ ~ & ~ ( ~ ) with some C > 0 independent of k. By density of C,OO in Hy, this inequality holds for all u E H ~ ( V ) . Note that the expression under the integral remains valid for all u E H,'(V) provided that we understand Vu as DU E L2(V).

Apply the same change of variables to the integral (2.3) for Ilu o qk - u ~ $ l l & ~ ( ~ ) and note that it converges to zero by the Lebesgue convergence theorem, since Du E L2 and the functions $k are uniformly bounded.

Lemma 2.4 Let U C RN be an open set and let $k E C1(U, U)) k E N, be open injective maps with a C1 inverse on $k(U), that satisfy $k 4 id in C1 (U, U). Let ik be the injection of Hi ($k(U)) into Hi (U) . Then the sequence of operators T k : u H ik(u o Ok)) as operators from Hh(U) to H i (U), is bounded and Tku -+ u for every u E Hi(U).

Proof. From Lemma 2.3 it follows that the operators u E Hi (qk (U)) ++

u o qk E Ht (U) are continuous with a uniform bound, which yields the first assertion of the lemma. The second assertion follows from the change of variables and the Lebesgue convergence theorem.


Definition 2.3 An open set R c RN is called a trace set if u E H ~ ( I w ~ ) , u = 0 a.e. on lRN \ 0 implies u E Hi(R) (If, in addition, R is connected, it is called a trace domain).

The last relation is of course understood in the sense of the standard injection of Hi ( 0 ) into H1 (RN).

Below we show that domains with C1-boundary are trace domains, starting with a specific case.

Lemma 2.5 Let w E Hi((-1, I ) ~ ) and assume that w = 0 almost everywhere i n (-1, l)N-l x (-1,O). Then w E H;((-1, x (0, l )) .

Proof. Let x E Co0(R, [0, 11) be such that ~ ( t ) = 0 for t 5 -$ and ~ ( t ) = 1 for t 2 0. Let E E (0, $). By Lemma 2.2, the operator u(x) H x(x~ /E)u (x ) is a continuous operator on Hi((-1, I ) ~ ) . Since (1 - x(xN/&))w = 0 almost everywhere, the function (1 - x ( x ~ / E ) ) w represents the zero element of Hi((-1, I ) ~ ) . Subtraction of this representation of zero from w gives

Let

By Lemma 2.4, T, is a bounded family of continuous operators Hi((-1, 1IN) -+ Hi((-1, I ) ~ ) and

Let wj E Cr( ( -1 , I ) ~ ) converge to w. Note that (1 - x ( x ~ / c ) ) w j -+ (1 - X(XN/E))W = 0, SO that x(xN/&)w~ 4 W. Then IITEX(XN/E)W~ -

TEwllH1 -+ 0 as j 4 oo. In particular, with E = j - l , we have Tj-1x(jXN)wj - 5"'-lw -+ 0. It remains to observe that since q - i w -, w, the function w is approximated by a sequence T j - i x ( j x ~ ) w j E

Cr(( -1 , I ) ~ - ' x (0, I)), i.e. w E ~ : ( ( - 1 , l)N-l x (0,l)).

Lemma 2.6 Let U E RN be an open set and let cp E C1((-1, I ) ~ , U) be a bijection with a C1 inverse. If w E H;(u) and w = 0 almost everywhere i n cp((-1, l)N-l x (-1,0)), then w E Hi(cp((-1, l)N-l x (0,l))).

Proof. By Lemma 2.3, v = w o cp E H;((-1, I ) ~ ) . Then by Lemma 2.5, v E Hi((-1, l)N-l x (0, l )) . Then by Lemma 2.3, w E ~;(cp((-1, l)N-l x (07 1)).


Theorem 2.1 If R C RN be an open domain and dR E C1, then R i s a trace domain.

Proof. Let w E (RN) be such that w = 0 almost everywhere in ItN \R. First assume that R is bounded. Since dR E C1, there is an open covering {U,, y E d o ) of dR and a family of C1-bijections with C1-inverse cpy :

(-1, l ) N Uy such that cp,(O) = y and cp,((-1, I ) ~ ) ~ R = p((-1, l)N-l x (0,l)). Since R is bounded, a is compact and there is a finite subcovering {Uk = Uy,),,Ean,k=l ,..., of dR. Let Uo CE R be an open set such that - R c Ur=oUk. Let { x k : k = 0 , . . . , m ) be a Cr(RN)-partition of unity on R subordinated to this covering and let wk = xkw. For k = 1,. . . , m from Lemma 2.6 it follows that wk E H;(R). By Lemma 2.2 we also have xow E H;(R) and then w = (CTz0 xk)w = C;fZ=l wk E HA(R).

Consider now a general domain 52. Let x E Cr(IRN, [O, 11) be such that ~ ( x ) = 1 for 1x1 5 1 and ~ ( x ) = 0 for 1x1 2 2. Applying the conclusion of this theorem for domains with C1-boundary that contain R n Bzk(0) and are contained in k E fl, we have x(k-'.)w E HA(n). Note that the sequence x(k-l.)w converges to w pointwise and that a simple calculation shows that it is Cauchy sequence in H;(R): for k > j ,

jly(k-'.)w - X( j - l . )wg . 5 1 (4j-11~X12w2 + 41vw12 + 2w2). xEn\Bj(0)

(2.11) Therefore, w E H;(R), and thus 52 is a trace domain.

Remark 2.1 Trace sets i n the sense of Definition 2.3 are involved i n Chapter 4 i n locating weak limits of sequences i n H1(ItN) that vanish on RN \ R for an open set R c RN as elements of HA(R), that is, almost everywhere.

Without a careful view it might appear that every open set i s a trace set due to the powerful Hedberg trace theorem (Theorem 9.1.3 i n [ I]) which asserts, i n restriction to the case studied here (and proved earlier by Deny in equivalent form as Theorem II:2, [44], cf. also [65]), that u E H ' ( I W ~ ) and u = 0 on RN \ R imply u E H i (R). It should be remembered however, that the function u i n the Hedberg trace theorem is a regularized representative of [u] E H1 and the relation u = 0 is understood as holding quasi-everywhere (i.e. up to sets of zero capacity). It i s also known that i f U i s an open set, u = 0 a.e. i n U implies u = 0 quasi-a.e, i n U. Consequently, from u = 0 a.e. i n RN \ R we infer only that v = 0 quasi-a.e. i n the interior of IRN \ R, which is not all the complement of R and the conditions of the Hedberg theorem are not satisfied.


Definition 2.4 An open set R c IRN is called an extension set, if there exists an open set 521 > and a bounded linear map T : H1(R) 4 ~ i ( R 1 ) such that for every u E H'(R), Tuln = u (if, in addition, 52 is connected, it will be called extension domain).

In particular, a bounded domain 52 is an extension domain if dR E C1, or, more generally, if there is an open truncated cone K such that for every point x E afl the domain S1 contains a truncated cone with vertex x congruent to K , in particular if R is a polyhedron. For proof of this statement we refer to [3]. Here we will give only a brief sketch of the proof when dR E C1. Similarly to the proof of Theorem 2.1, the proof can be reduced to construction of extension of u E C1(B+), defined on a half-ball B+ := {x E Bl(0), xN 2 O), to the whole ball Bl(0). This is done by setting for X N 5 0

where 3 = (xl, . . . , xN_l). One can immediately verify that the resulting function is in C1(B1(0)) and that there is a C > 0 such that for every such

U, I I ~ I I H ~ ( B ~ ( o ) ) 5 CII~IIH~(B+).

2.4 Friedrichs inequality

This section studies open sets R C JRN that admit the following inequality: for some C > 0

This is known as Fkiedrichs inequality. The class of functions where (2.13) holds true can be obviously extended by density to Hi(R). Since (2.13) fails for u = 1, it cannot be extended to H1(R).

Lemma 2.7 Let p, R > 0, p < R, and let 52 = {x E EXN : p < 1x1 < R), N E N . Then there is a C(p, R) > 0 such that for every u E Cm(R) with u(x) = 0 whenever 1x1 = p,


Proof. Let us use the spherical coordinates (r, w) E [0, co) x SN-I on R. From

follows by the Cauchy inequality, with constants C dependent on N, p and R, but not on u,

Integration in (r, w) over R gives (2.14).

Theorem 2.2 Let 0 c IRN be an open set and assume that there exists a p > 0 such that

& := sup d(x, IRN \ ( 0 + Bp(0))) < co. (2.15) % E n

Then Friedrichs inequality (2.13) holds true.

Condition (2.15) is satisfied, in particular, when R is bounded or when it is a cylinder with a bounded base.

Proof. Let us fix R E (Ro, Ro+ $ ). From (2.15) follows that the collection of balls BR(y), y E IRN \ (R + Bp(0)) is a covering of R. By Lemma A.l there is a countable subset J of RN \ (R+ Bp(0)) and a number M E N such that the collection B2R(y), y E J , covers R with multiplicity not exceeding M. Let u E C r ( R ) . Then by Lemma 2.7

since the balls B$(y) with y E J C BN \ ( 0 + Bp(0)) are disjoint from R. Adding terms in (2.16) over y E J, we have


A partial converse of this theorem is as follows.

Proposition 2.3 Let R c RN be an open set and assume that

sup d(x, EXN \ R) = +m. xEn

Then the inequality (2.13) does not hold.

Proof. For any k E N there is a yk E R such that Bk(yk) c Cl. Let u E

CF(Bl(0) ) \ (0) and let uk = k - + ~ ( k - ~ ( . - yk), SO that UI, E CF(Bk(yk)) . It is easy to see that ( ( ~ ~ ( ( ~ 2 ( ~ ) is independent of k, while Jn (vukI2 = k-2 J IVuI2. Consequently,

2.5 Compactness lemma

The following statement is subsumed by Theorem 2.8 below, but it is a crucial technical step leading to several essential properties of Sobolev spaces.

Lemma 2.8 Let R, R1 C EXN be open bounded sets, R Rl . Then the set J := { ~ l n : u E Cr(IKN), I I u I I ~ ~ ( ~ ~ ) 5 1) is relatively compact in ~ ~ ( 5 2 ) .

Proof. Let p E Cr(B1(0) , [O, 11) satisfy J p = 1 and let Mtu(x) =

J,v151 p(y)u(x - ty)dy, 0 < t < d(n,EXN \ Cll). We define Jt := {(Mtu)IE :

u E C r ( R N ) , I ~ u ~ J ~ , ( ~ , ) < 1). Then we have for u E J, x E a, due to the Cauchy inequality,

and

This shows that Jt for t sufficiently small is a bounded equicontinuous set in C ( n ) and so, by ArzelbAscoli Theorem 1.1, the set Jt is relatively compact in C ( n ) and consequently, Jt is also relatively compact in L2(R). For u E J


and x E R we have

Integrating the inequality, squared, over x c 0 and using the Cauchy inequality we obtain, for all t < d(0, IRN \ R1), the following estimate:

Let now u k E J. Consider a sequence t , -+ 0 such that, due to (2.21),

Let kjYl be a subsequence such that Mtlukj,, is a convergent subsequence in L2(R). In particular we may choose it so that

For every m E N we choose now a subsequent extraction kj,, of kj,m-l such that

Then, by the triangle inequality,

This implies that u k j , j is a Cauchy sequence, and therefore, it converges in L2 (0).

Chapter 2 Sobolev space9 41

Corollary 2.1 If R C RN is a bounded extension domain, then the set

J = {u E H1 (R), IluIIHi 5 1, } is relatively compact in L~ (0).

Proof. Let R1 c IRN be an open set such that R G R1 and let T :

H1(R) -, Hi(Rl) be an extension operator from Definition 2.4. Then the set T J will be bounded in HG(R1) and thus, relatively compact in L2(SZ1). Since T is an extension operator, we have J = {u = v 1 v E T J}, which is a relatively compact set in L2(R) since from Lemma 2.8 it is immediate that T J is relatively compact in L2(R1).

2.6 Poincarh inequality

Let R c IRN be a bounded domain. As we mentioned above, the Friedrichs inequality (2.13) becomes false if one considers it for u E H1(R) instead of Hi(R). In fact, it remains true for any function in H1(R) orthogonal to u = 1 on which it obviously fails. Below we prove a weaker version of the Poincar6 inequality (which nonetheless suffices for further arguments), with reference to the weak Lemma 2.8.

Theorem 2.3 Let R, R1 c RN be bounded domains, R G R1. Let $ E ~ ~ ( 0 ) be such that Jn $ # 0. Then there exist a C > 0 such that for every u E C r (RN),

Proof. Assume that (2.22) is false. Then there exists a sequence uk E

C r ( R N ) such that the left hand side of (2.22) converges to zero, while the right hand side of (2.22) equals 1. Then uk is bounded in H1(R1) (Indeed, with an appropriate cut-off function x E Cm(R1), XI,,,n = 1,

due to F'riedrichs inequality, Jn1,, u2 5 Jal (Xu)2 5 C Jnl ( V U ( ~ + C Ja u2) and therefore by Theorem 1.4 it has a weakly convergent renumbered subsequence uk A u in H1 (R1). Then / cpD (uk - u) -+ 0 for cp E C r (R1) and, since Duk is bounded in L2(01) and Cr(R1) is dense in L2(R1), Duk Du in L2(R1). Then by the weak lower semi-continuity of the norm,


Therefore, by Lemma 2.1, u is constant in R1. By Lemma 2.8, u k -+ u in L2 (R), which implies

which is a contradiction.

Problem 2.3

(a) Show, using Theorem 2.10 below that if R is a bounded extension domain, the inequality (2.22) holds with R1 replaced with 0 .

(b) Show that, for a bounded open 0, the inequality

is true for u E C r ( R ) . (c) Let R = (0, I ) ~ . Show that inequality (2.23) extended to u E C r ( R N ) ,

Jn u = 0, is false. (d) Let R = (0, I ) ~ . Show that the Hilbert space defined as completion of

C?(IKN) in the norm Ildx,ullz,n + I I u ~ ~ ; , ~ consists of functions u E L2- functions with d x l u E L2 and that it is imbedded continuously but not compactly into L2(R).

Lemma 2.9 There exist a C > 0, such that for every p 5 1 and u E

C?((-l, q N ) ,

Proof. Let us use the notations Q := (-1, I ) ~ , Q' := (-2, 2)N. Let m E W and let Q be partitioned into smaller cubes Qi, i = 1,. . . , 2mN with the side length p = 21-m. Let Q: be cubes of double diameter concentric with Qi. From Theorem 2.3 with R = Q, R1 = Q' and $ = 1 follows, with (2.22) rescaled to Qi by a linear change of variables,


Adding the relations (2.25) over all cubes and taking into account that no point of I K N belongs to more than 2N cubes Q:

we get (2.24) for p = 2-" and thus, for any p 5 1.

Corollary 2.2 There exist a C > 0 such that for every p > 0 and u E

C,-(RN),

Proof. Assume first that u E C F ( ( - 1 , I ) ~ ) . For p > 1 we have (2.26) from the Friedrichs inequality and for p < 1 it follows from (2.24). For a general u E cr(IKN) there exists a R > 0 such that suppu c (-R, R ) ~ and (2.26) is immediate from (2.24) with v ( x ) = ~ ( R x ) and p' = p /R .

Problem 2.4 Show that if R c IRN be a bounded extension domain, there exist a C > 0 dependent on R, such that for every p 5 1 and u E H1(R),

2.7 Space V ~ ~ ~ ( I W ~ ) . Sobolev, Hardy and Nash inequalities

Definition 2.5 The space V'12(IRN), N > 2, is the completion of C r ( R N ) with respect to the norm ( J R N ) V U ~ ~ ) ; .

Remark 2.2 The restriction N > 2 is due to the fact that for N = 1 , 2 the completion space is no longer a space of measurable functions, that is, there is no continuous imbedding of V112(IRN) into L:,,. Indeed, for N = 1 set u k ( x ) = 1 for x E [-k,k], u k = 0 for x 6 [-2k,2k], u k ( x ) = 2 - k-'x for x E [k, 2k] and u k ( x ) = 2 + k- 'x for x E [-2k, -k]. For N = 2 set u k = ~ x l ' / ~ for 1x1 I 1 and u k = lxl-llk for 1x1 2 1. In both cases i t is easy to see that JRN lVukI2 -) 0 , and therefore the sequence uk represents the zero element of the completion. However, uk converges, uniformly on bounded sets, to 1 # 0 .


When N > 2, the space D112(WN) is continuously imbedded into L2*. The exponent 2* is defined as

Theorem 2.4 (Sobolev inequality) For every N > 2 there exists a constant SN > 0 such that

whenever u E C T (RN). Consequently, the space D1>2(WN) is continuou~ly imbedded into L2' (WN).

Proof. Let us apply (2.26) to functions xj(lul), where xj(t) = 2 - j ~ ( 2 j t ) , j E Z and x E CT((4,4) , [O, 3]), such that ~ ( t ) = t whenever t E [I, 21 and lx'l 5 2. Then, observing that I x ~ I 5 2 and using Proposition 2.4, we have

Taking into account the upper and lower bounds of lul on the respective sets of integration, we have

If we substitute pj = 2 - A p , take the sum over j E Z, and notice that each of the intervals [2jP1, 2j+2], j E Z, overlaps with the others not more than four times, we get

Setting p = (J u2*)h and collecting similar terms we arrive at (2.29).

Chapter 2 Sobolev spaces

Remark 2.3 The best constant i n (2.29) is

where w~ = 27rwl?(v) is the area of the N-dimensional unit sphere. Calculation of the best constant is discussed i n Chapter 5.

Problem 2.5 Prove that D ' ~ ~ ( I W ~ ) is the space of measurable functions u (defined up to sets of measure zero) that have weak derivatives Du E

L~ ( I R N ) .

The following analog of Proposition 2.1 will be needed in later calculations.

Proposition 2.4 Let 1C, E C:,,(IW) have a bounded derivative and satisfies $ ( O ) = 0. Then the map T : u I+ 1C, o u , u E C r ( I R N ) , extends to a map D112(IWN) + D112(IRN), D ( $ J o u ) = $ ~ ~ ( u ) D u , and

where M = supw I1C,'I.

The proof is completely analogous to the proof of Proposition 2.1 and may be omitted.

Theorem 2.5 (Hardy inequality) For every function u E G'r(IRN \ 0 )

Proof. The inequality (2.32) follows from the identity

To prove (2.33) one can evaluate its right hand side, starting with the chain rule and expansion of the square:

= J 1v.1~ + (F)2 / 1x1-2u2 + 2~ 2 j I X I - ~ U X . vu.


The last term can be evaluated, using the calculation V . + = and:

and then (2.33) is immediate.

Consider now the space R N as a product space Rm x Rn, n = 0 , 1 , . . . , N - 1 , m = N - n with variables (x, y ) , x E Rn, y E Rm.

Corollary 2.3 For every function u E C r ( R N \ R n )

Proof. Fix x E Rn, write (2.34) in Rm and integrate over x.

Theorem 2.6 (Nash inequality) For every u E C r ( R N )

By density the inequality extends to u E H 1 (IRN) n L 1 ( R N ) .

Proof. The inequality (2.35) is immediate from (2.26) with

The best value of the constant in (2.35) is CN = 2~-~+~/~ ( 1 + N / ~ ) ~ + N / ~ K ~ ' w ~ ~ ~ , where W N - ~ is the area of a unit N - 1-dimensional sphere and K N is the smallest nonzero number such that the function d ( r l - N / 2 J d r ( N - 2 ) / 2 ( ~ r ) ) I r = 1 = 0 (see [29]). A proof that the best constant is attained is given in Chapter 10.


Lemma 2.10 Let IW? = RNP1 x (O,cm), N > 2 . There exists a C > 0 such that for every u E C r ( R N ) ,

Proof. Let u E C r ( R N ) , and set, similarly to (2.12),

(Tu)(?, X N ) = for X N 2 0;

(2.38)

It is immediate that T u E C 1 ( R N ) and

Then (2.37) follows immediately from the Sobolev inequality (2.29) applied to Tu.

Remark 2.4 Inspection of the proof of Theorem 2.6 shows that Nash inequality (2.35) is nothing but an equivalent form of (2.26). Similarly, the Sobolev inequality (2.29) in Theorem 2.4 is derived from (2.26) using only general properties of IVuI2 as a Dirichlet form (see [59] for definition), and from inspection of the proof one can conclude that (2.29) implies (2.26). Thus, Nash and Sobolev inequalities are equivalent. We also note that the exponents in (2.35) and (2.29) are consequences of the coefficient p-N in (2.26), which in turn can be traced to the scaling coefficient in the weak Poincare' inequality (2.22). For details on equivalent forms of Sobolev inequalities we refer to 1921, [ l l ] and [43]. The general structure in the theory of Sobolev spaces has an axiomatic generalization - a Sobolev spaces theory on metric spaces (see expositions in Hajlasz and Koskela [63] and Ambrosio and Tilli [5]).

2.8 Sobolev imbeddings

In this section we discuss continuity of imbeddings of Sobolev space H1(S1) into Lp(S1). When N > 3, the imbeddings can be derived from (2.29). For N = 1,2 we use the following statement.

Lemma 2.11 Let p > 2 and let N = 1,2. There exists a C > 0 such that


for every u E C r ( R N ) ,

Proof. The proof is an elementary modification of the proof of Theo- rem 2.4 when the exponent 2* is replaced by p, which leaves in the right hand side the integral of lu1q instead of a term similar to the left hand side side.

Theorem 2.7 Let R c RN be an open set. Let p E [2,2*] i f N 2 3 and p > 2 i f N = 1 ,2 . There exists a C > 0 such that for every u E Hi(S2),

S, IVu12 + u2 2 C (S, .lP) : Proof. For p = 2 the inequality is trivial. 1. Case N 2 3. If p = 2*, the inequality is immediate from the Sobolev inequality (2.29). If 2 < p < 2*, then from Holder inequality (1.1) follows

with s = 2. Applying to the right hand side the Sobolev inequality (2.29) and the elementary inequality atbl-t 5 Ct(a + b) , a , b > 0 , t = 2 2 ' 9 - p 2'-2 ( 0 , I ) , we arrive at (2.41). 2. Case N = 1,2 . The inequality for p > 2 follows from Lemma 2.11 once we notice that 2 < q < p, estimate J l u l q by the Holder inequality

S, 1.1~ 5 (S, 1 ~ 1 2 ) ' (Jn I ~ I P ) I-'

with s = s, and use the inequality

with E sufficiently small, so that the term with Ja l u l P could be carried over to the left hand side.

Corollary 2.4 Let R c RN be an extension domain. Then (2.41) holds for every u E H 1 ( R ) .


Proof. Let T : H1(R) -+ H;(Q1) be an extension operator and apply (2.41) for Tu on H,'(R).

Remark 2.5 If R is an open set offinite measure, then from the Holder inequality follows that relation (2.41) holds also for p E [I, 2).

Theorem 2.8 Let 52 c IKN be an open bounded set and let p E [I, 2*). Then the imbedding Ht(Q) in LP(R) is compact.

Proof. Assume first that p > 2 and let q E (p, 2*). From the Holder inequality follows

with s = s. Let u j be a bounded subsequence in H; (R). Then, by Theorem 2.7, u j is bounded in Lq(R). Moreover, u j has a renumbered weakly convergent subsequence u j - u E HA(R), and by Lemma 2.8, u j u E L2(R). Then from (2.44) follows S, luj - uIP 5 C (So luj - 2 ~ 1 ~ ) ' -+ 0.

Let now p E [I, 21. Since R is bounded, from Holder inequality follows that Lp(R) is continuously imbedded into Lq(R) with any q > 2. Since a bounded set in H;(R) is relatively compact in LQ(R) for q E (2,2*), it is also relatively compact in LP(R).

Theorem 2.9 Let u, uk E V112(RN), N > 3, and assume that uk - u in V1y2(RN). Then for any p E [I, 2*) and any set of finite measure A c IRN, uk 1 A -+ U ) A in Lp(A) .

Proof. By density we may assume that uk E CF(RN). Let let X j E

Cr (B j+ l (0 ) , [0, I]), j E N, be such that x I B ~ ( ~ ) = 1 and supj IIVxj I l m < m.

Then, taking into account the Sobolev inequality (2.29) and Theorem 1.3, one has


This proves that u H xju is a bounded linear operator in D112, and thus, by Lemma 1.4, xjuk is a weakly convergent sequence in for every j . Moreover, for every cp E C r ( R N ) ,

once we note that w H J w(xjAcp) is a continuous linear functional on L ~ * and thus on D112.

From here follows (with reference to Lemma 1.2) that xjuk - xju in D1v2 for every j . Due to the Friedrichs inequality, for every j the sequence xjuk is bounded in Hi(Bj+l(0)). Then from Theorem 2.8 follows that xjuk -+ x ju in LP(Bj+1(O)), 1 5 p < 2*. In particular, for every such p and every j E N

Then there exists a sequence jk -+ cc such that

By the Holder inequality,

The assertion of the theorem now follows from adding the inequalities (2.45) and (2.46).

Theorem 2.10 Let R C RN be a bounded extension domain. Let p E

[1,2*). Then any bounded set J c H1(R) is relatively compact in LP(R).

Proof. Let R1 c RN be a bounded domain such that 52 c R1. Let T : H1(R) -+ Hi(R1) be an extension operator and apply Theorem 2.8 to TJ.

Corollary 2.5 Every sequence bounded in H ' ( I w ~ ) has a subsequence that converges almost everywhere and weakly in H1 (RN) . Every sequence


bounded i n D ' > ~ ( I W ~ ) , N 2 3, has a subsequence that converges almost everywhere and weakly i n V112(RN) .

Proof. In either case, Banach-Alaoglu theorem yields a weakly convergent renumbered subsequence u k . Moreover, this subsequence, in restriction to any bounded set, is convergent in measure, due to Theorem 2.10 and to Theorem 2.9 in the respective case. It remains to note that a sequence that converges in measure on every bounded subset of R N has a subsequence convergent almost everywhere in R N .

We conclude the section with a compactness lemma ([104], Theorem 9.5).

Lemma 2.12 Let a E LN12(RN) , N > 2. Then the functional @(u) = JRN a(x)u2 is continuous with respect to the weak convergence in ~l>~(lR~).

Proof. Assume without loss of generality that a ( x ) 2 0. Note that from the Holder inequality follows immediately that @ E C ( L 2 ' ) . Let u k - u in V ~ ' ~ ( I W ~ ) . Applying an elementary identity and then the Cauchy inequality, we have

Thus it suffices to prove that @(uk-u) -, 0. Let E > 0 and let a, E C r ( R N ) be such that ( (a , - a ( ( N / 2 5 E. Then, by the Holder inequality,

2 @(uk - u ) 5 ] ae(u - u k l 2 + - allN/211u - ~ k 1 1 2 *

Cellu - ukllg,suppac +C//ac - a / l ~ / 2 -

By compactness of the local Sobolev imbeddings, u k 4 u in Lfo,, and thus,

Since E is arbitrary, @ ( u k - u) -, 0 and thus @ ( u k ) -+ @(u). 17

Corollary 2.6 If a E L ~ / ' ( J R ~ ) , N > 2, a > 0 , then the space D1t2(RN) is compactly imbedded into L ~ ( I W ~ , a ) .

2.9 Trace on the boundary

In this section we will consider the domain R y = RN- l x (0 ,00) , N > 2, denoting the respective variables as z = ( x , s ) .


Lemma 2.13 Let R, R1 c I R N - I be bounded domains, R G 511. Then the sets

and

J' := { ~ l ~ ~ { ~ ~ : u E c?(ltN), 1 I V U I ~ + Jnl u(., o12 5 1) nl x (o,m)

are relatively compact in L2 (0).

Proof. Consider a unit ball BI (0) in IRN-l. Let p E C r (B1 ( O ) , [O, 11) satisfy J p = 1 and let, for t E (0, d(n,RN-I \ R l ) ) ,

Mtu(x, S ) =

Define

and

Ji := { N t ~ l ~ ~ ~ ~ > : u E G',OO(IRN),

From this step the proof, for both sets J and J', is analogous to the proof of Lemma 2.8, once we establish the following analog of (2.21)

1 I ~ t u l ~ ( x , o ) d x 5 G't / I ~ u l ' d x d s (2.47) Ql x ( O w )

that can be verified by means of the following relation:


On the last step we used (2.21) in I R N - I for the first integral and (2.19) for d,u in RN-I for the second integral.

Lemma 2.14 Let R , R 1 be as above, let II, E ~ ' ( 0 ) be such that II, # 0. Then there exist a C > 0, such that for every u E Cr(IRN),

The proof is completely analogous to the proof of Theorem 2.3. One has the following analog of Proposition 2.9.

Lemma 2.15 There exists a C > 0, such that for every p 5 1 and

u E G='(Q)l

Proof. The proof follows the proof of Lemma 2.9 with self-explanatory modifications, e.g., starting with partition of Q = (-1, l ) N - l into cubes Qi, i = 1, . . . , 2m(N-1) , replacing (2.22) with (2.48) and adjusting to homogeneity of the N-1-dimensional Lebesgue measure instead of the N-dimensional one.

Corollary 2.7 There exists a C > 0 such that for every p > 0 and u E C ( y ( R N ) ,

Proof. The proof follows the proof of (2.26) starting with the case suppu C (-1, l )N-l x R . The only nontrivial modification is the use of following analog of the F'riedrichs inequality:


To prove this, let x E C F (R, [0, I ] ) , suppx = [ - I , 11, ~ ( 0 ) = 1. Then

In the last step we have applied the usual Friedrichs inequality (2.13) for u(., s ) on (-1, 1IN-l .

Theorem 2.11 Let N > 3 and let 2 = w. There exists a C > 0 such that for every u E C F ( R N ) ,

Consequently, the space D ~ ~ ~ ( I W ~ ) has a continuous trace on L~(IW~-~).

Proof. The proof of (2.51) repeats the one for (2.29) using the same functions xj(lul) = 2-jX(2jlul) , j E Z, as in Theorem 2.4, with the only modification that (2.50) is used instead of (2.26) and the scaling parameter

3 pj is chosen as pj = 2- N p.

The best constant in (2.51) is given by

where S N is the best Sobolev constant in (2.29) given by (2.31). Existence of the minimizer for the inequality is discussed in Chapter 5. The minimizing function is a scalar multiple of

cf. [46].

Problem 2.6 State and prove the analog of Nash inequality that estimates the L2(RN-')-norm by a product of appropriate powers of 1 1 VulI z,ay -norm and the Ilull l ,wtN-n~rm.

Theorem 2.12 Let N 2 2 and p E [2 ,2] . Then there exists a C > 0 such that for every u E C F ( R N )

Chapter 2 Sobolev spaces

Proof. Combine (2.51) and the Holder inequality.

Problem 2.7

(a) Show that (2.53) holds for N = 2, p > 2. (b) Show that one can replace JWN-, u(., 0)= in (2.53) with JwN u2. + Theorem 2.13 Let N > 2, p E [I,$], and let R c EXN-' be CL bounded extension domain. Then there exists a C > 0 such that for every u E

c,-(EXN),

Moreover, the imbedding defined by (2.54) is compact whenever p < 2.

Proof. Let O1 c IRN-l, R all be a bounded domain and let To : H1 (0) + H i (a1) be a correspondent extension operator. We leave it to the reader to verify that if T : H1(R x (0, co)) -+ H1(R1 x (0, co)) is given by (Tu)(., s) := TOu(., s), then T satisfies

Then applying (2.51) to Tu we have

which proves (2.54) for p = 2. This, combined with the Holder inequality gives (2.54) for p < 2.

To verify compactness of the imbedding, consider the set J =

{sax (o,m) (IVuI2 + u2) 5 1). When u E J , the integral Jalx(O,m) IV(T4I2 and, consequently the integrals Jal I T U ( . , O ) ~ ~ and Sol lTu(.,0)12 are bounded. The set {uln : u E J}, is contained for some C > 0 in


which, by Lemma 2.13, is relatively compact.

2.10 Differentiable functionals in Sobolev spaces

Sobolev imbeddings (inequality (2.29) and Corollary 2.4) allow to view LP- continuous functionals as functionals continuous in the Sobolev space. We start with examples considered at the end of Chapter 1.

Remark 2.6 Let R E J R N , be an open set. Let q > 2 when N = 1 1 2 and let q = 2* otherwise. Since HA(R) is continuously imbedded into LP(R), p E [2, q] , and D1t2(R) is continuou~ly imbedded into ~ " ( 0 ) for N > 2, the functional @(u) = Ja F ( x , u ) with F E C ( R x J R ) i s continuous i n H J ( R ) whenever

and i t is continuous i n D 1 > 2 ( R ) , N > 2, whenever IF(x l s)l 5 C l ~ 1 ~ ' . Sim- ilarly, the mapping (1.27) is continuous on H t (52) x HA ( R ) if (1.30) holds with p = 2 and q as above.

Lemma 1.7 combined with Corollary 2.4 gives:

Proposition 2.5 Let R c R N be an open set and let @ be as in Lemma 1.7 with p = 2, q > 2 for N = 1 ,2 and q = 2* for N > 2. If uk E H i ( R ) is a bounded sequence and uk -, u i n L r ( R ) for some r E (2 , q ) , then @(vk) -, @(v).

Theorem 2.14 Let R c IWN be an open set. Let f be as i n Lemma 1.9 with p = 2 and q as above. Let

and let

Then @ has a continuous gradient V @ : H t ( R ) 4 H;(R) :

Moreover, i f uk is a bounded sequence i n H t ( R ) and u k + v i n L r ( R ) , r E ( 2 , q ) , then V@(uk) 4 V@(U) i n H ~ ( R ) .

chapter 2 Sobolev spaces 57

Proof. By Lemma 1.8, the directional derivative of @ on L = CF(R) is given by the right hand side of (2.57) and by Lemma 1.9 it extends to a continuous map in ( L ~ n L4) x (L2 n L4) (with the same expression), which, by the Sobolev imbedding, is also continuous in H F ( R ) x HF(R) . The theorem follows then from Corollary 1.5. Continuity with respect to LT-convergence follows from Lemma 1.9.

2.11 Sobolev spaces of higher order

We give here some additional excerpts from the theory of Sobolev spaces, without proofs.

1. Let R c IRN be a domain and let D denote the weak derivative as defined by (2.1). We will say that a function u E L:,,(S2) admits locally integrable weak derivatives up to the order m if for each Ic = 0, 1, . . . , m there exists a collection of N k LL,(R)-functions u ( ~ ) , such that u(O) = u and u("') = DU("). We will say then that u ( ~ ) = D ~ U .

Problem 2.8 Show that if a function u E L:,,(R) admits locally integrable weak derivatives up to the order m, then for every cp E C,"(R)

2. The Sobolev space Hm(R), m E N, is a linear space of functions u E L2(R) with L2(R)-integrable weak derivatives up to the order m, which are usually denoted v k u , equipped with the norm

Here the notation I V ~ U I refers to any equivalent length of an Nk-tuple v k u . Since the term corresponding to Ic = 0 is S u 2 , the space Hm(R) is continuously imbedded into L2(R). It is known that the space Hm(R) is complete. An equivalent norm on Hm(R) can be given by

For m = 2 there is an equivalent Sobolev norm


The space Hr(S2) is a closure of C,"(R) in Hm(S2). 3. When 2m > N , there is a C > 0 such that, whenever 0 5 k <

m - $ < k + 1, for every u E Hm(RN) ,

for any bounded set B c RN the imbedding (in sense of restrictions to B ) into C ( B ) is compact, and

Iu(x)I -+ 00 for 1x1 + 00. (2.63)

4. The space Dm32(RN), 2m < N , is a completion of C,"(RN) with respect to the norm

An equivalent norm for m = 2, N > 4, is

There exists sLm) > 0 such that for every u E C r ( R N ) ,

This implies that the space Dm12(RN) is continuously imbedded into L*(R~) and the inequality holds for every function u E Dm'2.

5. The following inequality holds true when 2m < N , p E [2, *I, as well as when 2m=N and p 2 2:

This is a particular case of

whenever q E 12, e] and prn < N , or q 2 2 and pm 2 N , or q = m and

pm > N. The completion space of Cr(lKN) in the norm of the right hand side of (2.68) is called the Sobolev space wmJ'(lRN).

Chapter 3

Weak convergence decomposition

In this chapter we ~rovide a refinement of the Banach-Alaoglu theorem in a separable infinite-dimensional Hilbert space H equipped with a fixed set D of linear operators and consider its application in the model case, the space H ~ ( R ~ ) equipped with Euclidean shifts.

Consider a bounded sequence uk E H , k E N, and a continuous functional @ : H 4 R. If dimH < oo, then uk will have a limit point, say u, and, on a correspondent renumbered subsequence, @(uk) 4 @(u). If d imH = oo, then finding a point u such that, on a renumbered subsequence, @(uk) -+ @(u) cannot rely on finding limit points of uk, since bounded sequences may have no limit points. Banach-Alaoglu theorem asserts existence of limit points with respect to weak convergence, but the functional @ may not be continuous with respect to weak convergence: for example, the weak limit of a sequence of orthonormal vectors ek is 0 and the values of a(.) = 1 1 . I ( are @(ek) = 1. A class of continuous functionals is be- coming larger with respect to stronger convergence. The framework below introduces D-weak convergence, which is stronger than weak convergence. Of course bounded sequences do not necessarily have D-weak limit points; instead, one has to relate to a larger set of "dislocated" weak limits of uk, namely, the weak limits of gkuk where gk E D, k E N, as ones that determine the limit values of functionals. For example, if wl, wz E C r ( R ) , and uk(t) = wl (t - k) + wg(t + k), then uk 0 in H = H' (R), but at the same time, for all k sufficiently large, IIukll; = I I w ~ I I ; + IIw211P, 1 I p < m. Here lim llukll; is determined by the dislocated weak limits w limuk(. + I c ) = wl and w limuk (. - k) = WZ.

59


3.1 D-weak convergence and dislocation spaces

Let H be a separable infinite-dimensional Hilbert space. The following definition uses an observation that a linear operator of the form A*A, A E

L(H), is self-adjoint and Lemma 1.5 applies.

Definition 3.1 Let D be a set of bounded linear operators on H such that for every g E D, infuGH,llull,l ((gu(1 > 0. We will say that the sequence uk E H, k E N, converges to u D-weakly , which we will denote as

if for all cp E H ,

lim sup(uk - ~ , ~ ( ~ * ~ ) - l c p ) = 0. k+oo g~ D

D Equivalently, we can say that uk - u if and only if for any sequence gk E Dl

When the set D consists of isometries g*g = id, g E D, and is compact (in C(H) as the Banach space equipped with the operator norm), it is easy to see that D-weak convergence does not improve weak convergence, i.e., from uk - 0 follows (g;gk)-l giuk = g;uk - 0 for any sequence gk E D. Indeed, let uk - 0, and assume that there is a gk E D such that g;uk f\ 0. Let us select a renumbered subsequence such that gk converges strongly to some g E D and gkuk converges weakly to some w # 0. Then, for any

V E H l

(w, cp) = lim(gkuk, (P) = lim(uk, (gk - g ) * ~ ) + lim(uk, g*cp) = 0, (3.2)

a contradiction. On the other hand, if D is the group of all unitary operators D

on H, then uk --\ 0 implies uk -+ 0 in norm. Indeed, let e E H , ((e(( = 1, and let gk be unitary operators such that g;uk = IIuklle. For instance, define gk as an appropriate two-dimensional orthogonal rotation in the span of e, uk and as the identity on the orthogonal complement of the span and extended to whole H by linearity. Then llukll = (g;uk, e) -' 0.

Definition 3.2 Let H be a separable infinite-dimensional Hilbert space. A set D of bounded linear operators on H is a set of dislocations if

0 < y = inf llgu1I2 < sup 1 1 g ~ 1 1 2 < 00, g~D,llull=l g~D,ll~II=l (3.3)

Chapter 3 Weak convergence decomposition 6 1

and, whenever u k E H and gk, hk E D ,

If, in addition, D is a group under operator multiplication, we say that it is a group of dislocations.

The pair ( H , D) will be called a dislocatiolz space.

Remark 3.1 By (3.3) dislocations are injective maps. Definition 3.2 takes a much simpler form if the operators g E D are isometries, since conditions (3.3),(3.4) are obviously satisfied when g*g = id. Furthermore, when D i s a group of unitary operators, (3.5) takes an equivalent form

Proposition 3.1 Let D be a set of unitary operators on H . If

gk, hk E D , h;gk f\ 0 + h;gk has a strongly convergent subsequence,

(3.7) then D i s a set of dislocations.

Proof. Relations (3.3), (3.4) are immediate since every unitary operator g satisfies g*g = id. In order to prove (3.5), assume u k E H , gk, hk E D , h;gk + 0, and (taking into account that g;gk = id) gluk -\ 0. Note that h;gk f\ 0 is equivalent to g;hk f\ 0 , since both statements are equivalent to the negation of V u , v E H , (gku, h k v ) + 0. This allows to conclude from (3.7) that g;hk has a strongly convergent subsequence, which we renumber without changing the notation. Let v E H . Then g;hkv is convergent and then, by Corollary 1.4, (g;uk,g;hkv) -+ 0. Passing to the adjoint operator for g;hk, one has (h;gkg;uk, v ) --+ 0. Since gkg; = id and since v is arbitrary, this implies h;uk --\ 0 which verifies (3.5).

3.2 D-weak convergence in 1' with shifts

Proposition 3.2 Let H = 12(7ZN). Then the set

is a set of dislocations.


Proof. Note that the 12-norm is shift-invariant and, consequently, the elements of DZN are unitary operators. Since Dziv is a group, due to Proposition 3.1 it suffices to verify that any sequence of shifts 70, that does not converge weakly to zero, has a convergent subsequence. Note that if I,OkI + co then rlp, - 0. Thus, whenever the latter is false, Pk has a bounded subsequence, which has a constant subsequence, and thus the correspondent subsequence of r]p, is constant.

Proposition 3.3 Let H = l2(ZN) and let D = DZN. Then ck 4 0 in H if and only if ck -+ 0 in lm(ZN).

Proof. Suficiency. Assume that ck -t 0 in lm(ZN). Then for every a k E 2ZN, ck(ak) -' 0, or in other words, for every y E ZN and every sequence crk E ZN, (q;,ck)(y) -+ 0. Let Co(2ZN) denote a set of real valued functions on ZN with finite support. It is immediate that Co(ZN) is dense in 12(ZN). By taking finite linear combinations one has (q;,ck, cp) -+ 0 for every cp E Co(ZN) and every sequence a k E ZN. Therefore, by Lemma 1.2

D (qGkck1 cp) + 0 for every cp E 12(ZN), which yields ck - 0.

Necessity. Assume that ck 0. Let a k E ZN be such that Ick(ak)l 2 1 supaEziv (ck(a:)(. Then, using the notation e for the vector with compo-

nents e(0) = 1 and e(a) = 1, a: # 0, we obtain

Corollary 3.1 If ck D4-n 0 in 12(2ZN), then for evely q > 2, ck -+ 0 in lq(zN).

Proof. Note that ck - 0, which implies that the sequence ck is bounded in 12. Then

C I C ~ ) I ~ < sup lcf)q-2 C ~ c e ) ) ~ < ~ l l ~ ~ l l E ~ -' 0. a P € Z N a

3.3 Weak convergence decomposition

The following is a refinement of the Banach-Alaoglu theorem on a dislocation space.

Theorem 3.1 Let (H, D) be a dislocation space. Let y > 0 be as in (3.3). If uk E H be a bounded sequence, then there exists a set No C W,

Chapter 3 Weak convergence decomposition 63

w ( ~ ) E H , g p ) E Dl 9:) = id, with k E N, n E No such that for a renumbered subsequence,

gp)*gLrn) - o for n # rn, x 5 -y-'limsup lluk112, n E N o

where the series gP)w(") converges uniformly in k .

Proof. 1. We show first that (3.10) follows from (3.8) and (3.9). Consider the obvious inequality

Using the bilinearity of the scalar product, we have

The last term in (3.12) converges to zero due to (3.9). Let us estimate the third term, using (3.4) and (3.8).


Combining (3.12) and (3.14) we get:

M

lim sup lluk 1 1 2 2 limsup x 1lgp)w(") 1 1 2 , (3.15) n= 1

which, by (3.3), gives

M x I W ( ~ ) ~ / ~ 5 y-' limsup Ilukl12 < y-llimsup 11uk11~. (3.16)

This proves (3.10) when No is finite. If No is infinite, (3.10) follows from (3.16) by taking M -+ GO.

D 2. Observe that if uk - 0, the theorem is verified with No = 0. If not

so, consider the expressions of the form

The sequence ur; is bounded, D is a bounded set, the sequence (gC)*gf))-l is bounded by (3.3) and Lemma 1.5, so the sequence in (3.17) is bounded and thus, for any choice of gk E D, it has a weakly convergent subsequence. Since we assume that uk does not converge D-weakly to zero, there exists necessarily a renumbered sequence 9;) that yields a non-zero limit in (3.17).

Let

and observe, by (3.17) and because g;)*gf) have a uniformly bounded inverse due to (3.3), that

If vf ) 2 0, the theorem is verified with No = (1). If not - we repeat

the argument above - there exist, necessarily, a sequence gf?) E D and a w ( ~ ) # 0 such that, on a renumbered subsequence,

Let us set

Chapter 3 Weak convergence decomposition

Then we will have an obvious analog of (3.18):

If we assume that

then by (3.5), (3.19),

which, due to (3.18), yields

We now use (3.5) again to replace in (3.20) ( g * g ) l g * with

( g f ' * g f ' ) - l g f ' * , which results in

which cannot be true since we assumed w ( ~ ) # 0. From this contradiction follows

Since for bounded sequences of operators Ak - 0 implies A; - 0 , we also have

Recursively we define:

:= - g y W ( n ) = Uk - g f ) w ( l ) - . . . - gk (n) W (n) 7 (3.22)

where

calculated on a successively renumbered subsequence. We subordinate the choice of g r ) and thus extraction of this subsequence for every given n to the following requirements. For every n E N we set

wn = {w E H \ {O) : 3gj E D, { k j } c N : (g;gj)-lg:u$) 2 w} ,

65


and

Note that t , < co, since all operators involved at all steps leading to the definition of Wn have a uniform bound. If for some n , tn = 0, the theorem

is proved with No = (1, . . . , n - 1). Otherwise, we choose a E Wn such that

and the sequence g P f l) is chosen so that on a subsequence that we renumber,

An argument analogous to the one brought above for n = 1 shows that

gr )*gp) - 0 whenever p # q,p, g 5 n. (3.25)

This allows to deduce immediately (3.8) from (3.24). From (3.15) and(3.23) follows

Let cpi be an orthonormal basis in H . Then by definition of W,,

lim sup 2-' sup(uP), g(g*g)-1cpi)2 5 2t:, n E W. k ' g E D

Let k(n) be such that

This implies that

(.~;)7s*(s*s)-1Y) + 0

for any cp that is a linear combination of pi, and an elementary density argument extends this relation to any cp E H , so that


as n 4 co. Instead of k(n) selected for each n from the index set of a renumbered subsequence of uk (that was produced by successive extractions), we will now use the correspondent index (preserving the notation k(n)) from the original enumeration of uk. (This change of enumeration

(n) affects also the terms g$)), j = 1 , . . . . n, in the definition (3.22) of vk(,).) Then we conclude that

Note that kn can be chosen in (3.26) arbitrarily large, and in particular also so that the series xj g$)).w(j) is uniformly convergent due to (3.16), and therefore (3.11) follows. Since the final extraction is a subsequence of the sequence in (3.25), (3.9) follows. Finally, if w(') = w limuk # 0, we could have chosen 9:) = id at the first step. If w limuk = 0, we renumber the

terms in expansion by n = 2,3, . . . and set g?) = i d , w(') = 0.

Corollary 3.2 Assume, under conditions of Theorem 3.1, that D consists of isometries. Then the relation

holds with (3.9),

and

Proof. When operators gk are isometries, (3.8) becomes (3.28) and repeating the argument of (3.11) one obtains


3.4 Uniqueness in the weak convergence decomposition

Decomposition (3.11) is unique up to a permutation of indices, and up to constant operator multiples in the operators g r ) and the weak limits w ( ~ ) .

Proposition 3.4 Assume conditions of Theorem 3.1 and let UI, be the renumbered subsequence for which the assertions of the theorem hold. If this subsequence has two decompositions of the form (3.11) with respec-

( tive parameters g;), w ( ~ ) ) No, and gj,(""), ~ ' ( ~ 1 , N;, then there exists a renumbered subsequence of lc and a bijection No -, Nb, n H m,, such that

and

~ ( m ) * n) ( * w limyk = w limg;) gj;(lrL) = O for m # m,.

( (")* ( ) gf)* to the decomposition proof. Let n E N ~ . ~ p p l y gk g;)

uk - Em g$m)w'(m) 9 0 and calculate the weak limit:


(n)* ~ ( m n ) ~ t ( m , , ) Note that there exists m = m,, such that ( g r ) * g ~ ) ) - ' gk gk

has a subsequence with a non-zero weak limit. Then, on this subse-

quence, ( g ~ ) * g r ) ) g ~ ) * g $ ~ ) - 0 for all m # m,. Indeed, by (3.9)

g$mn)*g$m) -+ 0 and then, due to (3.4), (3.5),

Consequently, on a renumbered subsequence,

Let now k j ' ) be a subsequence of k extracted for n = 1 and k p ) be a

subsequence of kjn- ' ) extracted for n as above and set a monotone sequence

5 7 k y ) . It remains to use (3.4) and (3.5) to derive (3.30) from (3.31) and

g r ) g$m) - 0 from (3.32) for m # m,. A standard argument allows to return from extractions to the original sequence. 0

3.5 D-flask subspaces. D-weak compactness

Definition 3.3 Let (H, D) be a dislocation space and assume that D is a group of unitary operators. A closed subspace Ho of H is called a D-flask subspace if the set DHo is sequentially closed with respect to weak convergence, in other words, if for every sequence uk E Ho, gk E D, such that gkuk converges weakly in H , there exists a u E Ho, g E D, such that w limgkuk = gu.

Obviously, any closed D-invariant subspace of H, in particular, (0) and H itself, are D-flask subspaces of H.

Proposition 3.5 Let Ho be a flask subspace of a dislocation space (H, D ) . Then any bounded sequence uk E Ho has a subsequence satisfying the assertions of Corollary 3.2 and, in addition, w ( " ) E Ho.

Proof. By (3.8) and the definition of the flask subspace there exsit g(n) E

D, such that g ( " ) ~ ( ~ ) E Ho. Set dn) := g ( " ) ~ ( ~ ) E Ho and ?jF) :=

gj")g(")* . Then (3.27) and (3.29) hold with dn), fir). It remains to show

that ij?) satisfies (3.9). Assume that rn # n. Then for every u , v E H


follows

(m) (m) * (n) (n) \, v) (9 gk gk g

which immediately yields 3im)*$p) 0 by (3.9) for g p ) .

Proposition 3.6 Let (H, D) be a dislocation space. Assume that D is a group of unitary operators. Then a bounded sequence uk E H has a D-weakly convergent subsequence if and only if

on a renumbered subsequence.

Proof. Consider the renumbered subsequence of uk given by Corol- D lary 3.2. If (3.33) holds, then w(") = 0 unless n = 1 and uk - w(l). Since D w(') is by definition the weak limit of uk, assume now that uk - w('). This

implies that whenever gk E D, gk - 0,

3.6 D-weak convergence with shift operators in RN

Let H be the Hilbert space H = H1(RN) equipped with a set

where G is a closed subset of RN that forms a group with respect to the vector addition. Obviously, DG is a commutative group (g,+, = g,g, =

gygx with g(0) = id) of unitary operators (g;' = g-, = 9;).

Lemma 3.1 Let yk E RN. Then g,, - 0 if and only if lykl -+ m.

Proof. If lykl -+ CO, then for every u ,v E C ? ( R ~ ) , (gyku,v) = 0 once supp u + yk n supp v = 0, that is, for all k sufficiently large. Since C F (RN) is dense in H ' ( R ~ ) , Lemma 1.3 gives g,, --\ 0. Conversely, if yk has a bounded subsequence, it will have a renumbered convergent subsequence yk + y. If u, p E C r ( R N ) , then, by the uniform convergence theorem for integrals, (gyk U, p) + (g, u, p). Since C? (RN) is


dense in H1(IRN) and the operators in DRlv are bounded, the convergence extends to all u, cp E H1. Therefore, gYk A gy # 0.

Lemma 3.2 If DG = {u H u(. - y), y E G) where G is a closed subset of IRN, then (H1(RN), DG) is a dislocation space.

Proof. Due to Proposition 3.1, in order to verify that DG is a set of dislocations it suffices to show that if an operator sequence g,, : u H u(+- yk), - .

yk E KtN does not converge weakly to zero, then it has a strongly convergent subsequence. Assume that g,, does not converge weakly to zero. By Lemma 3.1, yk has a bounded, and therefore, a convergent renumbered subsequence yk -+ y. Let cp E CF (IRN), u E H; (BN). Then I (u(. - yk) -

u( . -Y)~ (P ) I = I(u, (P(. + ~ k ) - P(.-Y))I 5 l l ~ l l I I ~ ( . - ~ k ) -(P( .-Y)[[ + 0 by the uniform convergence theorem. Then u(.- yk) - u(.- y) by Lemma 1.2.

For comparison, consider a set of operators that is not a set of dislocations.

Example 3.1 Let EX: = IRN-l x ( 0 , ~ ) and Zy = ZN-' x N. Let us note that the set of shifts g,, y E Zy, satisfies (3.3) and (3.4), because its elements are obviously isometries on H;(R:), but it fails to meet the condition (3.5). Indeed, let hk = gke, gk = g(k+')e, where e = (0,. . . ,0, I), and let uk = v(. -Ice) with v E Cr(IRN-I x (0, l )) . Then h;gku = u(. - e) does not converge weakly to zero, giuk = v(. + e)JRY = 0, while h;uk =

v # 0.

The group G c KtN that defines D has to be sufficiently large in order t o have D-weak convergence significantly stronger than weak convergence in H' (IRN). Consider the following condition:

Remark 3.2 I t is easy to see that the group G satisfies (3.34) if and only if it contains a lattice of basis vectors in IRN. In particular, (3.34) is satisfied by ZN.

Lemma 3.3 Let G satisfy (3.34). Let uk be a bounded sequence in D H' (IRN) and let p E (2,2*). Then uk 0 * ((uk ( ( L p ( ~ ~ ) -+ 0.

Proof, For the only if statement note that due to the shift invariance of Lp-norm, for any choice of gk E D, the Lp-weak limits of the sequence gkuk will be zero. Since the sequence uk is bounded in H1-norm, by Lemma 1.3


the LP-weak convergence implies weak convergence in H1 and thus, the D-weak convergence of uk . To prove the if statement, note that in view of Remark 3.2 it suffices to consider without loss of generality the case G = ZN. Indeed, if Dl is a subgroup of D2, D2-weak convergence implies Dl-weak convergence, the group G contains a lattice and any lattice is linearly homeomorphic to ZN.

Let uk %N 0. Consider a unit cube Q := (0, I ) ~ . By the Sobolev inequality for bounded domains (Corollary 2.4), there is a C > 0 such that

By adding terms in (3.35) over y E ZN, and noticing that U y E Z ~ ( Q + y) is RN up to a set of Lebesgue measure zero, we obtain

(3.36) where yk E zN is any sequence satisfying

It remains to note that by compactness of imbedding of H1 (Q) into Lp(Q), one has gakUk -+ 0 in Lp(RN), SO that the assertion of the lemma follows from (3.36).

As a consequence of Corollary 3.2, Lemma 3.1 and Lemma 3.3, we have

Corollary 3.3 Let uk E H ~ ( I W ~ ) be a bounded sequence and let G C RN be a group satisfying (3.34) (in particular 7ZN or IRN). There exists a set No c M, w ( ~ ) E H , y r ) E G, y;) = 0, with k E M, n E No such that, on a renumbered subsequence,

w(") = w limuk(. + y?)), (3.37)

I y r ) - yim) 1 for n # m, (3.38)

l l ~ ( " ) ( ( & ~ 5 limsup llukll&l, (3.39) nENo

uk - w(")(. - y?)) + 0 DG-weakly and in L P ( R ~ ) (3.40) nENo


for any p E (2,2*), and the series in (3.40) converges in H1(IRN) uniformly i n k .

Lemma 3.4 Let H = H ~ ( R ~ ) , let G satisfy (3.34) and let ur, and w ( ~ ) be as in Corollary 3.3. If F E Cl,,(IRN x R) is as in Lemma 1.7 and F ( x + y, s) = f (x, s) for all x E RN, s E IR and y E G, then

Proof. From Theorem 3.1 and (1.22) in Lemma 1.7 it follows that

Moreover, from (1.22) follows that @ is uniformly continuous on bounded sets in H1(RN), and since the series En w ( ~ ) ( . + y?)) is convergent uniformly in k,

Therefore it suffices to prove that

Since G'r(IRN) is dense in H1(IRN), due to Lemma 1.7 it suffices to show (3.43) for w(") E C r ( R N ) . Let R = sup{lyl, y E s ~ ~ ~ w ( ~ ) , n = I , . . . , M ) andletko ~ h ' b e s u c h t h a t (yLrn)-yf)(> R f o r a l l m # n , m , n = 1, ..., M and k > ko.Then for k > ko, since the integrand in the left hand side is supported in the union of balls B R ( ~ ~ ) ) , n = 1, . . . , M ,

which proves the lemma.

Corollary 3.4 Let uk be a bounded sequence in H1(IRN) and let F : IR + R be as in Lemma 1.7. Then, on a renumbered subsequence, uk - w


in H 1 ( R N ) and

Proof. Consider H 1 ( R N ) equipped with the group D z ~ , apply Lemma 3.4 to uk and to ur, - w(') and take the difference between (3.41) for respective sequences.

This statement, known as BrBzis-Lieb lemma, is proved in [22] under a somewhat different set of assumptions. In particular, for F ( u ) = 1u(q, q > 1, it suffices to assume that the sequence U I , is bounded in Lq.

Theorem 3.2 Let q E [ I , oo) and let R c RN be a measurable set. Let uk - w in Lq(R) and assume that ur, converges to w almost everywhere in R. Then

Proof. Let E > 0 and

so that vi -+ 0 almost everywhere in R. Using an elementary inequality Ila + blq - lalQl 5 ~la lq + CElbIq, a, b E R , we have

Then by the Lebesgue dominated convergence theorem, Ja v; + 0. This implies

l i m s u p J n l ~ u ~ l q - ~ ~ ~ - w ~ q - / ~ / q ~ / ~ l i m s ~ p

Since E in the right hand side above can be arbitrarily small, (3.45) follows.

Remark 3.3 The statement in Theorem 3.2 remains true if Lq(R) is replaced by a weighted space L'J(R, w) , where w : 52 -+ (0, oo) is a measurable function.

Remark 3.4 One has the following elementary generalization of Lemma 3.4. Let uk be a bounded sequence in H ' ( R ~ ) and let F E

Cl,,(RN x R ) . Assume that s H up,,^^ IF(x, s)l satisfies (1.22) in Lemma 1.7 and that the limit

Chapter 3 Weak convergence decomposition

Fm (s) := lim F(x , s) 1x1-+m

exists. Let the group G c RN satisfy (3.34) and let w(") be as in Corol- lary 3.3. Then

3.7 Constrained minimization

We give here an elementary application of Theorem 3.1 to existence of minimizers in variational problems. Let p E (2,2*), let V E Lm(RN) be a ~ ~ - ~ e r i o d i c function and assume that inf V > 0. Consider

cp = inf J (lvu(x)I2 + V ( X ) ~ U ( X ) ~ ~ ) dx. (3.49) U E H ~ ( W ~ ) , ( ( U ( I , = I WN

Note that E(u) := JRN ( ~ V U ( X ) ~ ~ + ~ ( 2 ) 1u(x) 1 2 ) dx defines an equivalent Sobolev norm on H1(RN) and therefore cp > 0 by the Sobolev imbedding (2.41). If u is a minimizer in (3.49), then, due to Proposition 2.2, (u( E H1(R) is a minimizer, and then, by the strong maximum principle (Proposition C.2), lul > 0 and so u is sign definite and can be assumed to be positive. By the rule of Lagrange multipliers, u satisfies the equation

which is equivalent to -Au + Vu = Xup-l in the sense of weak derivaties. The constant X can be made 1 if one multiplies u with a suitable scalar multiple.

Proposition 3.7 Every minimizing sequence uk for the problem (3.49) has a (renumbered) subsequence such that uk(. - yk), with some yk E ZN, converges to a minimizer of the problem.

When V = 1, the minimizer of (3.49) is unique and satisfies

1x1 ~ e l x l u ( x ) 4 const.

For details see [61],[17], [79], [lo] and references therein.

Proof. Let uk be a minimizing sequence for (3.49), that is, Ilukllp = 1 and JRN ( I V U ~ ~ ( X ) ~ ~ + V ( X ) I U ~ ( X ) ~ ~ ) dx + cp. Note that Corollary 3.3 remains true for the dislocation space H1(RN) with the norm \lull = and

75


with the the group of dislocations DZ.v. Let the renumbered subsequence of uk, w ( ~ ) and y?) be as given by Corollary 3.3. By Lemma 3.4,

If we set J I ~ ( ~ ) l p d x = tn, then, comparing the Hilbert norm of the nor- 1

malized w ( ~ ) with the infimum value q,, we get Ilt ,b(n) n)l12 2 cp, which 2

gives I I w ( ~ ) ~ ( ~ 2 cpta. Substituting this into (3.52) we get

On the other hand, (3.51) can be written as C tn = 1 and since $ < 1, both relations can hold only if all tn but one, say trio, equal zero and tn0 = 1. This yields from (3.52) Ilw(no)112 = q,, because the left hand side cannot be less than the infimum of 1) w)I2, SO ~ ( ~ 0 ) is a minimizer. We conclude from (3.40) that uk-w("~)(--yp)) -, 0 in LP, or equivalently,

uk(.+y?)) -i ~ ( ~ 0 ) in LP. Moreover, u k ( . + y p ) ) - ~ ( ~ 0 1 , which together with the assumed convergence of Sobolev norms (11uk 1 1 2 + cp = llw("0) [I2) yields that uk(. + y p ) ) converges to ~ ( " 0 ) in ~ ' ( 1 ~ ) .

We give now a second proof of the same statement, based on the Brkzis-Lieb lemma.

Proof. Let uk be a minimizing sequence and note that for any sequence yk E ZN, uk(. + yk) is also a minimizing sequence. By Lemma 3.3 the

D relation uk - 0 is false, since otherwise ( ( u ~ ( ( ~ -' 0, a contradiction. Thus, for certain yk, on a renumbered subsequence, uk(. + yk) --+ w # 0. Let us rename the minimizing sequence uk(. + yk) as uk. By Theorem 2.9 uk converges in measure on every bounded measurable set, and thus, a renumbered subsequence of uk converges almost everywhere. By Proposition 1.6,

cp = lim lluk112 = llw112 + lim lluk - w I I 2 (3.54)

and by Brkzis-Lieb Lemma (Theorem 3.2),


Let t = JwN I w I P From definition of cp and (3.54) follows then that

which implies with necessity t = 0 or t = 1. I f t = 0 then w = 0, a contradiction. Moreover, i f lim (Iuk - w(I2 > 0, then ((w(I2 < cp, a contradiction. Consequently, uk -+ w in H .

Remark 3.5 Consider (3.49) with V = 1. Let R = ElN-' x (0,oo) and let

Obviously, cp(R) 2 c,. Let u be a minimizer for c, and let x E Cw((O, m)), ~ ( t ) = 0 for t E (0, i), ~ ( t ) = 1 for t > 1. Let uk := X ( X N ) U ( . - (0, k ) ) . An elementary calculation with uk shows that % ( a ) 5 c,, and therefore cp(R) = cp and uk is a minimizing sequence. Assume that cp(R) is attained on some w E HJ ( R ) . Then w, extended by zero to I R N , is a minimizer for (3.49). As such it satisfies the equation (3.50), contradicting the strong maximum principle (Proposition C.2). Thus the minimizer does not exist. The minimizing sequence uk constructed above has a dislocated weak limit w limuk(. + (0, k ) ) = u E H1 ( I R N ) .

3.8 Compactness in the presence of symmetries

Theorem 3.3 Let ( H , D) be a dislocation space and let D be a group of unitary operators. Let T be an infinite group of unitary operators on H and let

Assume that

(T) for every sequence gk E D such that gk - 0 there exists an infinite set T{gk) c T such that for every T I , 7 2 E T{gk) such that r1 # 7 2 ,

g;7;72gk - 0 on a renumbered subsequence.

Then HT is D-weakly sequentially compact in H , that is, any bounded sequence in HT has a D-weakly convergent convergence.

Note that i f D = Dwlv and T is an infinite subgroup of DO(N) := { u w

u o 7 , ~ E O ( N ) } , then the condition ( T ) is satisfied.


Proof. Let uk E HT be a bounded sequence. Assume without loss of generality that uk is weakly convergent to some u E H, and note that, since for every I- E T, r- luk = uk - u, by continuity of r , uk - TU, and then r u = u. Consequently u E ET. Assume that there exists a sequence gk E D such that gk - 0 and g;uk - w # 0. By (T), for every M E N there exist 71,. . . ,TM E T such that, on a renumbered subsequence, whenever i # j , gk*7;rjgk - 0. Then

which implies J J u ~ ) ) ~ >_ MJJw1I2 + o(1) for any M E N, which is a contradic- D tion. We conclude that, on a renumbered subsequence, uk - u - 0.

Consider now an application.

Lemma 3.5 Let HT = {u E H1(RN) : u o q = u, q E O(N)). The space HT is compactly imbedded into L P ( R ~ ) for any p E (2,2*).

Proof. Let H = H1(RN) D = DRN := {u H u(. - y), y E RN), T =

{u H u o q ,q E O(N)). Due to Lemma 3.1, and since for every yk E RN, (yk 1 + oo, and q # id, Iyk - qyk 1 -+ oo, conditions of Theorem 3.3 are satisfied. If uk E HT is a bounded sequence, then uk has a DRN-weakly convergent subsequence, which is also LP-convergent, p E (2,2*), due to Lemma 3.3.

Corollary 3.5 Let p E (2,2*). The problem

cP = inf U E H ~ ( R ~ ) : J J U J ~ ~ = ~ ll~ll;l


has a radially symmetric minimizer, and any radially symmetric minimizing sequence uk, JRN luklP = 1, 1 1 2 ~ ~ 1 1 ~ -+ C, has a subsequence convergent to a minimizer.

Proof. The space HT defined in Lemma 3.5 consists of all radially symmetric functions from HI. By (B.l), JwN Iu*IP = JwN lulP = 1, while from (B.3) and (B.l) follows I I u * ~ ~ $ ~ I IIUII&~, SO that c; = inf{u E HT : ]lullp =

~ ) I I U I I & ~ equals c,. Due to Lemma 3.5, @ : u H JRN lulP is a weakly continuous functional on HT. Let uz E HT be a minimizing sequence: @(u;) = 1 and ll~ill -+ cp and let, on a renumbered subsequence, u; --\ w. Then @(w) = lim@(u;) = 1 and, by weak lower semicontinuity of the Hilbert norm, llw112 5 limll~;11~ = c,. Since there is now with @(w) = 1 such that llw112 < c,, llw112 = c,, SO w is a minimizer.

3.9 The concentration compactness argument

One may state the basic objective of concentration compactness as proving, with a suitable operator set D, that a given sequence has a D-weakly convergent subsequence. Unlike the weak convergence, assured by the Banach- Alaoglu theorem, D-weak convergence is generally assured only in the sense of Theorem 3.1.

Characterization of D-weak convergence for a given dislocation space (H, D) in terms of a known space (such as L p ) , is an analytic problem in its own right. In the well understood case when H is a Sobolev space on a manifold M and D consists of actions of a transformation group G on M , the strength of the D-weak convergence is determined by how robust G is. A rule of thumb is that D-weak convergence implies LP-convergence in this case if the Sobolev imbedding over bounded subsets of M is compact and if M is co-compact relative to G (that is, if there is a compact set K c M such that UgEGgK = M). This is the case considered in Chapter 4 for M = EXN and in Chapter 9 for general manifolds. A dyadic partition of the range of a function, connected to dilation operators u(x) H 2aju(2jx), j E Z, provides, by means of reducing the functions to those with compact range, D-weak convergence that implies convergence in L ~ * with the critical Sobolev exponent.

Reasons vary for the series in (3.11) to contain at most one term (thus providing D-weak convergence for a dislocated sequence

( g p ) * g ~ ) ) - l g p ) * u k , or, if n = 1, for the sequence uk itself), depend-


ing on the applications. For instance, Proposition 3.7 uses a convexity reasoning that entails a variational penalty for the sequence splitting into separate bulks, while Lemma 3.5 exploits symmetry by proving that any dislocated weak limit w, escaping with translations yk, lykl -+ co, would be reproduced by rotation symmetry infinitely many times as w( - ryk, T E O ( N ) , adding up to the infinite Sobolev norm of the sequence. Many other mechanisms could force dislocated weak limits of sequences to become zero. Often the dislocated weak limit satisfies a different differential equation than the weak limit and reasons could be found in favor of the latter. This reasoning applies, in particular, when the original sequence is supported on a non-invariant domain 0: is may be treated as a sequence of functions on a larger invariant manifold, while its dislocated limits end up supported on a domain different from 0.

If it is not possible to verify D-weak convergence, the concentration compactness argument still renders a simpler verification whether the sequence converge D-weakly to zero or not. If it does not, there is a non-zero dislocated weak limit, which for a D-invariant problem could be a solution.

3.10 Bibliographic remarks

Weak dislocated limits based on dilations appear already in the 1981 paper [103] of J . Sacks and K. Uhlenbeck. A crucial model problem employing concentration compactness for the critical exponent has been studied in the 1983 paper [24] of H. BrQzis and L. Nirenberg. The term concentration compactness is due to P.-L. Lions, who presented a very broad array of variational non-compact problems that can be handled by concentration compactness approach in the celebrated series of papers [86], [87], 1881 and [89], preceded by several earlier publications and announcements, in particular ([85]). P.-L. Lions has presented concentration compactness in terms of behavior of sequences in LP (as vanishing, tight or dichotomic), which roughly corresponds, in the terms used here, to the expansion (3.11) with, respectively, none, one or several non-zero terms w ( ~ ) . M. Struwe ( [ l l l ] ) can be credited with the first LLmulti-bump" representation of bounded critical sequences related to (3.11), while H. Br6zis and J.-M. Coron have proved in [25] a similar expansion where the separation of dislocation parameters correspondent to (3.9) was specified. Both papers were dealing with the concentration of the critical exponent type. The multi-bump expansion of P.-L. Lions (Appendix in 1901) dealt with the translation invariant case,

Chapter 3 Weak convergence decomposition 8 1

but most of the use of concentration compactness in the first years (e.g. [47],[119], in addition to the already cited papers) concerned the limit exponent case. The focus of "multi-bump" expansions has been usually on sequences with specific properties - in the typical case, critical sequences of functionals.

The weak convergence decomposition (3.11) was published, in a slightly less general form than here, in [106]. This formulation of concentration compactness in functional-analytic terms, in addition to Struwe's global compactness and its Lions' counterpart is inspired by the results of H. Brkzis and E. Lieb: Lemma 3.3 is essentially [81], Lemma 6 (a similar statement is also found in [87]), Theorem 3.2 is the celebrated Brkzis-Lieb lemma [22] and [23] elaborates the use of concentration compactness in minimization problems based on the BrCzis-Lieb lemma.

Existence of minimum in (3.49) was proved in [86]. We present here two shorter proofs of the statement, a proof based on the weak convergence reasoning in [23] and the proof based on Corollary 3.3 that was first published in [105]. Compactness of imbeddings for the subspace of radial functions is due to P.L. Lions ([84]), but a shorter proof based on the concentration compactness argument was a common knowledge by the end of 1980's.

A summary of concentration compactness can be found in the books of J. Chabrowski [33], L.C. Evans [51], M. Flucher [56], M. Struwe [110] and M. Willem [122].

Chapter 4

Concentration compactness with Euclidean shifts

In this chapter we consider existence of minimizers in constrained minimization problems similar to (3.49), where convergence of minimizing sequences is studied in the Hilbert space H1(RN). We equip the space H1(RN) with a group of unitary operators DG = {gy : u H u(. - y), y E G), where G is an additive subgroup of EXN satisfying (3.34), for example, G = zN or G = RN. By Lemma 3.2, (H1(RN), DG) is a dislocation space.

Particular attention is paid here to problems in Hi(R), R c RN, where existence of minimizers depends on R, and to similar problems where positive solutions are minimizers in some modified sense. Related problems of the mountain pass type are considered later, in Chapter 6. Through- out the chapter the notations of scalar product and norm, unless specified otherwise, refer to the space H1(RN) and its subspaces Hi(R) with open R c RN.

4.1 Flask sets

The class of flask domains is based on Definition 3.3. If Hi(R) is a DG-flask subspace of H ~ ( R ~ ) , it is natural to call R C RN a G- flask set. We will use, however, a slightly more general definition. Together with G we also consider a subgroup T of O(N) and define (G, T)-flask sets, or flask sets with rotations.

Definition 4.1 Let G be an additive subgroup of RN and let T be a subgroup of O(N). An open set R c RN is called a (G, T)-flask set if for every sequence yk E G, lykl 4 oo, there exist z E G and r E T, such that whenever uk E H i (n ) and uk (. + yk) w in H' (RN), w E HA (7-Cl + z). If T = {id), we will say that R is a G-flask set.

83


Note that RN is a (G, T)-flask set for any choice of G and T

Theorem 4.1 Assume that the group G satisfies (3.34) and let O c RN be a (G,T)-Jtask set. If uk E Ht(O) is a bounded sequence, then there exist y r ) E G, dn) E T with yf) = 0 and = id, such that, on a renumbered subsequence,

(3.40) and (3.38) hold true, and

Moreover, if F E c(WN) is as in Lemma 1.7, then

and

Proof. Let uk be a renumbered subsequence given by Corollary 3.3 with corresponding y?) E G, w ( ~ ) E H1(WN). Due to Definition 4.1 there exist dn) E T and dn) E G, such that w ( ~ ) E HA(T(~)R + dn) ) and uk(. + y r ) + L ( ~ ) ) - ,(")(a + I(")) E H; (TO). Let us rename y r ) + An) as y r ) and w(") (. + ~ ( ~ 1 ) as w ( ~ ) , n > 2. Note that (3.40) and (3.38) remain true with the renamed y r ) and and that (3.37) holds with renamed w ( ~ ) E H;(T(~)R) which verifies (4.1). Relations (4.2), (4.3) and (4.4) follow then from (3.39), (3.41) and (3.44) respectively, once we take account of change of supports under transformation, invariance of the Lebesgue measure and of I V U ~ ~ with respect to translations and rotations.

Flask sets are characterized by their asymptotic behavior at infinity. Let X be any set and let X k C X, k E N. The lower and the upper

limit sets of the sequence X k are

Problem 4.1

(a) Show that lim inf X k C lim sup X k .

Chapter 4 Euclidean shi,fts 85

(b) Show that if Xkj is a subsequence of Xk, then lirn inf Xkj > lirn inf Xk and lirn sup Xkj c lirn sup Xk.

( c ) Show that if XA : X ++ {0,1) is a characteristic function of a set A C X , then Xlim inf xk = lirn inf XX, and Xlim sup xk = lirn sup XX,.

We shall say that two measurable sets Vl, V2 c lRN are equal up to a set of measure zero if Ifi \ VII + IVl \ fiI = 0. This establishes an equivalence relation and we will denote the correspondent equivalence class as V =

[Vl] = [Vz]. If IVl\V21 = 0 we say that Vl c V2 up to a set of measure zero. If f , g : R c lRN + lR are two measurable functions equal almost everywhere, we will denote the correspondent equivalence class as u = [f] = [g] . Let f : R c RN -+ R be a measurable function and define, taking into account that the right hand side is independent of f E [f],

Note that V([f]) is generally not [supp f] .

Lemma 4.1 Let uk be a bounded sequence in Hi(R) and let yk E RN. If uk(. + yk) -\ w in H1(RN), then there is a renumbered subsequence of yk such that, up to a set of measure zero, V(w) c liminf(R - yk).

Proof. By Corollary 2.5, a renumbered subsequence ur, (x + yk) converges to w(x) for every x E RN \ Z with some set Z of zero measure. Without loss of generality we may assume that w(x) = 0 when x E Z. When x @ Z and x 4 liminf(S2 - yk), by the definition of the lower limit set one has x 4 nk>,(R - - yk) for any n E N, which implies that x + yk $! R for some subsequence ykj Thus w(x) = limuk(x + ykj) = 0. Consequently, V(w) c liminf(S2 - yk) U Z.

The following is a sufficient geometric condition of a set to be a (G, T)-flask set.

Theorem 4.2 An open trace set R c RN is a (G, T)-flask set if for any sequence yk E G there exist z E G, T E T, such that, up to a set of measure zero,

lim inf (R - yk) c TR + Z. (4.6)

Proof. Let uk E H ~ ( R ) , w E H ' ( R ~ ) and yk E G be such that uk(. + yk) -\ w. By Lemma 4.1, there is a renumbered sequence of yk such that V(w) C liminf(0 - yk) modulo a set of zero measure. Therefore, by (4.6), V(W) C TR + z with some 7 E T, z E G. Since R is a trace set, one has w E H;(TR+z).


Let R, be an &-strip in R along its boundary:

R, = {X E R , ~ ( x , I R ~ \ R ) < E ) . (4.7)

When dR has a uniform regularity (namely, when the measure of R, vanishes as E + 0 uniformly within all balls of fixed radius), substitution of

lim inf (R\R, - yk) instead of lirn inf (0-yk) in (4.6) turns the sufficient condition into a necessary and sufficient one.

Theorem 4.3 Let R c IRN be an open trace set, and, moreover, assume that

m, := sup IBl(y) no,! + 0. yEWN

The set 52 is a (G,T)-flask set if and only if for every sequence yk E G there exist z E G, r E T, such that, up to a set of measure zero,

Proof. Let X, E C r ( R \ RE, [0, I]), ~ ( x ) = 1 for x E R \ R2€. All the set inclusions below are understood modulo a set of measure zero. Necessity. Assume that R is a (G,T)-flask set. Let, on a renumbered subsequence (that depends on E )

in H1(lRN). Since, on a renumbered subsequence, x,( + yk) + V, almost everywhere and X, = 1 on 52 \ R2,, from the definition of the lower limit follows that v, = 1 on lim inf(R \ R2, - yk). Consequently,

V(v,) 2 lim inf (R \ R2, - yk).

By the definition of the (G, T)-flask set there exist z E G, T E T such that for every E > 0, v, E H ; ( T ~ + z), and therefore

Combining last two inclusions we get

TR + z > lim inf (R \ 0 2 , - yk).

This inclusion remains true if one replaces the extraction yk with the original sequence (the liminf set of a sequence is a subset of liminf set of a subsequence).

Chapter 4 Euclidean shafis 87

Suficiency. Let uk E Hi(R), let w E H ' ( R ~ ) and let yk E G be such that, on a renumbered subsequence given by (Corollary 2.5), uk(. + yk) converges to w almost everywhere and weakly in H ' ( R ~ ) . For every E > 0 we define, for a further renumbered subsequence dependent on E,

By Lemma 4.1,

V(w,) C lim inf (R \ 0, - yk).

Let y E RN. We have, on a renumbered subsequence, using compactness of the Sobolev imbedding over Bl(y) and, at the last steps, the Cauchy inequality and (4.8):

L1(,) lw (x) - w, (x) 1 dx = lim L1(,) 1(1- X & ) U ~ I ( X + Y ~ P X

= lim / l(l - xE)lluxl 5 limsup/Rz, n BI(Y - yx)ltlluxllz B I ( Y - ~ k )

5 C sup n Bl(y)l; -+ 0 y€WN

as E + 0. Therefore w, + w in measure in any ball Bl(y), y E RN, and thus

and, since R is a trace set, w E H ~ ( T R + z), which proves the theorem.

Problem 4.2 Show that the following sets are (G, T)-flask sets.

(a) Any open bounded set R c RN (more generally, any Rellich set as defined in Section 4.3);

(b) Any open periodic set $2 c RN, i.e. R = UyEG(U + y), where U c EXN is an open set;

(c) If N = Nl + Nz and Ri c RNt, dRi E C1, i = 1,2, satisfy conditions of Theorem 4.3 with respective groups Gi,Ti then R1 x R2 is a (GI T)-flask set with G = GI x G2, TI x T2;

(d) A finite union of (G, T)-flask sets whose pairwise intersections are bounded;

(e) A set R = Ro U R1, where Ro is a (RN, O(N))-flask set, R1 c TOO with some T E O(N) and a bounded 00 n 01, is a (EN, O(N))-flask set.


The term flask set originates from the flask-shaped unbounded domains for which existence of Sobolev minimizers was verified first. Let w E RN-I be an open bounded set with dfl E C1 and assume that R 3 w x R and that for every E > 0 there is a R > 0 such that R \ BR(O) c (w x R) + B,(O). Then lim inf(R - yk) is, up to a set of measure zero and up to a translation, either 0 or R, or w x R, all of which are contained, up to a translation and up to a set of measure zero in R.

Proposition 4.1 The following sets are not (RN, O(N))-flask sets:

(a) An open set R c RN, # RN, which for every R > 0 contains a ball of radius R, in particular, an open cone, is not a (RN, O(N))-flask set;

(b) An open cylinder from which one has removed a closed bounded subset with non-empty interior;

(c) A product w x (0, m), where w c RNP1 is an open set.

Note that the condition in (a) cannot be weakened to R # RN, since it is easy to verify for N > 1 that C,"(RN \ (0)) is dense in H1(RN), and therefore H ~ ( I W ~ \ (0)) = H ' ( R ~ ) , which is a flask space.

Proof. (a). Let yk E RN be such that Bk(yk) E R. Let w E H1(RN), suppw = RN (e.g. w(z) = e-1x12), and let x k E C,"(Bk(O), [ O , l ] ) be equal to 1 on Bk-1 (0) and satisfy IVxkI < 2. It is easy to see that uk = xkw(.-yk) is a bounded sequence in Hi(R) (in particular, suppuk c suppx + yk c Bk(yk) c R), uniformly convergent on compact sets (and therefore, weakly in H1 (RN)) to w. Since suppw = RN, w $ H i (TO + Z) for any T E O(N) and z € R N . (b) Let R = w x R \ U , where w is an open set in RN-' and U is a proper bounded subset of w x R. Let w E Hi(w x R). Let M = sup{xN : x E U ) . uk = X(XN)W(. - k e ~ ) , where e~ = (0,. . . ,0 ,1) and x E Cm(R, [0, I]), ~ ( x ) = 0 for x 5 M, ~ ( x ) = 1 for x 2 M + 1, (x'( 5 2. Similarly to the argument above, uk(. + k e ~ ) --\ w. It is clear that

inf [ w x R \ ( ~ R + z ) l > O T E O ( N ) , ~ E W ~

which implies that there is a w E H i (w x R) that is not in H i (TR + z) for any T E O(N) ,z E RN. (c) The proof is analogous to the one in case (b).

Chapter 4 Euclidean shifts

4.2 Existence of Sobolev minimizers on flask domains

Consider the following constrained minimization problem

C(P, 0 ) = 2

inf ( ( ~ l l ~ ; ( ~ ) , p E (2,2*) u€H;(R),Il~llp=l

where R C !RN is an open, generally unbounded, set and p E (2,2*).

Theorem 4.4 Let R be a (G, T)-flask set and assume that T is a closed subgroup of O ( N ) and that G is a closed subgroup of !RN satisfying (3.34). Then every minimizing sequence uk for the problem (4.11) has a (renumbered) subsequence such that for some yk E D, uk(. - yk) converges in H1 (!RN) to a function w E HA (TO), where w o T is a minimizer of the problem and T E T.

Minimizers of (4.11) are positive (up to a scalar multiple) functions satisfying (3.50) with cp E H;(R), that is, -Au + u = Xu"-' in R with A understood in the sense of weak derivatives: the argument of Section 3.7 applies in the case R # RN with only trivial modifications.

Proof. By the Sobolev inequality, c(p, R) > 0. Let uk be a minimizing sequence, i.e. / ( u ~ ( / ~ = 1, ( ( u ~ ( ( $ ~ ( ~ ) + c(p,R), and apply Theorem 4.1.

Let uk, win), dn) and yp) be as provided by Theorem 4.1. By (4.3),

If we set Jn ~ w ( ~ ) o T ( ~ ) [ * = t,, then, comparing the Sobolev norm of w(") o 1

T(") with the infimum value c(p, R), we get ((t,%(n) (($A,,, > c(p, R), 2

which gives J/w(") o dn) //&A(n) 2 c(p, 0 ) t K Substituting this into (4.13) we

get

On the other hand, (4.12) can be written as En t, = 1 and since < 1, both relations can hold only if all tn but one, say trio, vanish and tno = 1.

89


This yields, due to (4.14), //w("o) 0 T("o) = c(p, a ) , since smaller

values contradict the definition of the infimum. Thus, ~ ( " 0 ) o ~ ( " 0 ) is a minimizer.

We conclude from (3.40) that uk - d n O ) ( . - ypO)) 3 0 in H1(RN),

or equivalently, uk(. + y p ) ) 3 w("~). In particular, uk(. + - w("o), which together with the convergence of norms (llukll$l + ~ ( p , 0 ) =

I ~ W ( " ~ ) ) $ ~ ) yields that uk(. + y p ) ) converges to ~ ( " 0 ) in H1(RN).

4.3 Rellich sets and compactness of Sobolev imbeddings

Definition 4.2 An open set R C RN will be called a Rellich set if HA (0) is compactly imbedded into LP(R), p E (2,2*).

It follows immediately from the imbedding of H;(R) into L ~ * (R) and the Holder inequality that compactness of imbedding of HA(S2) into LP(R), p E (2,2*), is independent of p.

Proposition 4.2 Let p E (2,2*) and let G be an additive subgroup of RN satisfying (3.34). An open set R C R N is a Rellich set if and only if for every bounded sequence uk E HA(R) and every sequence yk E G, lykl -+ a, there is a renamed subsequence such that uk(. + yk) -\ 0.

Proof. The assertion is immediate from Definition 4.2 due to Proposi- tion 3.6, Lemma 3.1 and Lemma 3.3.

Note that the condition in Proposition 4.2 is independent of G.

Proposition 4.3 Let G be an additive subgroup of R N satisfying (3.34). An open set R c RN is a Rellich set if for any sequence yk E G, lykl 4 GO,

the set liminf(R - yk) has measure zero.

Proof. Let uk be a bounded sequence and let uk(.+ yk) -\ w in H ' ( Iw~) . By Lemma 4.1, w = 0 a.e. in RN, i.e. w is the zero element of H ' ( R ~ ) .

In particular, any open bounded set is a Rellich set (which was already stated in Theorem 2.8).

Theorem 4.5 Let G be an additive subgroup of RN satisfying (3.34). Assume that R c RN is an open set satisfying (4.8) with RE defined by (4.7). The set R is a Rellich set if and only if for every sequence yk E G and every E > 0

Chapter 4 Euclidean shzfts 91

Proof. Let X, be as in the proof of Theorem 4.3. All the set inclusions below will be modulo sets of measure zero. Necessity. If R is a Rellich set, then x,(.+yk) - 0. Repeating the argument from proof of Theorem 4.3 with v, = 0, we arrive at

Sufcciency. Let uk E Hi(R), let w E H ~ ( I w ~ ) and let yk E G be such that, on a renumbered subsequence, uk(. + yk) - W. For every E > 0 we define (on a renumbered subsequence dependent on E),

By Lemma 4.1,

JV(w,) 1 5 I lim inf (R \ 0, - yk) I = 0.

From the proof of Theorem 4.3, that utilizes (4.8), we have that w, -+ w in measure within every ball Bl(y), y E JRN, and this yields w = 0.

An "infinitely narrow" flask-shaped set ((11 < *), where =

(xl, . . . , x ~ - ~ ) , is a Rellich set by Proposition 4.3.

4.4 Concentration compactness with symmetry

We prove the following generalization of Lemma 3.5.

Proposition 4.4 Let T be a closed infinite subgroup of O ( N ) and let R c lRN be an open set such that TR = R for every T E T. Assume that for every sequence yk E lRN such that I liminf(S2 - yk)l > 0,

Then the space HT : {u E HHR), u 0 T = u, T E T), is compactly imbedded into LP(R) whenever p E (2,2*).

Proof. Let uk be a bounded sequence in HT. Consider its renumbered subsequence given by Theorem 4.1. If for some n > 2, 1 lim inf(R - y$))l = 0, then w ( ~ ) = 0 by Lemma 4.1, so without loss of generality we may assume that y$) satisfies (4.16) for all n > 2. Fix n > 2 and assume that w ( ~ ) # 0.


Note that by (3.37) and since u E HT,

and that

I -l (n) - rF1 yp) 1 --t w whenever TI # 72 , 71 Yk

due to (4.16). Consider the following inequality with M E N and distinct T I , . . .,TM E T:

Expanding the expression, passing to the limit in k and taking into account (4.17) and (4.18) we obtain:

liminf lluk112 2 M I I W ( ~ ) ~ ~ & L ( , N ) ,

which can hold for every M E N only if w ( ~ ) = 0 for all n 2 2. We conclude

that uk D ~ N w(l), and thus, by Lemma 3.3, uk + w(l) in LP, which proves the proposition.

Example. Let T = S1, a group of plane rotations around the xs-axis in R~ and let 52 = {x E R3 : xz < x: + xi + c2), c E R. Then the subspace HT of H i (R) of axially symmetric functions is compactly imbedded into LP(R), 2 < p < 6 .

4.5 Concentration compactness and the Friedrichs inequality

According to Proposition 4.1, a (RN, O(N))-flask set R C RN, # RN, cannot contain a ball of an arbitrarily large radius. If for some p > 0, R + D,(O) does not contain a ball of arbitrarily large radius, by Theorem 2.2 the Fkiedrichs inequality (2.13) holds. These two geometric conditions are very similar, and, moreover, they are equivalent if dR possesses some additional regularity, in particular, if there exists E > 0 and S > 0, such that any ball B,(x), x E dR, contains a ball Ba(y) c IRN \ R with some y E IRN. In

Chapter 4 Euclidean shifls 93

other words, fiiedrichs inequality holds for typical flask sets and there are convergence results involving functionals of the form

Let

Then A1 > 0 if and only if (2.13) is true. Consider

' (0) = sup A1 (a + Bp(0)) 5 A1 ( R ) . (4.21) P>O

The equality A(R) = A1(R) is assured when R + Bp(0 ) -+ R in the sense of Mosco ([95]), which in turn is the case when a is stable (Hh (a) = HA (R) , [40], Proposition 7.4.), and in particular, whenever d R E C1. We extend the applications of Theorem 4.1 to problems where the quadratic form (4.19) with A < A(R) defines an equivalent norm. For the sake of simplicity we assume that T = {id).

Theorem 4.6 Let R c IRN be a G-flask set with respect to a group G c IRN satisfying (3.34). If R satisfies (2.15) and p E (2 ,2*) , then the problem

has a minimizer, and for every minimizing sequence uk there is a sequence yk E D such that u k ( . + yk) converges to a minimizer.

Proof. If the inequality (4.2) in Theorem 4.1 were established not for the standard H1-norm, but for the norm (4.19) on Ht(R), then the proof of this statement would be a literal repetition of the proof of Theorem 4.4. Thus, in order to prove the theorem, it suffices to show that the functions w(") given by Theorem 4.1 satisfy

Note that the expression (4.19) is not necessarily positive for every u E C? ( I R N ) , and therefore, while (4.2) is immediate from (3.39), this argument is not available for (4.23).

94 Concentrat ion Compactness

Let p > 0 be sufficiently small so that

X < X(R + B,(O)) = inf

Let x E C r ( R + B,(O)) be equal 1 on 0 . By Lemma 4.1, v ( w ( ~ ) ) C

liminf(R - yp) ) . Then for almost every x E v(w(")), one has x E 0- yp) for large k, which implies ~ ( x + y p ) ) -+ 1. Therefore, by the Lebesgue convergence theorem,

A similar calculation with the gradient norms yields

as k -+ 0. Let

(I, v)* = (VU . VV - XUV), (4.25)

and note that the bilinear form (4.25) is continuous (but not necessarily positive) on H ~ ( I w ~ ) and that it is positive in restriction to Hd(R+ B,(O)). In particular, for any M E N,

for all k sufficiently large. Expanding (4.26) by bilinearity, we have

Let us estimate terms in (4.27). By (4.24)

Chapter 4 Euclidecan shifts 95

Furthermore,

by (4.24) and (4.1). Finally, since ~ y r ) - yLrn)l -+ CQ, one has w(")(. -

yLm) + yf)) - 0 and thus

since the functionals involved in (4.30) are continuous in H ~ ( I w ~ ) . Using (4.24), we have

Collecting the evaluations (4.28, 4.29 and 4.31), we obtain from (4.27)

Since M is arbitrary, (4.23) follows.

4.6 Solvability in non-flask domains

The definition of the constrained minimum in (4.11) R H c(p, R) can be extended arbitrary sets A c IRN by

c(pl A) = sup c(pl a ) . (4.32) n > A , fl open

Proposition 4.5 Let R C RN be an open set and let p E (2,2*). If for every sequence yk E R, IykJ 4 oo,

C(P, lim inf ( a - yk)) > c(p, R), (4.33)

then the problem (4.11) has a minimizer and every minimizing sequence converges in HA (R) .


Proof. Consider a subsequence uk of the minimizing sequence provided by Corollary 3.3 with correspondent w(,) and y?), n E N. Let t, =

SwN I W ( " ) ~ P . Then from (3.4) we have C t , = 1. Let R, be an open trace

set such that liminf (R - y?)) c R, and c(p, R,) > c(p, R) (in particular,

such R, exist with dR, E C1). Since v(w(,)) c liminf(R - y?)) by Lemma 4.1, and v(w(")) c R,, w(") E Hof(R,), and by (3.39) we have

c(p, R) 2 c(p, ~ , ) t 6 . This implies with necessity, since c(p, 0,) > c(p, 0 ) and a < 1, that t l = 1 and t, = 0, n 2 2. Thus, uk + w(') in LP, and so,

weakly in Hi(R). Since lluk 1 1 2 + ~ ( p , 0 ) = l l ~ ( ~ ) 1 1 ~ , uk + w(').

Lemma 4.2 Let R1 c Rz, Rz \ # 0, be two open sets in RN. If c(p, R1) is attained, then c(p, 02) < ~ ( p , 01).

Proof. Since Hd(R1) c H;(R2), ~ ( p , R2) I ~ ( p , 01). Assume that c(p, Rz) = c(p, 01). Then the minimizer u E H; (a1) for c(p, 0 , ) will be a minimizer for c(p, Rz). In particular, -Au > 0 in R2, which by the maximum principle (Proposition C.2) implies that u > 0 in R2, which contradicts u E H;(R1). Thus, with necessity, c(p, R2) < c(p,R1).

Proposition 4.6 Let R1, R2 c RN be open sets and assume that the set RlnR2 is bounded. Let R = R1UR2 and let p E (2,2*). If c(p, Ri) > c(p, R), i = 1,2, then c(p, R) is attained.

Proof. Let uk E Hof(R) be a minimizing sequence for c(p,R), i.e. 11uk112 -t c(p,R) and llukJlp = 1. Let x E CF(RN, [O,l]) be equal to 1 on a closed set B containing ill n R2. Let, on a renumbered subsequence, uk w and uk + w a.e. Due to Lemma 1.4 and Lemma 2.2, xuk - xw - 0 and by compactness of Sobolev imbeddings on bounded domains, xuk -+ xw in Lp(suppx). Then

Let us show that

Chapter 4 Euclidean shzfts

Indeed

Note that the second integral in the last expression converges to zero by compactness of Sobolev imbeddings over bounded domains. By separation of non-compact and compact terms one arrives at

By (4.34) and (4.35), vk := uk - x uk - W) is a minimizing sequence. Note that one can write (1 - x)uk = 4) +up) where u r ) E H A ( R ~ \ B) and uf ) E G ( f 1 2 \ B ) are restrictions of (1 - x)uk to disjoint sets R1 \ B and R2 \ B. Let, on a renumbered subsequence, w(') = wlimur) and w(') = w limur). Then (1 - x)w = w(') + w ( ~ ) and we have

At the last step of the calculation we used the weak convergence of vk to w. Due to Theorem 3.2,

Therefore,

97


I f we set, o n a renumbered subsequence, ti = l im Jni luf) - w ( ~ ) ) ~ P , i = 1,2 , and to = Jn Iwlp, then,

and from (4.36) and definitions o f c(p, R i ) follows:

Since 2 / p < 1 and c(p, R i ) > c(p, R ) , i = 1,2, t he last inequality implies w i th necessity tl = t 2 = 0 and to = 1. B y weak semicontinuity o f t h e norm llw112 cannot be larger than l im 1 1 ~ ~ 1 1 ~ = c (p ,R) , and since llwllP = 1, w is necessarily a minimizer and lluk112 -+ l l ~ 1 1 ~ . T h e latter, together wi th u k W , implies uk -+ w in H i ( R ) .

Remark 4.1 The argument i n Proposition 4.6 indicates that one can weaken the condition (4.35') in Proposition 4.5 as follows: whenever l im in f (0 - y k ) i s contained i n a union of open sets Uj with disjoint closures, i t sufices to verify only

c(p, l iminf ( R - yk) n U j ) > c(p, 0).

The proof i s left to the reader.

4.7 Convergence by penalty at infinity

Proposition 4.7 Let V E L " ( W ~ ) , i n f w ~ V > 0 , and assume that

V ( x ) < V, := l im V ( y ) , x E IRN, IYI+"

with a strict inequality on a set of positive measure. Let

and let, with p E (2 ,2*) ,

Then cp is attained and any minimizing sequence for cp has a subsequence that converges to a minimizer i n H 1 ( R N ) .

Chapter 4 Euclidean shifls 99

Proof. Note that assumptions 0 < inf V and V E Lm imply that a(., a ) +

defines an equivalent norm on H1(RN). Let uk be a minimizing sequence and consider its renumbered subse-

quence given by Corollary 3.3 (with the standard Sobolev norm). From the identity

follows, with w = w(') = w limuk, vk = uk - w ,

c, = lima(uk,uk) = a(w,w) +lima(vk,vk) 1 cpIIwII~+lima(vk1vk). (4.39)

Once we show that

lima(vk, vk) > c p t 2 / ~ , where t = lim inf IlvkIIF, (4.40)

the proof of the proposition can be concluded as follows: observe from (3.45) that t = 1 - IIwII;, so that from (4.39) follows that, unless t = 0, c, > c, (t2/p + (1 - t)'/p), which is false for p > 2. Consequently, t = 0, S lwlP = 1 and since (4.39) implies that a(w, w) < c, and c, is the infimum value, w is a minimizer. Moreover, t = 0 together with a(wk, wk) 4 C, =

a(w, w) implies that uk + w in H1. Let us prove (4.40). Let E > 0 and let U, be an open ball such that

whenever x $ U,, V(x) > V, - E. Then by compactness of Sobolev imbeddings over bounded domains SUE V ( 2 ) 1 ~ ~ 1 ~ -+ 0 and

Since E is arbitrary, we have

lima(vk,vk) > C: limsup 11v,+IIi, (4.41)

where c r = infll,llp,l S IVuI2 + Vmu2. Let w E H'(RN), w > 0, be a minimizer for c? (see Proposition 3.7 and the preceding discussion of


positivity). Then J IVwI2 + v,w2 > J l V ~ 1 ~ + V w 2 2 cp which yields (4.40).

Remark 4.2 Assume that, under the rest of assumptions of Proposi- t ion 4.7, V(x) > V,. Then the infimum (4.38) equals c r (the value of c, for V = V , ) and is not attained. To see this, consider the minimizer w for c p , given by Proposition 3.7. Let yk E RN, lykl -+ 00. Since c, 2 c p , i t immediately follows that w ( . - yk) is a minimizing sequence for cp and cp = c p . If (4.38) were attained on some wo E H1(IRN), IIwOIJp = 1, we would arrive at a contradictory inequality c r < cp = c r .

4.8 Minimizers with finite symmetry

Proposition 4.8 Let b E L ~ ( ! R ~ ) , liml,l,, b(x) = b , > 0 and assume that there is a subgroup T of O ( N ) with m elements and that for every T E T \ {id), T - id is non-singular and that for every T E T , b o T = b. Let HT = { u E H1(RN) : VT E T U O T = u) . If

then the problem

has a minimizer.

Proof. Let

and let us show that

Indeed, due to the rearrangement inequalities (B.l) and (B.3) at the first step, and then using (4.42), we have

Chapter 4 Euclidean sh2ft.s

CP,m = inf u E H o ( N ) : S ~ N brnlulPd~21 llul12

2 in f u € H T : S ~ N brn )u (x ) )Pdx~ l

1 1 ~ 1 1 2

Let uk be a bounded sequence in HT and consider renumbered subsequences provided by Corollary 3.3 with G = RN in application to uk = u k o ~ , T E T. We have by T-invariance, for n > 1,

Our assumption on T E T \ {I) implies that l ~ y p ) - T'~P)I 4 co whenever T # TI, SO we get in the expansion (3.40) m distinct terms of the form w ( ~ ) ( . - T ~ P ) ) , T E T, which allows us to write (3.40) in the form

From (3.39) we have

At the same time, due to Remark 3.4,

Let t l = J b(x) lw(l) lp and en = J b, lw(") lp, tn = me, n > 1. From (4.47) follows that

From (4.47) we deduce that

101


which, together with (4.45), yields tylp + m2/p En,, &IP 5 1, or in other

words, ti'" 5 1. This and (4.48), given p > 2, hold simultaneously only if all but one oft,, say n = no, equal zero. However, if no > 1, since (4.42) is strict, the inequality in (4.48) is strict, so w("o) cannot be a minimizer. Therefore (4.46) implies with necessity that u k -, w ( l ) in Lp. Since the norm of w ( l ) cannot be less than cp,b, we conclude that u k -+ w ( l ) in H 1 and that w ( l ) is a minimizer.

Remark 4.3

(a) Proposition 4.8 includes the case without a symmetry, that is T = {id), m = 1, with condition (4.42) reduced to b ( x ) > b,.

(b) It i s easy to show, repeating the argument of Section 3.7, that a minimizer for (4.43) satisfies the equation (3.50) with V = 1 and is positive (or negative).

(c) Condition (4.42) i n Proposition 4.8 can be relaxed to b ( x ) 2 ml-f b, with strict inequality on a set of positive measure. The proof requires to observe that the inequality i n (4.48) remains strict when w(") # 0 , which i n turn requires a proof that w ( ~ ) > 0 . This can be verified by the following steps. ( i) Replace the minimizing sequence u k with Iukl, which remains the minimizing sequence, so that w(") 2 0. (ii) Show that w(") is, up to a constant, a minimizer i n (4.44), for i f not, i t could be replaced i n the expansion (4.46) with another function that will decrease the value of llukll below infimum. (iii) Conclude from the equation (3.50) and strong maximum principle that w(") > 0 SO that J b(x)(w(")lp > ml-f J b, lw(")(p.

4.9 Positive non-extremal solutions

We consider first the problem, analogous to the one in Proposition 4.8 with T = {id), but replace the penalty condition b(x ) < b, with an "averaged penalty" as follows.

Theorem 4.7 Assume that b E LW(IRN), b, := limlZl,, b(x ) > 0, and that there exists a Bore1 measure p on R N , p ( ~ N ) = 1, such that

with a strict inequality on a set of positive measure. Then for every p E

Chapter 4 Euclidean shifts

(2,2*) there exists a positive solution u E H ' ( R ~ ) to the equation

Proof. 1 . L e t

and

c, := sup 1 b,lu(x)lpdx. ll1~11~=1 RN

A positive minimizer for (3.49) with V = 1 is, up to a multiple, a positive maximizer w(,) for c, . 2. Let uk E H 1 ( R N ) be amaximizing sequence, that is, Ilukll 5 1, S blulp -+

cb. Since lukl is also a maximizing sequence, we assume that uk L 0 and consider the renumbered sequence of uk, w(") 6 H 1 and yp) E RN given by Corollary 3.3 with G = R N . Let

where t 2 := IIw(n)112 and yk E RN is any sequence such that lykl -+ m. By (3.39) we have

((vk / I 2 I I w ( ~ ) ~ ~ ~ + t 2 = 1lw(")11~ j limsup l l u k 2 = 1,

n

while, due to Remark 3.4 and since p/2 > 1,

from which follows that vk is a maximizing sequence for cb and

3. Let us show that w ( l ) # 0. If t = 0, this follows from (4.51). Assume that t > 0. Considering u H S b(ulP on a translated sequence vk(. + yk - y) =

103


w(')(* + yk - y) + w(")(. - y), y E RN, we have from Remark 3.4

Comparing (4.51) and (4.52), we obtain

Integrating this inequality with respect to the measure p over y E RN and using the Fubini theorem, we have, due to (4.49),

which proves that w(l) # 0. 4. The theorem is proved once we show that w(l) is a (weak) solution of (4.50). Assume that w(l) is not a solution. Then there exists a v E H1(RN) such that ( ~ ( ' 1 , v) < 0 and 6 := (p - 1) J b ~ ( l ) ~ - l v > 0. By density, we may assume that v E C r ( R N ) . Let u i := vk + sv, s > 0. Then for all s sufficiently small, E, := -(2s(v, w(')) + ~ ~ 1 1 ~ 1 1 ~ ) > 0 and

Thus for every s sufficiently small there is a k, E N such that llui112 5 1 whenever k 2 k,. Similarly, with f (A) := (p - 1) I Alp-2A,

provided that s is sufficiently small and k 2 k, with some (renamed) k, E N. In this calculation we have used the facts that vk + tv A w(') + tv in H1 and thus vk +tv + w(l) +tv in LP(suppv), that the map u H JsuPp, b f (u)v is continuous in Lp, that the sequence t H JSUPPV bf (vk + tv)v is bounded on (0, s) and that the change of the order of integration, obvious for smooth functions, extends to H1 by density.

Chapter 4 Euclidean shzfts 105

We arrive at the conclusion that for all sufficiently small s the sequence ui has terms with 1Iu;II < 1 and J bluilP > cb, a contradiction. Therefore w ( l ) > 0 is a solution to (4.50).

The next statement is existence of positive solution with the penalty condition (4.42), case m = 1, reversed.

Theorem 4.8 Assume that b E C(RN) is such that

b, > 0 and p E (2 ,2*) . Then there exists a finite set Y c iZN and a solution to

where by is a convex combination of functions b(. - y ) , y E Y .

Note that (4.53) holds as well for the function by and that b: = b,.

Proof. 1 .Le t

Since p E (2 ,2*) , by Theorem 2.14 we have g, E C 1 ( H 1 ( R N ) ) . Let

cb := SUP inf g y ( u ) . II1~11~<1 Y E Z N

Note that for any u # 0 and any z E zN,

and therefore the infimum in the definition of cb is always attained on a finite set Y(u ) c Z N . For the same reason the minimum of g y ( u ) over y E zN \ Y is also attained, and

6 ( u ) := min gy (u) - min gy (u) > 0. y€ZN\Y Y € Z N

Let u k E H1(IRN) be a maximizing sequence for cb (that is, I(ukll 5 1 and maxy,~N gy (uk ) 4 cb) and let yk E Y ( u k ) . Then


In what follows we assume that uk 0, since whenever uk is a maximizing sequence, so is 1 uk 1 . Moreover, since uk (. - yk) is also a maximizing sequence corresponding to yk = 0, we rename it as uk corresponding to yk = 0, that is, we may assume that 0 E Y ( u k ) . Consider the renumbered subsequence uk, w ( ~ ) E H 1 and y p ) E lRN, given by Corollary 3.3 with D = Z N . Note that w(") 2 0 since uk 2 0. 2. Passing to the limit in (4.56) with y = yim) + z, z E 2ZN, m E N , we obtain from Remark 3.4

which yields

Note that w ( l ) # 0, for if it were zero, from (4.57) would follow that w ( ~ ) =

0 for every m, which yields cb = 0, a contradiction. Let Y := Y(w( ' ) ) . By (4.57) with m = 1, 0 E Y. 3. Assume that the vector w(') does not belong to the positive cone in H 1 ( R N ) generated by { v ~ , ( w ( ~ ) ) ) ~ ~ ~ . Then there exist a vector v E c r ( R N ) , llvll = 1, and a E > 0 such that ( w ( l ) , v ) < - 2 ~ and (gh(w( l ) ) , V ) > 2 ~ . Consider now a sequence uk + t v , t > 0. We can see immediately that lluk + tv1I2 < 1 - 4 ~ t + t2 5 1 if t 5 4 ~ . hrthermore, for all t sufficiently small and y E Y , using Remark 3.4, we have

Then for all t sufficiently small there is a k ( t ) such that for every Ic 2 k ( t )

1 gy(uk + t v ) L cb + - ~ t , y E Y. 2 (4.58)

Chapter 4 Euclidean shifts 107

Let 6 := b(w(')) > 0 be as in (4.55). Then, for y E zN \ Y, using Remark 3.4, we have

For all t sufficiently small, in particular, so that 6 - Ct > !j6, we have that for k > k(t) with some renamed k(t) E N, g,(uk +tv) L cb + i6 . Therefore, there is a t > 0 such that for all k sufficiently large, minYEzlv gy(uk + tv) > cb, a contradiction. We conclude that w(l) is in the convex hull of {gh)yEy, which yields (4.54).


Geometric conditions of the flask type, when the limit set of R + yk is a subset of R up to a translation, in the particular case of domain contained in a cylinder and approaching at infinity a smaller cylinder, originate in the work of M. del Pino and P. Felmer 1981. The sufficient geometric condition of Theorem 4.2 (with the trivial rotations group T = {id)) is due to [106]. A weaker form of Theorem 4.4, when the domain is periodic (invariant with respect to a lattice-containing group G c I K N ) is due to W.-C. Lien, Sh.- Y. Tseng and H.C. Wang, [83], Theorem 4.2. A simpler proof was given by J. Chabrowski [32], Theorem 2.

The purpose of Section 4.3 is to connect the compactness in Sobolev imbeddings on unbounded domains with geometry of more general flask domains. Characterizations of Rellich sets as "thin a t infinity" have been known for several decades (e.g. R.A. Adams [2], C. Clark [38]; see V. Maz'ya [93] for further references and a necessary and sufficient conditions in terms of capacity).

Compactness of Sobolev imbeddings for subspaces of symmetric functions is due to M. Esteban and P.-L. Lions ([48]). Proposition 4.4 is an elementary generalization of their results.

Proposition 4.7 is due to P.-L. Lions, [86], and Proposition 4.5 is its


immediate counterpart in view of Lemma 4.1. Proposition 4.6 is due to M. K. Kwong, [79]. A refined use of comparison of energies of the type of (4.33) with geometric conditions on R other then flask domains can be found in [78].

Proposition 4.8 is due to [126], whose proof is a "multi-bump" argument in H 1 ( R N ) , is similar to the one given here. Existence of positive solutions to semilinear problems is constrained both by geometry of the domain (Remark 3.5) and the behavior of coefficients at infinity (Remark 4.2). Existence when the penalty condition b(x) 2 b, of Proposition 4.8 in the no symmetry case rn = 1, is relaxed to b(z) >_ b, - c ~ - E I " I , has been shown by A. Bahri and P.-L. Lions ([lo]), but instead of surveying this long paper we present here two other related results: Theorem 4.7 proves existence of positive solutions to (4.50) under an averaged penalty condition, and Theorem 4.8, [116], does so under the reversed penalty condition b(x) < b,. Another notable existence result not presented here, for a cylindrical domain with a hole (which is not a flask domain and where minimizer does not exist) is due to H. C. Wang, [120]. Although most of applications of concentration compactness based on shifts deal with nonlinear problems, some recent work concerns eigenvalues of elliptic operators on unbounded domains ([log], [91], [97]).

Chapter 5

Concentrat ion compactness with dilations

5.1 Semilinear elliptic equations with the critical exponent

In this section we consider the Hilbert space z ) ~ ? ' ( I R ~ ) , N > 2 (cf. Defini- tion 2.5), equipped with the group of operators

where Sw is the group of dilations

Note that hs+t = h,ht for all s, t E IR. We will also consider a subgroup 6~ = { h j E bw, j E Z) and the correspondent product group D N , ~ = DWN x 6 ~ . Each element of D N , ~ (resp. D N , ~ ) can be written as u c-, (h,u)(. - y ) as well as u H h, (u(. - y) ) ) with y E R N , s E R (resp. s E Z) and z = Y y . The integrals SwN \VuI2 and SwN \uI2* are invariant with respect to the dilation operators h, and the shift operators u I--+ u( . - y), and therefore the elements of DN,W are unitary operators in I D ~ , ' ( I W ~ ) as well as isometries (under the Sobolev imbedding) on L'* ( I R N ) .

Lemma 5.1 Let u E ; D ~ - ' ( R ~ ) \ (0) . The sequence (hsk )u( . - y k ) ,

(yk,sk) E RN x IR , k E N , converges weakly to zero if and only if ( ~ k l + l ~ k l 00.

Proof. Since cr(IRN) is a dense subspace of D1~'(RN), it suffices to prove the lemma for u E cr(IRN) \ (0) . Necessity. Assume that the sequence ( s k , y k ) has a bounded subsequence. Then it has a renumbered subsequence such that sk -+ so E R and yk +

YO E R N . Let cp = (hS,u)( . -~o) . Then ( (h , ,u ) ( . -~k) , v) -+ ((h,,u)(-yo), v ) = llv1I2 = 1 1 ~ 1 1 ~ > 0.

109


Suficiency. Let y, E C r (RN). Consider first a renumbered subsequence with SI, + +m. Then, changing the variables under the integral and integrating by parts, we have

Consider now a renumbered subsequence where sk -+ -m. Since the operators in DN,W are unitary,

and the preceding argument applies with interchanged u and y,. Finally take a renumbered subsequence where sk + so E R and lykl -+ m. Then for Ic sufficiently large, the supports of u(2'O(. - yk)) and of y, become disjoint, thus turning the scalar product into zero. We conclude that every subsequence of (hs,u)( - yk) has a subsequence that weakly converges to zero, from which (h,,u)(. - yk) -\ 0 is immediate.

Lemma 5.2 Let N > 2. The pairs (V112(RN), DNPz) and ( D ~ I ~ ( I W ~ ) , D N , ~ ) are dislocation spaces.

Proof. Due to Proposition 3.1, since DN,W is a group of unitary operators, it suffices to prove that

gk E DN,w, gk f\ 0 * gk has a strongly convergent subsequence (5.3)

from which (3.4) is immediate. By Lemma 5.1, if gk f. 0, then the corresponding parameter sequence (sk, yk) is bounded and has convergent (renumbered) subsequences yk -+ yo E RN, sk -, so E R. Let gou = hs,u( - yo) Then gku ---\ gou for u E Cp. However, since the operators gk, go, are unitary, JJgkuJJ = JJuJJ = 11gouJJ and therefore gk -+ gou. By density this extends to all u E V ' ~ ~ ( I W ~ ) .

The analytic meaning of DN,z-weak convergence in V1,2(RN) is L ~ * - convergence:

Lemma 5.3 If uk is a bounded sequence in V112(RN), N > 2, then

Chapter 5 Concentration compactness with dilations 111

Proof. Since C r ( I R N ) is dense in V112(IRN) and the latter is continuously imbedded into L2(IRN), we may assume without loss of generality that u k E C r ( I R N ) .

D3, z 1. The implication u k DsR 0 + uk 0 is trivial. 2. Assume IIukllLz* + 0. Then for every gk E D N , ~ , llgkukllLz* = IIukllLz* + 0. Then for every cp E C?(IRN)

and since I l g k ~ k l l v l ~ = IIukllvl.~, by Lemma 1.2 gkuk - o in v1>2, i.e.

U k D3R 0. L2* 3. It remains to prove that u k Dfi.z 0 + u k + 0. Assume uk D3Z 0. Let

x E C r ( ( i , 4 ) , [ O , 3 ] ) , such that ~ ( t ) = t whenever t E [ I , 21 and Ix'I 5 2. By Sobolev inequality (Corollary 2.4), for every y E Z N ,

from which follows, if we take into account that X(t)2* 5 C t 2 ,

Adding the above inequalities over y E Z N and taking into account that ~ ( t ) ~ < cltI2*, SO that by (2.29)

we obtain


Let yk E ZN be such that

1-2/2* 1-2/2*

Since uk D3z 0, uk(. - yk) 0 in 0 ' t 2 ( l R N ) and due to Theorem 2.9,

Substituting this into (5.4), we obtain

Let

xj ( t ) = 2jX(2-jt)), j E Z.

Since for any sequence jk E Z, hjkuk D2z 0, we have also, with arbitrary

j k E Z,

Note now that, with j E Z,

which can be rewritten as

Adding the inequalities (5.6) over j E Z and taking into account that the sets 2j-I 5 lukl 5 2j+2 cover R with uniformly finite multiplicity, we obtain

Let jk be such that


and note that the right hand side converges to zero due to (5.5). Then from (5.7) follows that u k -, 0 in L2* , which proves the lemma.

Corollary 5.1 I f u k is a bounded sequence i n H1(EXN) and p E (2 ,2*] ,

then u k D3z 0 + ~ ~ u k ~ ~ L p -+ 0 .

Proof. If u k D3z 0 , then by Lemma 5.3, uk -+ 0 in L2*. Since the

sequence u k is bounded in L 2 , the convergence in Lp, p E (2 ,2*) follows immediately from the Holder inequality.

We can now state the concretization of Theorem 3.1 for the dislocation space (Dl>' ( E X N ) , D N , ~ ) .

Theorem 5.1 Let u k E v ~ ~ ~ ( E X ~ ) , N > 2, be a bounded sequence. There exist w(") E D112(RN), y p ) t E X N , j p f E iZ with k , n E N, and disjoint sets No U N+, U N-, = N, such that, for a renumbered subsequence of u k ,

N - 2 .(n) .(n) w(") = w l i m 2 - 7 j k uk(2-jk . f y p ) ) , 12 E N , (5.8)

l j p ) - jLm'l + 1 2 2 ) ( y p ) - yLm))l -+ m for n # m, (5-9) 2 1 1 1 ~ ( ~ ) 1 1 & 1 . 2 5 limsup 1 1 ~ k / 1 ~ 1 , 2 ~ (5.10)

n E N N - 2 . (n)

u k - 2 7 , k w(")(2jF' (. - y p ) ) ) -+ 0 D N , Z - ~ e a k l y , (5.11) n E N

(the latter is equivalent to L'* -convergence), and, moreover, the series above converges uniformly in k .

Moreover, 1 E No, y f ) = 0; j p ) = 0 whenever n E No; j p ) -+ -m

(resp. j p f + + m ) whenever n E N-, (resp. n E N+,); and y p ) = 0

whenever 12jr) y p ) 1 is bounded.

Proof. 1. Relation (3.8) written with g p ) specified as

yields (5.8). Relation (5.9) is an equivalent form of 3.9) due to Lemma 5.1. Relations (5.10) and (5.11) and the identification g:'J with y f ) = 0 , j?) = 0

are specific cases, due to (5.12), of (3.10), (3.11) and 9;) = id respectively.

2. Note that every unbounded sequence j p ) may be replaced by its renumbered subsequence convergent either to +m or to -m and every bounded sequence may be replaced by a constant subsequence (since infinitely many sequences are possibly involved, the usual diagonalization is


also required). Moreover, when jp) = j, E Z one may set j, = 0 and

rename w lim uk(. + y r ) ) = h-jn w(,) as dn). 3. Let 6 be the set of n E W such that 2jp' y?) has a bounded subsequence.

( For every n E 6 there is a renumbered subsequence (j;), y?)) and a y, E

RN such that 2j:^'yf) -+ y,. Due to (5.8),

Let us rename w(,)(. - y,) as w(,), which corresponds to setting y p ) = 0. Since the set may be infinite, the extraction of subsequences is successive and has to be concluded by the standard diagonalization. It is easy to see that (5.9) remains true.

Let

v ~ ~ ~ ( R ~ ) = {U E V ' ~ ~ ( I W ~ ) : 'v'q E O(N), ~(77.) = u). (5.13)

Proposition 5.1 Every bounded sequence uk E V:,2(RN) has a subsequence satisfying the assertions of Theorem 5.1 with y p ) = 0 and

w(,) E V:12(RN) for all n E N. In particular, ljp) - jLrn)1 -+ m whenever m # n and

Proof. Consider the renumbered subsequence provided by Theorem 5.1. Note that if H' is the set of all n E H for which 2jc) lyp)/ -+ m and fi is

the complement of N', then for every n t fi, y r ) = 0. Then, for n t fi and

77 E O(N),

and therefore w(,) E D ~ ~ ~ ( R ~ ) , n E fi. TO conclude the proof it suffices to show that w(") = 0 whenever n t N'. Indeed, from (5.8) follows


Since 2jP' 1 y p ) ] + oo, for every 7) E O(N) \ {id), one has 23pr lvyp) - yP)l --t oo, which, by Proposition 3.4 and (5.10) implies that for any collection of distinct qi E O(N), i = 1,. . . , M, M E W,

limsup 1Iukllbl.2 > \lw(") O l l l b l . 2

i

= C llw(")ll&,., = ~ l l ~ ( ~ ' l l $ l . z .

i

Since M is arbitrary and the left hand side is finite, we have with necessity w(") = 0, n E N'.

Note that a similar statement for bounded sequences in H1(RN) of radial functions, Lemma 3.5, is a statement of compactness, and that the terms in (5.14) corresponding to n > 1, when w ( ~ ) are bounded in H ' ( R ~ ) , become in (3.40) a part of the LP-remainder, p E (2,2*). This follows immediately from applying to the sequence from Proposition 5.1 the following lemma identifying the No-terms in (5.11) as the terms in (3.40).

Lemma 5.4 Let uk be the sequence given by Theorem 5.1. If, in addition, (Iuk 112 is bounded, then w(") = 0 for all n E N-, . Moreover, for p E (2,2*),

the series in (5.16) converges absolutely in lI1(lRN) uni;formly in k, and w(") are the weak limits of uk(. + Y p ) ) in H ' ( R ~ ) .

Proof. The last assertion is immediate from Lemma 1.2. If n E N-,, then jP) -, -00 and from the Fatou lemma we have

which implies that N-, = 0. Note now that if n E N+,, then a similar calculation gives


Let cp E c ? ( R N ) and let yk E R N , k E N. Then from (5.11) and (5.17), and since N-, = 0, follows

By Lemma 1.2 and since w ( ~ ) E H 1 ( R N ) , we have then that

Convergence in LP is due to Lemma 3.3.

Problem 5.1 State and prove the generalization of Lemma 5.4, comparing decompositions (3.11) for two dislocation spaces, ( H I D ) and ( H I , D l ) when HI is dense and continuously imbedded into HI D is a group of unitary operators in H I Dl is a group of unitary operators in H I and Dl is a subgroup of D.

Corollary 5.2 Let F : R --t IR be a continuous function satisfying conditions of Lemma 1.7 and let uk be as i n Lemma 5.4. Then

Proof. Lemma 5.4 and Lemma 3.4.

5.2 Oscillatory critical nonlinearity and the minimizer in the Sobolev inequality

Consider the class of functions F E Cl,,(R) satisfying

This class is characterized by continuous functions on the intervals [1,2?] and [-2?, -11, satisfying F ( 2 y ) = 2 N ~ ( 1 ) and F(-2?) =

2NF(-1) , and extended to (0, m) and to (-CQ, 0 ) by (5.19). It is immediate then that


and thus F extends by continuity as zero at zero. It also follows from (5.19) that

2' RN 1 ~ ( h ~ u ) = 1 F(U) for all j E Z, u E L ( ). (5.21)

The functional J F(u) is continuous in L2* (and, thus, in V112) due to Lemma 1.6, (5.20), and the Lebesgue convergence theorem.

Problem 5.2 Assume that F is a locally Lipschitz function that satisfies (5.19) and let M E W. Show that there exists a C(M) such that for every a l , . . . , aM > 0 ,

Hint: Prove the statement for M = 2 (for al/a2 very large or very small) and then use induction.

Lemma 5.5 Let uk be a bounded sequence in D112(RN) and let w ( ~ ) , y r ) , and jp) be as provided by Theorem 5.1. If F is a locally Lipschitt function on R satisfying (5.19), then, on a renumbered subsequence,

N-2 .(n) Proof. Since uk and En 2 TJ* w(") (2jP) (. - y r ) ) ) are bounded in V1y2 by Theorem 5.1, and u H J F(u) is continuous in D112, it follows that

N - a .(n) Moreover, since the series En 2 7 ' * w(")(2jP) ( a - y r ) ) ) is convergent in 'D'v2, uniformly in k, without loss of generality it suffices to prove that for any M E N,


Due to (5.21) and (5.22), it suffices to show that for all m # n,

This relation can be rewritten, by an elementary change of variable x' = .(n)

2jk ( X - Y p ) ) as

It is easy to see that the expression in (5.24) involves a continuous functional v H JRN ( w ( " ) ~ ~ * - ~ v . By Theorem 3.1, gz*gr --\ 0 in V112 and so in L ~ * which verifies (5.24) and therefore the lemma.

Corollary 5.3 Let uk + w be a bounded sequence in V1y2(IWN). If F is as in Lemma 5.5, then, on a renumbered subsequence,

Proof. Apply Lemma 5.5 to uk and to uk - w , noting that respective limits w(") coincide for n > 1 and w(') takes respective values w and 0. Subtraction of respective series (5.23) for uk and for uk - w yields (5.25).

Lemma 5.6 Let F E Cl,,(IW), let N > 2, and assume that the following limits exist:

b+, = s++, lim ~ ( s ) l s l - ~ * ;

b-, = lim ~ ( s ) ) s ) - ~ ' ; 3+-00

lim ~ ( s ) l s l - ~ * ; b+o = s++O

b-0 = lirn ~ ( s ) l s l - ~ ' . s+-0

Let F,(s) = b+,lsI2* for s 2 0, F,(s) = b-, l~)~* for s < 0. and let Fo(s) = b+o)sI2* for s 2 0 , Fo(s) = b-ols)2* for s < 0. Let uk E V1s2(IWN), w("), y p ) E IWN and let jp) E Z, No, N+,, N-, c N be as provided by Theorem 5.1. Then


Problem 5.3 This lemma is a straightforward generalization of Lemma 5.5, and the proof is left to the reader.

Theorem 5.2 Let N > 2. Assume that F : W -t R is a locally Lips- chitz function satisfying (5.19). If there exists a uo € D ~ ~ ~ ( R ~ ) such that JRN F (uO) = 1, then the problem

has a point of minimum w E D112(WN). Moreover, there exists a sequence jk E Z and yk E WN such that, on a renumbered subsequence,

Note that existence of minimizer implies existence of solution to -Au =

F1(u): a Lagrange multiplier in the equation satisfied by a minimizer w can be removed by setting u(x) = w(tx) with suitable t > 0.

P~oof . Let uk be a minimizing sequence, that is, lRN ( v u k l 2 -$ S N , ~ and JRN F(uk) = 1. Due to Lemma 5.3 and the continuity of J F ( . ) , the

latter condition cannot hold when uk D*z 0. Therefore there exists a w E D ' ~ ~ ( W ~ ) and a sequence gk E D N , ~ such that, on a renumbered subsequence, g;uk - w # 0. Let vk := giuk - w. It follows from Corollary 5.3 that, on a renumbered subsequence,

1 Let t = J F ( w ) and set 6 = w(t N .). Then from the change of variable in the integrals follows that J F ( 6 ) = t-' J F ( w ) = 1 and

Similarly, if Sk = v k ( ( l - t ) h . ) , then F(Sk) 1 and

On the other hand,


Comparing this with (5.30) and (5.31), we have

which is false unless t = 0 , l . Since w # 0 , with necessity t = 1 , which corresponds to J F ( w ) = 1. Since g;u - w and I I w ~ ~ ; ~ , ~ = S N , ~ = lim 11g;~k11$~,~ imply g;uk --t w in D l r 2 , w is a minimizer and the theorem is proved.

Corollary 5.4 Let N > 2. The problem

has a point of minimum w E D112(RN). Moreover, there exists a sequence jk E 2% and yk E R N such that, on a renumbered subsequence,

From the rearrangement relations (B.l), (B.3) follows that the minimal value S N in (5.33) is attained even if minimization is restricted to radially symmetric functions, so that the correspondent Euler-Lagrange equation is an ordinary differential equation. Explicit calculations yield the radial solution

unique up to dilations h,, s E R, which corresponds to the value S N from (2.31).

Theorem 5.3 Let R c R N , N > 2, be an open set such that RN \1# 0. Then

S N ( R ) := inf ] I v u ( ~ = S ~ , U E V ' . ~ ( R ) : J ~ 1 ~ 1 2 ' =I 0

and the infimum is not attained.

Proof. Without loss of generality assume that 0 E 0. Let x E C r ( R : [O, 11) and assume that ~ ( x ) = 1 for 1x1 < 6, for some 6 < d(O,RN \;a). Let w be as in (5.35) and set

Chapter 5 Concentration compactness with dilations

From the Lebesgue convergence theorem follows that, as t -, m,

Similarly,

Sn IVwt12 = Sn x21vwt12 + S ~ W : I V X I ~ + 2 V w t . V X (5.38)

= Jn X ( ~ - ' . ) ~ ~ V W / ~ + 0(1) 4

Comparing (5.37) and (5.38) with definition of S N , one finds that S N ( R ) < S N , and since the reverse inequality is immediate, S N ( R ) = S N .

Assume now that the minimum in (5.36) is attained on some wn. With- out loss of generality we may assume that wn > 0. Then wn is also a minimizer for (5.33), and therefore satisfies the Lagrange multiplier equation

2'-1 in RN -Awn = Awn in the week sense; substitution of p = wn into the weak form of the equation,

yields X = S N . Since w = 0 on an open set R N \a, by the strong maximum principle, Proposition C.2, w n = 0, a contradiction. Therefore, the minimum is not attained.

5.3 The BrBzis-Nirenberg problem

Let R c R N be an open bounded set and let Xl(R) be given by (4.20). We recall that Xl(R) > 0 if R is bounded.

Theorem 5.4 Let R C R N , N > 3, be an open bounded set. If X E (0 , X1(Q)), then the minimum in problem

is attained and every minimizing sequence has a subsequence convergent in

H,1 ( a ) .

121


Lemma 5.7 Let cp E C r ( R N : [O, I]), cp(x) = 1 in a neighborhood of 0 and let

Then, for N > 4, there exist C1, C2 > 0 dependent only on N, such that, as t -+ 0,

and

+ 0(1) , for N > 4; C2I logtl + 0(1) , for N = 4.

Proof. Let b > 0 besuch that cp(x) = 1 whenever 1x1 5 6 . 1. Proof of (5.40). Expanding the gradient of wt we have

Then

since

the estimate for the mixed term is

and

Therefore,

Chapter 5 Concentmtion compactness with dilations 123

Thus we have (5.40) with

2. In order to verify (5.41) consider

The second integral can be estimated as follows, if we recall that cp - 1 vanishes for 1x1 5 S with some S > 0:

I (max cp2* + 1)6-2N lsuppcpl

It remains to evaluate

where w := ,-, is, up to a real multiple, a minimizer for (5.33). By (1+1xI2)-

(5.43) we have

J,. (0wI2 = SN(IW(( ;* = SNCI, (5.45)

and (5.41) follows. 3. Proof of (5.42). Consider first the case N > 4:


Using estimates analogous to the previous step of the proof, one can easily see that the second term above is bounded. Thus,

1 Thus we have (5.42) for N > 4 with C2 = LN (1+1z12)N-a.

Let now N = 4 and let R > 0 be such that suppcp c BR(O). Then

It remains to take into account that

where wg is the area of S3.

Lemma 5.8 Under assumptions of Theorem 5.4, < SN.

Proof. Without loss of generality assume that 0 E R and let wt be as in Lemma 5.7 with suppcp C R. From Lemma 5.7 we have

and

In both cases, the right hand side is less than SN when t is sufficiently small.

Proof of Theorem 5.4. Let

Let uk E C r ( R ) be a minimizing sequence for (5.39), that is, 1 ~ ~ 1 ~ ' = 1 and Q x ( u k ) -t K X By the assumption on A, the expression Q x defines an equivalent Sobolev norm on H: (0). Since uk is bounded in H i ( R ) , it has a renumbered subsequence u k - u E HA(R), which, by compactness of the imbedding into L2(R) , uk -t


u in L 2 ( R ) . Without loss of generality we assume that u k + u almost everywhere. Then by the Brbzis-Lieb lemma (Theorem 3.2),

2' I I U I I $ + lim I I U - ukll;: = lim I I ~ k l l g * = 1. (5.46)

Applying Proposition 1.6 to the sequence uk E V ' ~ ~ ( I R ~ ) ) and taking into account that So U: 4 So u2, we obtain

fix = l i m Q x ( u k ) = l imQx(uk - U ) + A IVu12.

Then from definitions of fix and SN we have

K A L fix lim lluk - ullg. + SN IIull;. . (5.47)

Let t := Ilull$. Then, substituting (5.46) into (5.47), we have

Since K A < S N by Lemma 5.8 and & < 1, we have with necessity t = 1, that is, IIu112* = 1. This and the weak lower semicontinuity of Qx ( f i x < Q ( u ) < l i m Q ( u k ) = f i x ) implies that u is a minimizer. Since u k --\ u and Q(uk) + Q ( u ) , u k converges to u E HA(R).

The following statement is a counterpart of Theorem 5.39 in I R N .

Theorem 5.5 Let a E L ~ I ~ ( I W ~ ) , N > 2, and let

)ro := inf u e ~ ~ ~ ~ . ~ au2=l /IN IVul2

(this is a positive number due to Lemma 2.12). If A E ( 0 , X o ) and a > 0 on a set of positive measure, then the minimum i n the problem

is attained and any minimizing sequence has a subsequence convergent i n

Proof. By Lemma 2.12, the functional S a ( x ) u 2 is continuous in L ~ ' and continuous with respect to the weak convergence in v1v2(IRN). Since X E ( 0 , X o ) , the integral JRN (IVuI2 - Xa(x)u2) defines an equivalent norm in V112(IRN). Let u k be a minimizing sequence for (5.48) and consider a renumbered subsequence of u k given by Theorem 5.1. By Lemma 5.5,


From Proposition 1.6 and Lemma 2.12 follows

Let w be as in (5.35) and assume, without loss of generality that 0 is a (weighted) Lebesgue point in the set (x E RN : a(x) > 0) in the sense that

Then there exists a s > 0 such that Sa(x)w2(sc1x)dx > 0. Let w, := N-2

h-,w = s - z ~ ~ ( s - ~ . ) . Then Saw2 > 0 and

ca 5 LN (jvwSl2 - ~ ( x ) w ? )

= SN - h lN a(x)w? < SN.

Therefore, from (5.50), (5.49) follows with necessity that IIw(")112* = 0, Ilw(l)l12* = 1 and I I W ( ' ) ~ ~ & ~ , ~ 5 ca. Consequently, uk -+ w(l) in D112 and w(') is a minimizer.

5.4 Minimizer for the critical trace inequality

In this section we consider the Hilbert space H ( R ~ ) , the space of the restrictions of functions from D1'2(RN) to IW: = R"-' x (0, oo), equipped with the norm

and with the group D N - ~ , ~ (or, in some instances, D N - ~ , ~ ) , generated by dilations (5.1) and the group of N - 1-dimensional shifts gy : u H u(. - y), y E ElN-'. We keep the notation RN-I for IRN-l x (0).

The space H ( R ~ ) is continuously imbedded into L2*(IWy) by Lemma 2.10 and has a continuous trace on L2(RN-l) by Theorem 2.11. The elements of the group D N - l , ~ are unitary operators in H ( R ~ ) and they are also isometries in L ~ * (EX?) and in LZ(RN-').


Lemma 5.9 The groups Djv-1,~ and DN-I,W on H(WT) are groups of dislocations.

Proof. It suffices to prove the lemma for the larger group D N - 1 , ~ . Since the group consists of unitary operators, by Proposition 3.1 it suffices to verify (3.7) for which it suffices to show that if gk E D ~ - ~ , ~ , g k f\ 0, then gk has a strongly convergent subsequence. Note that Lemma 5.1 applies also in the present case with only trivial modifications. Therefore, if gk f. 0, then the corresponding parameter sequences yk E KtNp1, s k E W, have convergent (renumbered) subsequences yk -+ yo E WN-I , s k -+ SO E W. Let gou = htou( - yo). Then gku - gou for u E C c However, since the operators gk, go, are unitary, ((gku(( = ( ( ~ ( 1 = ((gou(( and therefore gk 4 gou. By density this extends to all u E H(R:).

Theorem 3.1 applied in this case (with the group D N , ~ replaced by the subgroup D N - i , ~ ) is repetitive of Theorem 5.1, and in particular em- ploys the same index sets No, N+, and N+,. It should be noted only that in the present case the analogue of (5.11) should not claim L2* (a:)- convergence: the group D N - l , ~ does not remedy the lack of compactness resulting from shifts in the N-th variable. Instead, the DN-1,~-weak convergence in H(W:) implies ~'-conver~ence on IRN-'.

D N ~ I , Z Lemma 5.10 If uk is a bounded sequence in H(JRy), then uk 0 + IIuk(., O)IILZ(~N-~) 4 0.

Proof. 1. Assume without loss of generality that uk E C p (IRN). Let x E C p ( ( $ , 4), [0,3]), such that ~ ( t ) = t whenever t E [I, 21 and Ix'I 5 2 and define

Observe that if vk - 0 in H(R:), then

Indeed, if .J, E Cp((-2,2)N), . J , I ~ O , l ) ~ - ~ = 1, then, by Lemma 2.2, $vk is bounded in Hd((-2, 2)N). By Lemma 2.13, $vk -, 0 in L2((0, I ) ~ - ' ) and thus vk(.,O) 4 0 in L2((0, I ) ~ - ' ) .

D N ~ I , Z 2. Assume now that uk 0. From (2.51) applied to .J,x(uk)) follows,


for every y E ZN-l,

Taking into account that ~ ( s ) 5 2s2, we then have

Adding the above inequalities over y E ZN-l, we obtain

Note that, due to the definition of x and (2.37),

which implies

1-2/2

J x(uk(', o ) ) ~ 5 c SUP ( O , l ) N - l + ~ y ~ z N - 1

(5.52) Let yk E ZN-' be such that


Since uk D%l'Z 0, uk(. - (yk, 0)) -\ 0 in H ( W y ) and, due to (5.51),

uk(' - Y k , ~ ) 2 4 0. io71)N-1+yk


Moreover, since for any sequence jk E Z,

we also have, with arbitrary jk E Z, k E N,



. (5.55)

Adding the inequalities (5.55) over j E Z and taking into account that the

sets 2j-I 5 Iukl 5 2i+2 cover R with a uniformly finite multiplicity, we obtain

Let jk be such that

and note that the right hand side converges to zero due to (5.54). Then from (5.56) follows that uk(., 0) 4 0 in L'(Iw~-').

129


Theorem 5.6 Let N > 2. The problem

has a point of minimum w E H(Ry) . Moreover, there exists a sequence jk E Z, yk E RNP1, such that, on a renumbered subsequence,

Proof. Let uk be a minimizing sequence, i.e. JRN-1 Iu(., O)lZ = 1, D ~ ~ 1 . z

JRN J V U ~ ~ -+ n. Note that the relation uk 0 is false, since otherwise, by Lemma 5.10, Iu(., O)lZ 3 0, a contradiction. Thus, for certainjk E

N-2 - Z, yk E RN-I, on a renumbered subsequence, 2-jkuk(2jk .+(yk, 0)) --\ w, w(., 01 # 0, and, since 2 y ' h U k ( 2 ' h ' + ( ~ k , 01) is also a minimizing sequence, we assume without loss of generality that

Note also that uk(., 0) converges in measure on RN-l. Indeed, for every x E C,00((-2, 2)N), x = 1 on (-1, I ) ~ , xuk is bounded in H1((-2, 2)N-1 x (0,2)) and converges weakly to xw in H1((-2, 2)N-1 x (0,2)). By Lemma 2.13 this implies that, on a convergent subsequence, uk(.,O) -+

w(.,O) almost everywhere in (-1, I ) ~ - ' , and thus, due to translation invariance, uk --+ W, on a renamed subsequence, almost everywhere in EXN-'.

By Proposition 1.6 and the BrBzis-Lieb lemma (Theorem 3.2) we have, respectively,

n = lim lluk112 = llw1I2 + lim lluk - w I I 2 (5.59)

and

Let t = I I W I I ~ . F'rorn (5.59) and the definition of K, follows that

which implies with necessity t = 0 or t = 1. If t = 0 then w(., 0) = 0, which contradicts (5.58). Furthermore, if lim JJuk - w)J2 > 0, then JJwJJ2 < n, a contradiction. Consequently, uk -+ w in H(R$!) and w is a minimizer.


5.5 A singular subcritical problem

Let N E W, N > 2, n E {O,l,. . . , N - 11, and m = N - n. In this section we will denote the variables z E RN as pairs (x, y) with x E Rn, y E Rm. The Hilbert space considered in this section is V112(RN \ Rn), that is, the closure of CF(RN \ Rn) with respect to the norm

In order to keep the notations uniform for all n, we ignore the fact that for n = N - 1 the set RN \ Rn is a disconnected union of two half-spaces and that for n < N - 1 the space V1v2(RN \ Rn) coincides with V1v2(RN) (the latter can be easily shown by inspecting approximating truncations). As a subspace of V1~2(RN), the space V112(RN \ Rn) is continuously imbedded into L " ( R ~ ) . If we interpolate, using the Hijlder inequality, between the Hardy inequality (2.34) and the Sobolev inequality (2.29) by taking a B E

(0,l) and p = 28 + 2*(1 - B), we obtain

Setting

we conclude that

C, := inf 1 1vul2 > 0. (5.61) U E Z ) ' ~ ~ ( R ~ \ R ~ ) : ~ ~ ~ lyl-aPIuIP=l RN

We equip the space \ Rn) with the group DnYz (resp. Dn,w) of linear operators:

N - 2 . u(x, y) H 2 ~ ~ u ( 2 j ( x - a, 2jy), a E Rn, j E Z (resp. j E R).

131


The argument repeatedly presented in previous sections, when subjected to trivial modifications, yields that these operators are unitary on v112(RN \ Rn) and preserve Swnr Iyl-aplulPdxdy for any p E [2,2*]. Furthermore, by Lemma 5.1, a sequence (ak, jk) E Rn x R has a bounded subsequence if and only if for the corresponding operator sequence,

A literal repetition of Lemma 5.2 yields that the pair ( D ' > ~ ( R ~ \Rn), Dntz) (resp. ( v ' > ~ ( R ~ \ Rn), D n , ~ ) ) is a dislocation space.

Lemma 5.11 Let p E (2,2*), and let uk E V1'2(RN \ Rn) be a bounded sequence. Then

Proof. 1. The implication uk D ~ R 0 j uk DAZ 0 is immediate. 2. If uk 4 0 in Lp(RN \ Rn, (y(-ap), then for every sequence gk E Dn,w, gkuk -\ 0 in Lp(RN \ Rn, I yl-"P). Since uk is bounded in the D1t2-norm,

we have gkuk - 0 in D112 and therefore, uk D ~ R 0.

3. It remains now to show that if uk DlfiR 0, then uk 4 0 in L P ( R ~ \ Rn, lyl-ap). Let Qj = {y E Rm : 2j < lyl < 2j+l), j E Z, B = (0, l)n x QO, and let

Note that U,EZ,,jEz Baa = WN up to a set of measure zero. From the standard Sobolev inequality on B, since the weights used in the integrals are bounded on B from above and from below, we have


By replacing the variable x with x + a , a E Zn, and rescaling both variables (x, y) by the factor 2j, j E Z, we have:

Adding the inequalities (5.63) over a E Zn and j E Z, we arrive at

Using the Hardy inequality (2.34) and choosing an appropriate "near- supremum" sequence ( a k , j k ) E (Zn x Z), we get from the last inequality the following estimate:

Since gkuk -\ 0 in D112(RN \ Rn) for every sequence of dislocations gk E Dn,z, from Theorem 2.9 follows that gkuk --+ 0 in LP(B) and thus in Lp(B, l ~ l - ~ p ) . Therefore

so that the assertion of the lemma follows from (5.64).

Theorem 5.7 Let N > 2, n E (0,. . . ,N - 1) and p E (2,2*). The infimum in (5.61) is attained.

Proof. The proof follows the second proof of Proposition 3.7 for the Euclidean case. Let uk be a minimizing sequence, that is, IVukI2 -+ c, and JwN I~~lPlyl-~pdxdy = 1. The latter, due to Lemma 5.11, implies that

uk D ~ z 0 false. Thus there is a w E D ' . ~ ( R ~ \ Rn) \ (0) and gk E D,,z, such that on a renumbered subsequence, gkuk W. Since gkuk is also a minimizing sequence, renaming it as uk, we have uk - w # 0. By


Proposition 1.6, IlVuk - Vwllg + llVwll; + cp, and by Brkzis-Lieb lemma (in this case, Remark 3.3),

From this and the definition of c, in (5.61) we have cp 2 % t t + c,(l - t): where t = JwN IwIPlyl-"pdxdy. Since p > 2, we have with necessity t = 1 or t = 0. Since w = 0, we have t = 1. By the weak lower semicontinuity of norms, I ~ V W 11; 5 cp, which implies that w is a minimizer.

Theorem 5.7 does not include the values p = 2,2*. When p = 2* and n < N - 1, cp = SN and the minimizer is (5.35). In the case p = 2*, n = N - 1 the problem (5.61) reduces to the problem (5.36) in the halfspace that does not have a minimum. When p = 2, the minimum is not attained: a minimizer has with necessity to satisfy -Au = cz = ( F ) ~ . This equation is known to have a unique (up to a constant) positive solution u = lyl9 , which is not in V1t2(RN \ Rn) \ (0).

We have the following analog of Lemma 3.4.

Lemma 5.12 Assume that uk E H, gc) E Dn,z and w(") E H are as provided by Theorem 3.1 for v ~ > ~ ( R ~ \ Rn), Dn,Z. For every p E (2,2*),

Proof. Due to Lemma 5.11 the proof can be reduced to sequences of the form uk = En gf)w(n). The subsequent argument is entirely analogous to that of Lemma 5.5, with the only difference being that translations in the y-variable are not involved.

We consider now the analog of (5.61) on Dn,z-flask sets. We say, similarly to Definition 4.1, that an open set R is a D,J-flask (or simply flask) set, if V112(R) is a flask subspace of D112(RN \ Rn) according to Defini- tion 3.3, or in other words, if for every sequence gk E Dn,z and uk E V ~ > ~such that gkuk converges weakly in D112(RN), there exists a g E Dn,z such that g w limgkuk E V112(R).

Theorem 5.8 Let R C RN \Rn, N > 2, n = 0, . . . , N - 1, be a Dn,z-flask set. Then the infimum


where p E (2 ,2*) and a, = N ( l - p/2*) E [O, 21, is attained.

Proof. The proof is repetitious of the proof of Theorem 4.4 (with T =

{id)) and is left to the reader. Lemma 5.12 is to be quoted where the proof of Theorem 4.4 uses Lemma 3.4.

The geometric characterization of flask sets in this case can also draw on the Euclidean case.

Lemma 5.13 Let R be a V112-trace set in the sense that

If for every ( a k , j k ) E Rn X Z , there exist (a , j ) E Rn x Z such that, up to a set of measure zero,

lim inf 2jk (0 + ( a k , 0 ) ) C 2j (0 + ( a k ) , (5.67)

then the set R is a Dn,Z-fEask set. N - 2 .

Proof. Let ( a k , j k ) E W n x Z and assume that 2 - 2 ' k ~ k ( 2 - j k . +(ak, 0 ) ) --\ w . Without loss of generality we may assume that the convergence to w is almost everywhere. Then, repeating with only trivial modifications the argument of Lemma 4.1 we conclude that, up to a set of measure zero,

V ( w ) c lim inf 2jk (a + (ak , 0 ) ) . (5.68)

Then, by (5.67), there exist (a , j ) E Rn x Z such that V ( w ) C lim inf 2 j (R+ (a,O)). Since R is a trace set, w E V 1 > ' ( 2 j ( R + (a,O))) and thus 2 F w ( 2 j ( . + ( Q , 0 ) ) ) E v112(fl).

Examples of DnYz flask sets:

(a) R = { ( x , y ) : Iyl < $ J ( x ) ) where $J is a continuous function on [0, m],

$J 2 limlzl+m ; (b) R = ( ( 2 , ~ ) : 1x1 < ~ I Y I I , > 0 ; ( c ) R = { ( x , y ) : 0 < a < J y ) < b < m ) ; (d) any bounded open set whose closure is contained in RN \ Rn.

Proposition 5.2 Let R C RN \ W n be an open set such that for every sequence ( a k , j k ) E (Rn x Z ) , l a k l + ljkl -+ 00,


Then for every p E (2,2*), the space V1>2(R) is compactly imbedded into

LP(R, l ~ l - ~ p ) .

Proof. Let u k be a renamed subsequence with correspondent w ( ~ ) and gp) (defined by (a?), j p ) ) E Rn x Z) as in Theorem 3.1. For n > 1

the relation (3.9) yields )ar)l + I jp)l - 00 and from (5.69) and (5.68) follows that v(w(")) has measure zero. Thus w ( ~ ) = 0 as an element of

V ' ~ ~ ( I R ~ \ Rn). Consequently, on a renumbered subsequence, u k D ~ Z w('), which implies, by Lemma 5.11, u k + w(') in LP(R, lyl-ap), which proves the proposition.

The condition (5.69) is satisfied, for example, by a set R = {(x, y) : lyl < $(x)) where $ is a continuous function on [0,00) and $(x) -+ 0 when 1x1 - ia and by a set R = {(x, y) : 1x1 < $(y)) with - 0 when lyl - 0 or 1y1 --, 00.

5.6 Minimizer for the Hardy-Sobolev-Maz'ya inequality

We continue with the notations of the previous section, where z E RN, N > 2, was denoted as (x, y) E Rn x Rm. We consider the Hardy-Sobolev inequality of V.Maz'ya that refines the Hardy inequality (2.34) for n =

1, . . . N - 1 as follows:

with some Cm,, > 0. Obviously the case n = N-2, i.e. m = 2, is the usual Sobolev inequality. Due to the inequality (5.70), the Hilbert space Hm,,

defined as completion of C r ( R N \ Rn) with respect to the norm Q&,, is a space of measurable functions. There is no immediate reason why the integrals in the left hand side will remain defined on the elements of H,,,, but the question of finding a minimizer in (5.70) is still meaningful.

Theorem 5.9 Let N > 3 and let n = 1, . . .N - 3. For all u E C r ( R N \ Rn),


The Hilbert space H,,, defined by completion of C r ( R N \Rn) with respect 1

to the norm Q&,, consists of measurable functions with measurable weak derivatives such that the integral (5.71) is finite. The minimum in

is attained.

Proof. The identity (5.71) follows from the definition of Q,,, and integration by parts and is left to the reader. We also leave to the reader to verify the second assertion, assuming that (5.70 holds): similarly to the argument in Section 2.1, if uk is a Cauchy sequence in H,,,, ~ ( l ~ l q u k ( x , Y)) is a Cauchy sequence in Lfoc and thus has a Lfoc-limit, which allows to pass to the limit in (2.4) (with l y l v u k instead of uk).

By the rearrangement inequalities (B.l) and (B.3) applied in the x- variable only, the infimum in (5.72) does not change if we consider it over functions in HkC(RN \ Rn) with compact support, radially symmetric in y E RN, SO we may restate the problem, regarding lyl as a radial variable in R2, in the form

c,,, = inf - WN-l / IVU(X, y)~2dxdy u E r 2~ Rn+Z\Rn

where

2~ 2(m-2) r := {U E v 1 ' 2 ( ~ n + 2 \ ~ n ) : - Iu(x, Y)I= 191- N-2 dxdy = 1).

We note that, since n + 2 < N, the value of the critical Sobolev exponent in RN, is below the critical value 2* = &k4 for Rn+'. Therefore N-2 n

the assertion cm,, > 0, and thus (5.70), follow from c, > 0 in (5.61) for Rn+' \Rn, with p = E (2,2*). Existence of the minimizer follows then from Theorem 5.7.


The term concentration compactness owes its name to the dilations case studied in this chapter. Convergence reasoning using unbounded dilation sequences has been used by J. Sacks and K. Uhlenbeck [lo31 and by H. Brbzis and L. Nirenberg, whose result from [24] we consider in Section 5.3. A generalization of Theorem 5.4 in the case of DR~-flask asymptotically cylindric domains was proved by M. Ramos, Z.-Q. Wang


and M. Willem [102]). The method and the applications of concentration compactness involving dilations were extensively elaborated by P.-L. Lions ([88],[89]). J. Chabrowski in [32] provided a generalization of Lions' version of concentration compactness that combined both cases, of the unbounded domain and of the critical exponent. A multi-term expansion of sequences, similar to Theorem 5.1, but associated with particular equation, was proved by M. Struwe [ l l l ] . H. Br6zis and J.-M. Coron ([25]) have produced the first multi-bump expansion, also for a particular class of sequences, where the separation of the dislocation parameters correspondent to (3.9) in the abstract case and to (5.9) in this chapter. Multi-bump expansions for critical sequences accounting to both translations and dilations in RN can be found in many publications, to mention just two, in [14] and [26].

Section 5.1, following ([106]), identifies the functional-analytic grounds of concentration compactness based on dilations as a realization of Theo- rem 3.1 in the Hilbert space D ~ ~ ~ ( I W ~ ) equipped with the group of dislocations D N , ~ . It is transparent from the proof of Lemma 5.3 that the identification of DN,z-weak convergence as L~*-convergence originates in partition both of the domain and of the range of functions into compact cells. Such partition allows to benefit from compactness of correspondingly restricted and truncated functions. For comparison, in problems with a subcritical exponent p < 2* compactness of Sobolev imbeddings on bounded domains allows to partition only the domain of functions without any truncations in the range.

The existence result in Corollary 5.33 (which, due to radial symmetry, can be proved also without the concentration compactness argument) is due to G. Talenti([ll2]) with the explicit solution (not yet shown to be a minimizer) calculated earlier by G. Bliss ([21]). Uniqueness of the Bliss- Talenti solution is due to G. Gidas, W.-M. Ni and L. Nirenberg, [61] and references therein. Theorem 5.5 is a simple partial case of Theorem 1 from [32]. Existence of positive solutions in semilinear problems with critical nonlinearity is generally more volatile than in the subcritical case, as tes- tified, in particular by the elementary Theorem 5.3 or by the celebrated non-existence result of S. Pohozhaev [loo]. Most of the known positive solutions to semilinear elliptic problems involving the critical nonlinearity are not obtained by constraint minimization (see the further discussion in the next chapter). An instance of "well-balanced" nonlinearity where the constrained minimum is attained is Theorem 5.2.

Theorem 5.6 in Section 5.4 is due to J. Escobar [46], where it is also


proved that the minimizer in this problem is connected to the Talenti minimizer.

Section 5.5: Questions of existence of minimizers and of exact constant in the minimization problem (5.61), with focus on the case n = 0, were studied by E. Lieb [81], P.-L. Lions [89] (existence Theorem 2.4), Z.-Q. Wang and M. Willem [121], F. Catrina and Z.-Q. Wang [31], P. Caldiroli and A. Malchiodi [27], to mention just few. The case n = N - 1 was studied in [114] and n = 1, . . . , N - 2 was considered by M. Badiale and G. Tarantello, [12]. The proof in this section, adapted to the general case, follows [114].

The Hardy-Sobolev inequality (5.70), (first appeared in the book [93] of V. Maz'ya), which we consider in Section 5.6 for n 5 N - 2 holds true also for n = N - 1 and, on a punctured ball, for n = 0. The proof of the inequality and of the existence in Section 5.6 follows a remark in the paper [113]. A version of this problem for the case n = 0 and with the exact

constant ( Y ) ~ replaced by X E (0, ( Y ) ~ ) (which also can be handled

by the argument of Section 5.6) was considered by P.-L. Lions (Theorem 1.3 in [88]) and a sketch of a proof was given. A full follow-up of this sketch has been carried out in [113], dealing with the existence in the case n = N - 1, N 2 4, where reduction to locally subcritical problem cannot be applied. Existence of minimizers in the Hardy-Sobolev inequality when n = 2 and N = 3 is still unknown.

The constrained minimization argument can be used in D'j2 and in H1 also with nonhomogeneous nonlinearities, in which case the minimizers satisfy an Euler-Lagrange equation with a Lagrange multiplier whose value is not easy to determine. An analog of Theorem 5.5 with a nonhomogeneus nonlinearity is Theorem 1.5 of P.-L.Lions in [88]. Minimizers of the counterpart of (5.33) for N = 2 (Moser-Trudinger inequality) were studied in [30] by L. Carleson and A. Chang.

Chapter 6

Minimax problems

Solutions of semilinear elliptic equations that were obtained in the previous two chapters by constrained minimization are also critical points of functionals of the form a (u ) = SO IVuI2 - SO F(u). In this chapter we consider critical points in unconstrained variational problems without compactness. Minimization problems for the functional a when R is bounded and F is a continuous function with subcritical growth l ~ ( s ) l l s l - ~ * 4 0 as Is1 + co; or when -F is convex, can be solved by the standard weak lower semicontinuity argument (in the first case, invoking compactness of Sobolev imbedding).

If we consider typical cases when the functional is continuous in H ' ( R ~ ) but not weakly lower semicontinuous, we find that @ is unlikely to be bounded from below. In particular, if the function F has a critical growth, that is, lim suplsl,, ~ ( s ) l s l - ~ * > 0, then it is easy to see that the

is unbounded from above and from below. This is also the case when F is subcritical, R = RN and l i m s ~ p , , ~ F(s)s-~ > 0. If we reverse the latter condition, i.e. assume l i m s ~ p , , ~ F(s)s-2 5 0, the point u = 0 becomes, by the Friedrichs inequality, a local minimum of @ considered on every finite-dimensional subspace of C r (RN).

Consequently, the natural unconstrained problems without compactness for the functional @ concern the case of saddle points for a functional that is unbounded from above and from below. In this case, however, there matters are complicated by an intrinsic mechanism that assures existence of divergent bounded critical sequences. For example, in a subcritical problem

M on RN a sequence uk = w(,)(.- y r ) ) , where w(,) are (not necessarily distinct) nonzero critical points of cP ( v @ ( w ( ~ ) ) = 0 and @(w(")) = c,) and l y ~ ) - y ~ m ) l + co for m # n, is divergent whenever M 2 2, but V@(uk) -+ 0 and @(uk) + z @ ( w ( ~ ) ) = C c n . The role of concentration compactness

141


argument is therefore to identify critical sequences whose expansion (3.11) consists only of one term, or at least to assure that when w(l) = w limuk is a critical point (not necessarily at the level lim @(uk)), it is not zero.

6.1 The mountain pass theorem

Let H be a separable Hilbert space.

Lemma 6.1 Let @ E C1(H), A C H , and assume that for all u E A V@(u) # 0. Then there is a locally Lzpschitz vector field X : A + H satisfying for every u E A,

and

(X(.)l V@('ZL)) 2 l lV@(~> 1 1 2 . (6.2)

Proof. Consider the following covering of A, equipped with the norm of H, by the open sets 0, := {u E H : (V@(u), V@(V)) > ;IIV@(U)~)~). Since A is a metric space, it is paracompact, and thus there exists a subset V of H and an open covering {Nv)vE~ of A, such that Nu C O,, v E V. It is easy to see now that the required vector field

satisfies the required properties.

For a set S c H and a 6 > 0 we will use the following notation:

Lemma 6.2 Let H be a Hilbert space, @ E C1(H) , let S C X be a closed set, c E R, E , 6 > 0 and assume that

Then there exists a r] E C([O, 11 x H, H ) such that

(a) r]t(u) = u i f t = 1 or if u $ @-l[c - 2 ~ , c + 2 ~ ] n SZa, (b) r]l(@-l((-00, c + El) n S C @-'((-00, c - El), (c) r]t(.) i s a homeomorphism of H , t E [0, 11,

Chapter 6 Minimax problems

(dl Ilvt(u) - 41 I 6, '1L E HJ t E [O, 11, (e) t H @(vtu) is non-increasing, u E H , (f) @(vt(u)) < c, u E @ - l ( ( - ~ , c ] ) n ss, t E [(),I].

Proof. Let

Observe that the function x is locally Lipschitz on H , suppx c A and x = 1 on B. Let X : A -+ H be the locally Lipschitz vector field provided by Lemma 6.1. We define

for u E A and Z(u) = 0 otherwise. Note that by assumption (6.4), 11Z(u)11 I & for all u E H. As a bounded, locally Lipschitz, vector field, Z generates a continuous flow u H ut(u) E C(R x H, H) , defined as the unique solution of the evolution equation

Then, by (6.4) and Lemma 6.1,

and

Verification of (a) and (c-f) is elementary and is left to the reader. Let us show (b). Consider u E @-'((-w, c + E ) ) fl S. If there is a t E [O, 8.4 such that @(at(u)) < c - E, then @(asc(u)) < c - E and (b) is satisfied. If, for

143


every t E [O,~E], @(ut(u)) E [c - E , c + E], then from (6.5) and (6.6) follows

and (b) is also satisfied.

Definition 6.1 Let @ E C1(H). A sequence uk E H is called a critical sequence for @ at the level c E R if

One says that @ satisfies the Palais-Smale condition at the level c E IR (the (PS),-condition for short), if any critical sequence at the level c has a convergent subsequence.

It follows that if a functional @ satisfies the (PSc)-condition and has a critical sequence on the level c, then a subsequence of uk converges to a critical point u, satisfying V@(u) = 0 and @(u) = c.

Theorem 6.1 Let @ E C1(H) and let c := infUGH @(u) > -00. Then @ has a critical sequence at the level c.

Proof. If no sequence satisfies (6.7), then there exists an E > 0 and a 6 > 0, such that whenever @(u) < c + 2&, IIV@(u)II > F. Then by Lemma 6.2 with S = H, @(qlu) 5 c - E , which contradicts the definition of c.

Note that the critical sequence in Theorem 6.1 may be divergent: consider H = R and @(x) = ex.

Theorem 6.2 (Mountain Pass Theorem) Let @ E C1(H). Let eo, el E H , Q : {v E C([O, 11, H ) : v(0) = eo, v(1) = e l ) and let

c := inf max @(v(t)). v€Q t € [ O , l ]

If c > @(eo) and c > @(el), then @ possesses a critical sequence at the level C.

Proof. Let EO = c - max{@(eo), @(el)). If @ has no critical sequence at the level c, then there exist E E (0,;) and b > 0 such that whenever 1Q(u) - cl 5 2 ~ , IlVG(u) 1 1 > 9. Then, together with S = H, the conditions of Lemma 6.2 are satisfied. If q E C([O, 11 x H, H ) is as provided by Lemma 6.2,

Chapter 6 Minimax problems 145

then for every v E Q and t E [O, 11, qlv(t) E Q (in particular, q1ei = ei, since at the points eo, el is less than c - 2 ~ ) . Let us fix a v E Q such that @(v(t)) 5 C + E for all t E [0, 11. Then by Lemma 6.2, (b), @(r]lv(t)) < c- E

which contradicts the definition of c.

One of the points ei in the mountain pass theorem can be placed at infinity:

Theorem 6.3 Let @ E C1(H). Assume that the set

Qo := {V E Cloc([O,oo), H) : v(O) = 0, lim Ilv(s)II = oo, lim @(v(s)) = -00) s+m S-+,

(6.9) is nonempty, and let

c := inf sup @(v(t)). vEQo t>0

If c > @(O), then @ possesses a critical sequence at the level c.

Proof. The proof is repetitive of that of Theorem 6.2 with eo = 0 and with el in the neighborhood of infinity. The only modifications are: the choice of €0 = c - a(0) and the observation that qlv(s) E Qo for every v E Qo and s 2 0: in particular qv(s) = V(S) for all s large enough since lim,,, @(v(s)) = -00 implies @(v(s)) < c - 2~ for all sufficiently large s.

I3

6.2 Functionals for the semilinear elliptic problems

In this section we give general sufficient conditions for the functionals associated with the semilinear elliptic problems to be continuous, to have mountain pass geometry, to have bounded critical sequences and to have critical sequences that converge if they are bounded. Let R c RN be an open set. Let q > 2 if N = 1 ,2 and q = 2* for N > 2. Let f E ClOc(R x R) and let F (x , s) = J: f (x, t)dt. If for every E > 0 there exist C, > 0 and p, E (2, q) such that

I f (x, s)I L &(Is1 + IsIq-') + c ~ I s I P ~ - ' , s E R, x E 0 , (6.11)

then by Theorem 2.14, @ E C1(~l ( IRN)) , where

Let (Pa, for an open set R, denote the restriction of the functional (6.12) to HA(R). From the definition of directional derivative it follows that every


critical point u of an satisfies the relation

which corresponds (in the sense of weak derivatives) to the equation

-nu(%) + ~ ( x ) = f (x, ~ ( x ) ) , x E fl (6.14)

with the boundary condition ulan = 0 in the sense of the trace on the boundary. We give a general sufficient condition to the functional (6.12) to have bounded critical sequences.

Lemma 6.3 Assume that there is a p > 2 such that

f (x, s)s 2 pF(x, s), x E R, s E R. (6.15)

Then for every c E R there exists a Mc > 0 such that for every critical sequence uk of an at the level c, limsup I ) U ~ I ) ~ ; ( ~ ) 5 Mc.

Proof. Let c E R. Since uk is a critical sequence, then on a renumbered tail of the sequence

and

from which we derive

2 ~ - ' \ l u k \ / ~ ; ( ~ ) f P-' f (xl uk)"k < P-I ll"kll~;(n)- (6.17)

Adding (6.16) with (6.17) and using (6.15) we have:

5 c + 1 +p-'II~kllH;(n),

from which follows the lemma.


Lemma 6.4 Let R C RN be an open set, and assume (6.1 I ) , (6.15) with some p > 2, for all x E 52, s E R. Then the class \ko given by (6.10) for an in H = Hh(R), is nonempty and the correspondent constant c is positive. (Consequently, by Lemma 6.3 and Theorem 6 .9 the functional an has a bounded critical sequence at the level c.)

Proof. By (6.11) and Theorem 2.14 an E C1 (HA (0)). From (6.11) follows

For all u such that 11u11& = t with a sufficiently small t E (0, l ) we have

Let t be now fixed. Since (6.15) for s > 0 is equivalent to $F(x, s)s-P > 0, taking v E C r ( R ) , v >, 0, s 2 1, we have immediately J F(x , sv) 2 J F(x, v)sP and thus an(sv) -+ -oo as s 4 +oo. Therefore the class Qo in (6.10) is nonempty and, since any curve v E \ko crosses the sphere I/ull& = t, the correspondent constant c satides c 2 > t > 0.

In what follows we will consider the following conditions for an open set R c RN, relative to an additive group G c RN:

(A) For every sequence yr, E G there is a z E G such that for every x E R there exist E(X) > 0 and N(x) E N satisfying

B,(,)(x) - yk C R + z whenever k 2 N ( z ) ;

(B) For every sequence yk E G, lykl -+ 00, there exist a z E G, an open set U c RN and a set Z C RN such that

Lemma 6.5 If R is a trace set satisfying condition (A), then it is a G- flask set (that is, if uk E Hi(S2) is a bounded sequence, yk E G and w E

H1(RN) a, such that uk(.+yk) - w in H1(llUN), then w(.+z) E H i (R) for some z E G). Moreover, if (6.11) holds true, V@n(uk) -+ 0 in H;(R) and uk(. + yk) - w in H1(RN), then w(- + z) E Hi(R) and VcPn(w( + z ) ) = 0.

Proof. 1. By Theorem 4.2 and the definition of the trace set it suffices to show, in order to prove the first assertion, that lim inf(R - yk) c R + z. It remains to observe that liminf(R - yk) c UxEn,n>N(r.(BE(x)(x) - - yk).


2. Let p(u) = JRN F ( x , u). By (6.11) and the compactness of local Sobolev imbedding V q is weakly continuous in H1(RN), namely

Let v E C r ( R ) . Consider a (finite, by compactness) covering of suppv by the balls BE(,,) (xi). Then it follows from (A) that suppv( - yk - z) E R for all k sufficiently large. Due to the invariance of V p with respect to G-shifts and (6.18) we have, using the scalar product of H ~ ( I w ~ ) ,

(V@(w(. + z), v) = (V@(w limuk (. + yk + z)), v)

= lim(V@(uk), v(. - yk - z)) = lim(V@n(uk), v(. - yk - z ) ) ~ ; ( ~ ) = 0.

In the last step both the arguments in the inner product of H1(RN), for k sufficiently large, are elements of Hd(R). Thus the ~ l ( R ~ ) - i n n e r product can be identified as a directional derivative of an in H,'(R), and so as the Hi (0)-inner product involving the gradient of an in that space, which converges to zero. Since v is arbitrary, we conclude that w(.+z) is a critical point of a n .

Remark 6.1 While conditions ( A ) and ( B ) are stronger than the flask property, it is easy to show that bounded trace domains satisfy (A) and (B) and that G-periodic sets (cf. Problem 4.2) satisfy (A). Another example of a trace domain satisfying ( A ) and ( B ) is R = {x E RN : xq + . . . + X K - ~ < $(xN)) where $ E C1(loc) satisfies $(s) > limsupltl,, $ ( t ) > 0.

Proposition 6.1 Let G C RN be an additive group satisfying (3.34), let f E Cloc(RN x R) satisfy (6.11) and assume that f (x + y, s) = f (x, s) for all x E RN, s E R and y E G. Let R c EXN, N > 2, be an open trace set, let be the functional (6.12) on H ' ( R ~ ) , and let an be its restriction to HA (R). Assume that uk E HA(R) is a bounded sequence satisfying

If R satisfies condition (A) , then uk converges to a non-zero critical point of an weakly i n HA(O). If, additionally, R satisfies condition (B) , and f (x, uk(x)) 2 0 i n R, then uk converges to a critical point of @n i n Hd(s2).

Proof. By the first assertion of Lemma 6.5, R is a G-flask set. Consider the renumbered subsequence of uk and yp) E G , w(") t H,'(R), given by Theorem 4.1. By the second assertion of Lemma 6.5, w ( ~ ) for a11 n are critical points of V a n (the shifts z = z ( n ) are absorbed into renamed w(") by the argument of Theorem 4.1). If w(") = 0 for all n, then by (3.40) and


Lemma 3.3, uk -+ 0 in LT(RN) for any r E (2,2*). Then, by Theorem 2.14 V9(uk) + 0 in H1(JRN), and consequently, since V@(uk) 4 0, we have uk -+ o in H ~ ( I W ~ ) and so in HA@). Since a n is continuous, @n(uk) +

0 # c, a contradiction. We conclude that a n has a non-zero critical point w ( ~ ) # 0 with some n E W.

Assume now that f (x,uk) > 0. Then f (x, w ( ~ ) ) > 0 and, since w ( ~ ) satisfies (6.13), from the weak maximum principle (Proposition C.l) follows w ( ~ ) > 0. If condition (B) is satisfied, then by Lemma 4.1, for each n > 1, w(") = 0 on some open set in 0. Then the strong maximum principle (Proposition C.2) yields w ( ~ ) = 0 for all n > 1 and we conclude that uk -$ w(l) in LT(R) for all r E (2,2*). Then by Theorem 2.14 V9(uk) 4

V~(W( ' ) ) in H1(RN), and consequently, uk converges to w(') in H ' ( R ~ ) and so in Hd (R). Since an E C1(HA (R)), we have a n ( ~ ( ' 1 ) = c. I3

6.3 Critical points of the mountain pass type

In this section we prove convergence of critical sequences for the functional by elimination of the dislocated weak limits for problems with periodic

nonlinearity on the strict flask sets and for problems where the function F imposes a variational penalty at infinity.

Theorem 6.4 Let G c RN, be an additive group satisfying (3.34), let f E Cloc(RN x R) satisfy (6.11) and (6.15) with some p > 2, and assume that f ( x + y , s ) = f (x , s ) f o r a l l x ~ ~ ~ , s E R a n d y ~ G. LetR c R N be a trace set, let be the functional (6.12) on H1 (RN), and let a n be its restriction to Hi (R) . Let

c = inf sup an ( ~ ( s ) ) , (6.20) v@o 8 2 0

and

v(0) = 0, lim IIv(s)IIH~ CX), lim @(v(s)) -) - 8-00 8-00

If R satisfies the condition (A) relative to G, then an has a nonzero critical point. If, furthermore R satisfies condition (B) and f (x, s) > 0 for all x E R and s E R, then an has a critical point w E Hi(R) satisfying an(w) = c.

Proof. The functional an has a bounded critical sequence uk (6.20) with an(uk) + c by Lemma 6.4, Lemma 6.3 and Theorem 6.3. Conclusions of


the theorem follow therefore from Proposition 6.1.

Theorem 6.5 Let f E C ~ , , ( R ~ x R ) satisfy (6.11), (6.15) and suppose that f,(s) := limlxl,, f ( x , s ) exists (which implies that limlxl+03 F ( x , s ) = F,(s) := J: f,(t)dt). Assume that

s --i - f 0 3 ( S ) , is an increasing function on R, (6.21) Is1

and that for every x E I R N , s # 0

If @ is the functional (6.12) on H ~ ( I w ~ ) , then @ E C1(H1 ( R N ) ) and has a critical point w E H 1 ( R N ) satisfying @(w) = c where

c = inf sup @ ( v ( s ) ) v a ' o 320

and

q o = { v E CI,,([O, m), :

v(0) = 0, lim 11~(s)11~1 --t 00, lim @ ( v ( s ) ) --, -00). 8-03 3-03

Conditions of the theorem are satisfied, for example, by a function f ( x , s ) =

b ( x ) l s l ~ - ~ s , b E Lm(IRN) with b(x) > 0, 2 < p < 2* and 0 5 b(x) < b, < 00.

Proof. Step 1. By Lemma 6.4, @ has a bounded critical sequence at the level c. In order to verify its convergence in H 1 , consider a subsequence of uk, w ( ~ ) E H and E Z N provided by Corollary 3.3 with the group G = Z N . Since @ ( u k ) -+ C, one can estimate c from below by means of (3.39) and Remark 3.4:

From V @ ( u k ) -+ 0 follows ( v @ ( u ~ ) , cp( . - yp ) ) ) -+ 0 for every cp E C r ( R N ) , n E N . Passing to the dislocated weak limits we obtain

and


In the calculation of (6.25) we use compactness in Sobolev imbeddings for bounded domains and the continuity of u w JB f (x, u)cp on balls B > suppcp, which follows from Lemma 1.9. A similar argument leads to (6.26), once one takes into account, with the help of Lemma 1.6, that j" I f (x, uk) -

fa (uk) 1 lp(. + y r ) ) 1 -t 0. Relations (6.25) and (6.26) extend by continuity to all cp E H1(IRN). In particular we have

llw(l) 1151 = f (x, w ( ~ ) ) w ( ~ ) , J

Substituting (6.27) into (6.24) we get

Note that

1 1 - f (x, S)S - F(x, S) > 0 and - f,(s)s - F,(s) > 0, s # 0. (6.29) 2 2

Indeed, if (x, s )s - F(x, s) 2 ( 5 - l )F(x , s) by (6.15) and the right hand side is positive by (6.22), whenever s # 0. The inequality at infinity is similar. Step 2. Assume that

w ( ~ ) # 0 for some n > 2. (6.30)

Let us estimate c from above by observing that for every w E H ~ ( R ~ ) \ {0), the path s H sw is of the class XUo, and thus

c < sup @(sw). sE[0,00)

Let now substitute w = w ( ~ ) ( . - yk) with yk E z N , l Y k l + M. Then


By taking k -+ co and using (6.11) and the Lebesgue convergence theorem in J F ( x + yk, w ( ~ ) (x)), we have

where

Let us now evaluate the supremum in (6.31). Since @,(sw(~)) is negative for s large, is zero for s = 0 and has a positive value, the function s H

~ , ( s w ( ~ ) ) , s E (0, co), attains a maximum at some so > 0, and

Due to (6.21), the function s H s-'$@,(sw(~)) is monotone decreasing on (0, oo), and thus the function s H @,(sw(~)) has a unique critical point so on (0, co), which is necessarily so = 1 by (6.27). Therefore

c 5 max @,(sw(~)) = @,(w(~)) s € ( o , ~ )

Comparing this with (6.28), we see, due to (6.29), that for m # n, w ( ~ ) = 0 with necessity, and therefore

On the other hand, by (6.22),

sup @ ( S W ( ~ ) ) < sup @,(sw(~)) = @,(w(~)) = C, s € [ O , ~ ) s€[O,W)

which contradicts the definition of c. Thus 6.30 is false. Step 3. We arrive therefore at w ( ~ ) = 0 for all n 2 2. Therefore uk D ~ N

w('), and therefore, by Lemma 3.3, uk -, w(') in Lr for any r E (2,2*). Then from Theorem 2.14 and V@(uk) -, 0 follows that uk is a convergent sequence in H ' ( I w ~ ) . Then limuk = w limuk = w(') and, since @ E C1, v@(w(')) = 0 and @(w(l)) = c.


Remark 6.2 One can reduce the number of assumptions in the theorem by replacing (6.15) and (6.21) by a single stronger condition: there exists a p > 2 such that for all x E R N

S H - (x' is an increasing function on R. (6.34) IsIP

Indeed, (6.21) follows from (6.34) immediately, (6.15) for s > 0 follows from the estimate p F ( x , s ) = S: y p t p d t 5 m s r - ' SP = f ( x , s ) s . The computation for s < 0 is similar.

Remark 6.3 Let R C RN satisfy condition (A) relative to some group G c R N satisfying (3.34). Then the assertion of Theorem 6.5 extends to the restriction @n of @ ( u ) to H i ( R ) . The same proof applies, with only trivial modifications borrowed from the proof of Proposition 6.1. A similar argument is used also in the proof of the next statement.

Theorem 6.6 Let R C IKN be an open set satisfying condition (A) with an additive group G C R N satisfying (3.34). Suppose that f ( x + y, s ) =

f ( x , s ) for all x E R N , y E G , s E R , and that F ( x , s ) > 0 for all s # 0. If (6.1 1) (6.15) hold and for all x E R N ,

S H - (x' , is an increasing function on 8, (6.35) Is1

then there is a u E H i ( R ) such that V @ ~ ( U ) = 0 and @ n ( u ) = c, where c

can be evaluated as infVEuo maxSEro,ll @ ( v ( s ) ) .

Proof. The proof of the theorem is repetitive of the proof of Theorem 6.5 combined with Proposition 6.1, and we give here only a sketch. Step 1. Consider the renumbered subsequence of uk and yp) E G , w ( ~ ) E

HA ( R ) , given by Theorem 4.1. Due to (A ) , by Lemma 6.5 (with the shifts z = z ( n ) set to zero by the argument of Theorem 4.1), v @ ~ ( w ( ~ ) ) = 0 for every n. Application of Lemma 3.4 together with (3.39) give the following counterpart of (6.24)

Instead of (6.25), (6.26), one has

J, V w c n ) - V p + w(.)p = f ( x , ~ ( ~ ) ) p , n E PI, p E H ~ ( R ) , (6.31)

154

and consequently,

Concentration Compactness

Substitution of (6.38) into (6.36) gives

f (x, w ( ~ ) ) w ( ~ ) - F ( ~ , w("))) . n E N

By the positivity assumption for F and (6.15), if (x, s )s - F(x, s) 2 ($ - l )F(x , s) > 0 whenever s # 0, which implies

@ ~ ( w ( ~ ) ( x ) ) > 0 whenever w(")(x) # 0. (6.40)

Step 2. If we assume that w(") = 0 for all n E N, then by Theorem 4.1, uk --+

0 in LP, p E (2 ,2*) , and consequently, by (6.11) and Theorem 2.14, and since V@n(uk) -t 0, uk 4 0 in HA(R). Then a n ( u k ) --+ 0, which implies c > 0, a contradiction. We may assume then without loss of generality that w(') # 0. Repeating the analogous argument of in Theorem 6.5 on the estimate of c from above gives

c 5 sup @(SW(')), (6.41) ~€(0,cQ)

and as before, (6.35), yields that s -+ @(sw(')) has one and only one critical point on (0, I ) , which, due to (6.38) is so = 1. Step 3. Comparing (6.39), (6.41) and (6.40), we conclude that w ( ~ ) = 0,

n # 1. Consequently, that uk 3 w('), and thus uk --+ w(') in LT(R) for every r E (2,2*). Then, as in the proof of Theorem 6.5, uk -+ w(') in H t (R), and thus Q ~ ( W ( ' ) ) = c.

Remark 6.4 Assume additionally in Theorem 6.6 that F ( x , s ) > F(x , -s) for s > 0. Let w(') be the critical point of an provided by the proof of the theorem. Then, unless w(') > 0,

sup @(slw(')l) < sup @(sw(')) = c , (6.42) s€[O,cQ) s€[O,@J)

which contradicts the definition of c. By the strong maximum principle we have then w(') > 0.

Chapter 6 Minimax problem

6.4 Mountain pass problems with the critical exponent

We return to the problems with critical oscillatory nonlinearity considered, in the framework of constrained infima, in Chapter 5. Let F E Cf,,(IW) satisfy (5.19) and let f (s) = F1(s). Obviously, there is a C > 0 such that

Let

By Theorem 2.14, @ E C1(V1l2(IWN)). The critical points of @ satisfy the equation -Au = f (u), in the sense of weak differentiation, over IWN.

Theorem 6.7 Assume, in addition to (5.19), that with some p > 2, for s E [I, 2?] u [ - 2 7 , 1 ]

and that

S H - is increasing, Is1

Then there is a u E V'y2(IWN) such that V@(u) = 0 and @(u) = c with c > 0 given by (6.10).

Proof. Notations of the norm and of the inner product in this proof refer, unless specified otherwise, to the space D ~ > ~ ( I w ~ ) . Step 1. From (6.45) and (5.20) follows that (6.45) extends to all s # 0 and thus that F(s) > ~ l s 1 ~ * , with some X > 0 and thus for every w # 0, @(sw) < 0 for all s sufficiently large. Thus the set 9 0 given by (6.9) is nonempty. By (5.20) and the Sobolev inequality (2.29), there exists a T- > 0 such that whenever = r,

Then the constant c in (6.10) is positive and by Theorem 6.3, there is a sequence uk E D1?'(IWN) satisfying V@(uk) + 0, @(uk) -+ 0. Repeating literally the proof of Lemma 6.3, with the only difference that the notation of the norm refers now to the space V112, and that instead of (6.15) we


quote (6.45) extended to s E R, we conclude that uk is a bounded sequence in 'D112(RN). Consider now a renumbered subsequence of uk, yc) E RN, j(n) E Z and w(") E 'D1v2, given by Theorem 5.1. Then, by (5.10) and Lemma 5.5, we have

By compactness of local Sobolev imbeddings and Lemma 1.9 with p = q =

2* - 1, for every v E c r ( R N ) the map u w JwN f (x,u)v is continuous with respect to weak convergence in 'D1y2(lRN). Thus v @ ( w ( ~ ) ) = 0, and consequently,

and, due to (6.45),

(6.48) unless w ( ~ ) = 0. Step 2. Without loss of generality we may assume that w(') # 0: if = 0 for all n, then by Theorem 5.1 uk -+ 0 in L ~ * ( I w ~ ) , and therefore, since (V@(uk),uk) + 0, uk -+ 0 in V ' , ~ ( I W ~ ) leading to c = lim@(uk) = 0, a contradiction. In this argument we used the continuity of p(u) and of J f (U)U in L ~ * , following from (5.20), (6.43) and Lemma 1.7.

Let us now estimate c from above, starting with the path (s H SW(')) E

QO:

Similarly to the argument in Theorem 6.5,

From (6.7) (which extends by (5.19) to all s E R) follows that s w

s - ' $ @ ( s ~ ( ~ ) ) is a decreasing function and thus has at most one zero. This unique zero is provided by s = 1 due to (6.47). Since @(sw(l)) is zero at s = 0, negative for large s and has positive values, it means that its only critical point on (0, co) is a maximum attained at s = 1. Then


Comparing this with (6.46) and taking into account (6.48), we come to D

conclusion that w(") = 0 for n # 1. Then uk --\ w('). Consequently,

uk w(') in L ~ * and then from V @ ( u k ) 4 0 follows that uk -, w(') in D112. Consequently, v@(w(')) = 0 and @(w(')) = c.

Remark 6.5 Let w(') be as in the proof of Theorem 6.7. If F(-s) < F ( s ) for s > 0 then, unless w(') 2 0, @(s1w(')1) < @(sw(')) < C , which contradicts the definition of c. Consequently, w(') 2 0 and by the strong maximum principle w( l ) > 0.

6.5 Critical problem with punitive asymptotic values

Theorem 6.8 Let f E Cl,,(R) and assume that the following limits exist and satisfy

f < rn f (3). 2* lim - 0 < b:= 2* lim - - -2~ = s+O Is1 Isl+, lsI2*

Moreover, assume, for some p > 2,

and

Then the functional (6.44) is in c ' ( v ' ~ ~ ( R ~ ) ) and there is a u E V112(RN) such that V @ ( u ) = 0 and @(u) = c with c > 0 given by (6.10).

Proof. We sketch the proof, referring the reader to the proofs of Theo- rem 6.7 and Theorem 6.6 for similar details of the argument. 1. By (6.50)

( f ( s ) l < CIS(^*-' and ( F ( s ) ( I C ( S ( ~ * , s E R , (6.53)

which easily yields @ E C'(V112(RN)). By the L1H6pital rule,

F ( s ) lim - - F ( s ) - l imy = b < r n , s ~ R .

s-00 )sl2* s-o 1st

The proof of the existence of a bounded critical sequence U I , at the level c given by (6.10) is identical to the proof in Theorem 6.7 and can be omitted. Consider'a renumbered subsequence of uk and the index sets No, N+, and N-, given by Theorem 5.1.


2. From (5.10) and Remark 5.6 follows

where

It is easy to see that v@(w(")) = 0 for rn E No, v@,(w(")) = 0 for n $ No, and consequently,

f ( ~ ( ~ ) ) w ( ~ ) , r n E NO; l l ~ ( ~ ) l l % l , r =

(6.56) Due to (6.51),

m) - / (A (w(m))w(m) - ~ ( w ( m ) ) > O,m t '0 @(w( ) - 2 (6.57)

and

@ ( w ( ~ ) ) = (2*/2 - 1)b ~ w ( ~ ) ) ~ * 2 0 ,n $ No. (6.58)

3. Assume that w ( ~ ) # 0 for some n @ No. Repeating the argument similar to the one in Theorem 6.6, we have

C I sup (~ , ( sw(~) ) , s>o

and tlle maximum is attained at s = 1 due to (6.56). Then

Comparing this with (6.55) and taking into account (6.57) ,(6.58) we come to conclusion that w("') = 0 for n' # n and, with necessity, c = @,(w(~)) . This is, however, impossible, since by (6.52)

D Thus w ( ~ ) = 0 for n > 1 and uk --\ w('). Consequently, uk 4 w(') in L ~ * and then from V@(uk) + 0 follows that u k 4 w(') in v@(w(')) = 0

and @(w(')) = c.

Chapter 6 Minimax problems


In the early 1960's R. Palais and N. Smale developed the theory of the critical points of functionals in Hilbert space (see the book [96] and references therein), providing the general framework for proving existence of critical sequences, associated with minimax values, by employing the deformation argument. The latter we produce here in the version due to M. Willem ([122], Lemma 2.3) as Lemma 6.2, and use it to prove the elementary mountain pass theorems. Theorem 6.2 was first applied to semilinear elliptic problems by A. Ambrosetti and P. Rabinowitz in [4] and Theorem 6.3 is an elementary modification.

Weak convergence of a bounded critical sequence to a critical point is trivial, and in many cases one can show that this critical point is nontrivial, by proving that a critical sequence uk does not converge D-weakly to zero, since otherwise from Lemma 3.3 (or its analog) and V@(uk) -+ 0 follows uk -' 0 in the Sobolev norm. The first assertion of Theorem 6.4 for R =

E l N is immediate from that argument, which was employed originally by P. Rabinowitz, [loll and repeated many times throughout the literature in solutions for a variety of non-compact variational problems. The second assertion of Theorem 6.4 is a generalization of Theorem 1.2 from [98].

Convergence of bounded critical sequences presents, on the other hand, a major technical difficulty, which in literature is often bypassed by considering constrained minimization that gives the same Euler-Lagrange equation. In addition to the case of homogeneous nonlinearity, constraint minimization is also used with the Nehari constraint inf(o~(,),,)=o) @(u), when a convexity condition yields that the Lagrange multiple of V(VG(u), u) is zero. For applications of concentration compactness in the context of Nehari constraint we refer the reader to the books of M. Willem ([122]), M. Flucher ([56]) and J. Chabrowski ([33]) and references therein. While the convexity condition for use of Nehari constraint is restrictive, condition (6.21) in Theorem 6.5, which in practice allows to use the Nehari constraint argument for @,, does not put such restriction on the functional @ itself.

We could trace the observation in the beginning of this chapter, that the Palais-Smale contdition in non-compact problems can hold only for a subset of the functional's range, to the 1983 paper of H. Br6zis and L. Niren- berg ([24], p.463). Existence of critical points can be proved by matching a quantitative estimate of a validity interval for the Palais-Smale condition with a quantitative estimate of the minimax level. On the other hand, when the mountain pass problem is concerned, Palais-Smale condition can

159


be verified in many cases by making only an implicit use of the critical value as a mountain pass value. Theorem 6.5 here is a model existence statement for the subcritical case, related to main result (Theorem 3.4) in the paper [45] of W.-Y. Ding and W.-M. Nil with the difference that [45] requires (6.35) (which, in particular, allows to use the Nehari constraint), while Theorem 6.5 replaces (6.35) with its asymptotic counterpart (6.21). The critical case statements, Theorem 6.7 and Theorem 6.8 partly overlap with Chabrowski's Theorem 1, [32]. M. del Pino and P. Felmer, [99] have proved an existence result based on a local, rather than global, penalty condition, by exploiting the exponential decay of positive solutions. It provides technical tools to the study of multi-peak solutions to semilinear elliptic problems, a topic that that was studied intensely in the recent years and deserves a more focused survey than this book.

We would like to point to two related results that lie out of the scope of this chapter. A paper of V. Coti Zelati and P. Rabinowitz, [39] proves existence of critical points of (6.12) on the mountain pass level c, provided that there are finitely many distinct critical points with CP 5 c + a and that the nonlinearity F is ~ ~ - ~ e r i o d i c with a positive minimal period. A remark- able existence result by V. Benci and G. Cerami, [14], on a positive solution to the equation -Au + a(x)u = u ( ~ ~ ~ ) ( ~ - ~ ) with a > 0, is emphatically apart from Theorem 6.7 and Theorem 5.5 (which provides solvability when inf a < 0) due the sign of the coefficient a. This is the problem where the straightforward mountain pass gives a minimax value where every critical sequence is bounded and divergent, and the authors develop a different variational construction in order to obtain a critical point. The opposite case, a < 0, allows a straightforward mountain pass statement in the spirit of Theorem 6.7 (the problem is formally not covered by the conditions of the theorem), but instead it is handled by a simpler constraint minimization argument of Theorem 5.5.

Chapter 7

Differentiable manifolds

In this chapter we give a concise introduction to differentiable manifolds. Statements given without proof can be either regarded as elementary exer- cises or we indicate in the bibliographic references at the end of the chapter where in the literature they are to be found.

Our presentation is essentially based on the books [76] and [70] as main references; only for the F'robenius Theorem 7.1 we refer to [36], p. 94.

7.1 DifferentiabIe manifolds

A topological n-manifold is a metric (more generally, a Hausdorff topological) space M, where every point has an open neighborhood homeomorphic to an open subset of Rn. Such a homeomorphism is also called a chart. In order to do analysis on manifolds we have to consider systems of compatible charts:

Definition 7.1

(1) An n-chart on a metric space M is a pair (U, cp) with an open subset U c M and a homeomorphism cp : U -+ V C Rn onto an open subset V c Rn. Two n-charts (Ul, cpl) and (U2, 9 2 ) are called compatible, if the transition map

is a diffeomorphism between the open sets VI2 c Vl and Vzl c V2 , i.e. Cm in both directions.

(2) A differentiable (n-)atlas on a metric space M is a system A :=

{(Ui,cpi)iEI) of mutually compatible charts, such that the open sets (Ui)iEI cover M.

161


(3) A differentiable n-manifold is a pair (M, A) with a metric space M and a maximal (with respect to inclusion) differentiable n-atlas.

Remark 7.1 Given a differentiable n-atlas A on M there is exactly one maximal atlas containing A: it consists of all charts on M , which are compatible with all charts in A. Thus, in order to define a differentiable n- manifold, it is suficient to give one (not necessarily maximal) differentiable atlas.

Example 7.1

(1) Any open subset U c Rn is a differentiable n-manifold with the one- chart atlas A = {(U, id)}.

(2) Embedded manifolds: Let W C Rm be an open subset and F : W 4

a m - n (with n 5 m) be a differentiable map. Let M := F-~(o) . Then,

if the Jacobi matrix DF(x) has maximal rank m - n for all points x E M , we can endow M in a natural way with the structure of a differentiable manifold: for every point a € M we choose a chart (U,, as follows. We may assume that :::::PI,: (a) # 0. Consider now

Its Jacobi matrix DG(a) is non-singular at a and thus there is a neighborhood U c Rm of a, such that

is a diffeomorphism onto the open set V c Rm. Now set Ua := M n U, cpa := p r o j p oGIU, Then A := {(U,, cp,); a E M} is a differentiable atlas on M.

(3) The n-sphere Sn c Rnfl is the most important example of an embedded manifold:

Take W = Rn+' and F(x) = 1xI2 - 1. (4) The above example can be generalized to any differentiable manifold M .

Call a closed subset N Q M a submanifold of M , if N can be covered by open sets U c M , such that there is a chart cp : U -+ V C Rn, such that cp(N n U ) -, V is an embedded manifold as in item (2) above. We leave it to the reader to check that N ~1 M inherits in a natural way the structure of a differentiable manifold.

Chapter 7 Differentiable manifolds 163

(5) Any open subset W c M of a differentiable manifold M is itself a differentiable manifold: given an atlas A := {(Ui, on M, then l3 := {(Ui n W, (cpilu,nw)iEl) is an atlas for W C M.

(6) The cartesian product MI x M2 of differentiable manifolds Mi, i = 1 ,2 , of dimension ni, is again a differentiable manifold, of dimension nl +n2: the charts (Ul x Uz, (cpl, cp2)), where (Ui, cpi) is a chart for Mi constitute an (nl + n2)-atlas for MI x Mz.

Definition 7.2 Let M be a differentiable manifold.

(1) A continuous function f : M -+ R is called differentiable, if for all charts (U, cp) E A in a (not necessarily maximal) atlas A on M , the function f o cp-l : V := cp(U) 4 R is differentiable. We shall denote Cw(M) the set of all differentiable functions on M. In fact CW(M) is an R- algebra: it is closed with respect to addition of functions, multiplication by scalars X E R and multiplication of functions.

(2) A continuous map F : M -t N between differentiable manifolds (not necessarily of the same dimension) is called differentiable, if for all differentiable functions f on N the pullback F* ( f ) := f o F again is differentiable, i.e. f E Coo(N) =. F * ( f ) E Cw(M).

Remark 7.2 The above definition of a differentiable map F : M -+ N is the most satisfactorg one from a systematic point of view, but for practical purposes note that a continuous map F : M -+ N is differentiable if given atlasses A on M and t? on N, for all charts (U, cp) E A and (W, $) E I3 the

map

is a differentiable map between open sets i n Rn and Rm, where n :=

dim M, m := dim N .

For explicit calculations on a manifold one often has to refer to charts (U, cp), but usually one does not mention explicitly the data U and cp, but instead, if e.g. f E Coo (M), writes simply f (xl, ..., x,) in order to denote f (cp-l(xl, . . . , 2,)) and calls it the representation of, or expression for f with respect to the local coordinates 21, ..., x,, where the choice of cp is understood. In fact often XI, ..., x, are identified with the component functions cpl , .. ., cp, E Cw (U).


7.2 Tangent vectors and vector fields

On a differentiable manifold there is no natural notion of derivatives, independent from the choice of a chart. But we can define homogeneous differentiable operators of degree one - usually called vector fields. We start with the notion of a tangent vector at a point a E M :

Definition 7.3 Let a E M be a point in the differentiable manifold M . A tangent vector X , of M at the point a E M is a linear map

satisfying the Leibniz rule

The set of all tangent vectors of M at a E M forms a vector space T,M, called the tangent space of M at a E M .

Remark 7.3

(1) We have X,(R) = 0 for every tangent vector X , E T a M , since X a ( l ) = Xa(12) = Xa(1) + X a ( l ) .

(2) Take a chart cp : U -+ V c Rn with a E U and cp(a) = 0. Then the maps

are tangent vectors at a. In fact the tangent vectors a?, ..., form a basis of the tangent space TaM. To see this we remark first that X a ( f ) = 0, if f vanishes near a: take a function g E Coo(M) with g = 1 near a and f g = 0. Then 0 = g ( a ) X a ( f ) + f ( a ) X a ( g ) = X a ( f ) . In particular X a ( f ) is determined only by the values o f f near a, that is, for any neighborhood U of a, the map X , : Coo(M) -+ R uniquely factorizes through the restriction C w ( M ) -+ C w ( U ) and a tangent vector Cm(U) -t R of the differentiable manifold U: use the fact that every function in Cw(U) coincides near a with some Coo-function on M . If now (U, cp) is a chart near a as above, then X , = Cy=l Xa(xi)dr. Indeed, take any f E C m ( M ) . After, may be, a shrinking of U we may assume f = f ( a ) + Cy'l xi f i with f i E Cm(U) and then obtain X a ( f ) = En 2= 1 Xa(x i ) f i (a) = EL, X,(xi)aT(f) . Another, may be more geometric, construction that avoids the choice of charts is the


following: to any curve, i.e. diflerentiable map, y : Z + M defined on an open interval Z c R with y(to) = a for some to E Z we can associate the tangent vector ?(to) E TaM defined by

The vector ?(to) is called the tangent vector of the curve y : Z -, M at to E Z. In particular the tangent space TaRn is naturally isomorphic to Rn itself: associate to x E Rn the tangent vector qx(0) with the curve yx(t) := a+tx. The adjective "natural" means here that it only depends on Rn as vector space, not on the choice of a particular base (e.g. the standard base) of Rn.

Definition 7.4 Given a differentiable map F : M 4 N between the differentiable manifolds M and N, there is an induced homomorphism of tangent spaces:

defined by

It is called the tangent map of F at a E M.

Obviously we have for a curve y : ( - E , E ) -t M with y(0) = a that

F,(j(O)) = 8(0), where 6 := F o y . For explicit computations we note that, if F = (Fl, ..., Fm) : U + W is a differentiable map between the open sets U c Rn and W c Rm, and b = F(a) for a E U, then with respect to the bases dr , ..., d: of Tau and d t , . .. , d z of Ti, W the linear map TaF has the matrix:

the Jacobi matrix of F at a E U. Furthermore it is immediate from the definition, that the tangent map

behaves functorially, i.e. if F : MI + M2 and G : M2 -+ M3 are differentiable maps, then G o F : M1 -+ M3 is again differentiable and the chain rule


holds. All the tangent vectors at points in a differentiable n-manifold M form

a differentiable n2-manifold:

Definition 7.5 Let M be a differentiable n-manifold. The tangent bundle T M is, as a set, the disjoint union

of all tangent spaces at points a E M. Denote n : T M + M the map, which associates to a tangent vector Xa E TaM its "base point" a E M . Now given a chart cp : U 2 V C Rn on M , we consider the bijective map (t7-ivialization)

where we use the natural isomorphism TbRn N Rn as explained above and 9*ITaM = Tap We endow T M with a topology: a set W c T M is open if Tcp(W n pP1(U)) c Rn x Rn is open for all charts (U, cp) in an atlas A for M . Finally, the charts (nP1(U), T p ) with (U, cp) E A define an atlas on TM.

Now we can generalize Definition 7.4: given a differentiable map F : M 4

N the pointwise tangent maps TaF : TaM -+ TF(,)N combine to a differentiable map TF : T M + T N , i.e.

Indeed, the map T F fits into a commutative diagram

TM TF' TN 1 1 ,

F M - N

i.e. n p ~ o T F = F o n~ holds with the projections TM : T M - M and n~ : T N 4 N of the respective tangent bundles.

Definition 7.6 A vector field X on a differentiable manifold M is a differentiable section of the projection n : T M + M , i.e., a differentiable

map

Chapter 7 Daflerentiable manzjolds 167

satisfying .rr o X = idM, with other words, X ( a ) E TaM for all a E M . In that case we also write Xa := X ( a ) . We denote O ( M ) the set of all vector fields on M .

Remark 7.4

( 1 ) The set O ( M ) carries, with the argument-wise algebraic operations, in a natural way the structure of a real vector space. In fact the scalar multiplication

can be extended to a multiplication by functions:

where

(2) Vectorfields can be identified with derivations D : C w ( , M ) --, C m ( M ) , i.e. linear maps satisfying the Leibniz rule D ( f g ) = D ( f ) g + fD(g) for all f , g E C w ( M ) . Given a vector field X E O ( M ) the corresponding derivation X : C m ( M ) 4 C w ( M ) , f ++ X ( f ) is defined by ( X ( f ) ) ( a ) := X a ( f ) . In fact, evey derivation D : C w ( M ) -+ C w ( M ) is obtained from a vector field: Take X E O ( M ) with

(3) For an open subset U C M the tangent bundle TU is identified, in a natural way, with the open subset .rr-'(U) c T M . In particular there is a natural restriction map O ( M ) -+ O(U) for any open subset U c M .

(4 ) Let F : M -+ N be a differentiable map. Given a vector field X E O ( M ) , we can consider T F o X : M 4 T N , but that map does not in general factor through N , e.g. if F is not injective. But it does if F : M -, N is a diffeomorphism: then we may define a map

Example 7.2 If ( x l , .., x,) are local coordinates on U c M , we call the vector fields dl , ..., 8, E O(U) with


the coordinate vector fields of the local coordinates X I , ..., x , (Note here that di does not depend just on xi E C w ( U ) , since one has to differentiate with respect to x i , while the remaining xj, j # i are kept constant!). Every vector field X E O ( U ) then has a unique representation

with differentiable functions fi E C w ( U ) .

R e m a r k 7.5 In general i t is not possible to find on an n-manifold n vectorfields X I , ..., X , E O ( M ) , such that ( X I ) , , ..., ( X n ) , is a frame at a , i.e., a basis of T a M , for all a E M . If such vector fields exist, the manifold M is called parallelizable. As we have seen i n Example 7.2, coordinate neighborhoods are always parallelizable.

De f in i t ion 7.7 Let X E O ( M ) be a vector field. A smooth curve y :

Z -+ M defined on an open interval Z C J K is called an integral curve of the vector field X , if j ( t ) = X,(,) for all t E Z.

Remark 7.6 The basic theorem i n the theory of ordinary differential equations says that, given a vector field X E 0 ( M ) on a differentiable manifold M and a point a E M , there is an integral curve y : Z -+ M defined on an open interval Z 3 0 such that y ( 0 ) = a , and if -?. : % -t M is a second such curve, then y and ;j. coincide on the intersection Z n 2. If M is compact, we can always assume Z = R . Moreover, there is a differentiable map p : U 4 M defined on an open neighborhood U c M x JK of M x ( 0 ) ~ - f M x R , such that U n ( { x ) x R ) is an interval for all x E M and t -, p ( x , t ) an integral curve of X satisfying p ( x , 0 ) = x . I f , moreover, M is compact, this map is defined on all of M x R . The flow of the vector field X then is the family ( p t ) t E W of the differentiable maps

In fact, we have

Po = id^, P,+t = P, 0 Pt,

since integral curves are uniquely determined by their initial values and t H y ( s + t ) i s an integral curve with the value y ( s ) at t = 0. Since PO = id^, i t follows that each map pt : M -+ M is a difleomorphism with inverse p-t.


The vector space O(M) carries a further algebraic structure: though the compositions X Y and YX of two derivations X , Y : Cw(M) 4 Cw(M) are no longer derivations, their commutator is:

(XU - YX)fg = XY(fg) - YX(fg)

= X ( f Yg + gYf) - Y(f Xg + g X f )

= f x y g + ( X f + gXYf + (Xg)Yf

- f y x g - (Yf )(Xg) - gYXf - (Yg) ( X f = fXYg- fYXg+gXYf -gYXf

= f (XU - YX)g + g(XY - YX) f .

Definition 7.8 The Lie bracket [X, Y] E O(M) of two vector fields X , Y E O(M) is the commutator of the derivations X , Y : C w ( M ) -+

Coo (M), i.e.

[X, Y] := X Y - YX,

or, in other words, the vector field [X, Y] satisfying

for all differentiable functions f E Cw(M) at every point a E M .

Note that the tangent vector [X, Y], is not a function of the values Xa, Y, E

T,M only, since the local behavior of the vector fields X , Y near a E M also enters in the computation rule. If XI, ..., xn are local coordinates on U c M , and X , Y E O(U) have representations

then

So, in particular, [di, dj] = 0 for coordinate vector fields. On the other hand we mention:

Theorem 7.1 (Frobenius Theorem) Let X I , . .. , Xn E O(M) be pairwise commuting vector fields, i.e. [Xi, Xj] = 0 for 1 < i, j < n. Then every point a E M , such that (XI),, ..., (X,), is a frame at a (i.e. a basis of the


tangent space T,M) admits a neighborhood U C M with local coordinates XI, ..., x, E Cm(U) such that

The proof relies on the fact that given the flows (pt) and (fit) of commuting vector fields X , .% E O(M), one has p, ofit = fit o p, for all sufficiently small s, t E R .

The Lie bracket defines on the vector space O(M) the structure of a Lie algebra, a notion which plays a key role in the investigation of continuous transformation groups (Lie groups):

Definition 7.9 A (real) Lie algebra g is a real vector space endowed with an antisymmetric (or alternating) bilinear map [.., ..] : g x g 4 g, i.e.

[X, Y ] = -[Y, XI, V X, Y E g, in particular [X, X ] = 0,

such that the Jacobi identity holds:

7.3 Cotangent vectors a n d 1-forms

Definition 7.10 Let M be a differentiable manifold. The cotangent space of M at a E M is the dual space

T,*M := (T,M)* = {f : TaM + R linear functional),

the elements in T,* M are called cotangent vectors. A 1-form on a differentiable manifold M is a fiberwise linear differentiable map w : T M 4 R, i.e., such that

for all a E M . We denote R(M) the set of all 1-forms on M .

Remark 7.7

(1) There is a multiplication offunctions with 1-forms Cm(M) x n ( M ) 4

R(M) defined by


(2) There is a natural identification of 1-forms on M and linear maps

commuting with the multiplication by functions, i.e. a ( f X ) = f a ( X ) : associate to w E R ( M ) the map X H w ( X ) := w 0 X .

(3) A 1-form w E R ( M ) can be integrated along a smooth curve y : [a, b] -+

M as follows:

(4) Given a differentiable map F : M + N between the differentiable manifolds M and N , there is an induced homomorphism

of cotangent spaces. (5) In the above situation there is a pullback homomorphism

where

Example 7.3 On a differentiable manifold we do not have the notion of a gradient vector field for a function f E C w ( M ) , but instead we can associate to it a 1-form, its differential: given a function f E C w ( M ) we define the 1-form df E R ( M ) by

If in particular X I , ..., xn 6 Cm(U) are local coordinates on the open subset U c M , we see that

i.e. (dxl),, ..., (dx,), is the basis of T,*M, dual to the basis a?, ..., 8: of the tangent space TaM. Now the chain rule gives an explicit formula for the differential df , namely:


Any function f E C w ( M ) with w = df is called a primitive function of w ; if it exists and M is connected, it is unique up to an additive constant. Finally, since for a smooth curve y : [a, b] + M we have

a 1-form w has a primitive function if and only if S7 w is path-independent, i.e., depends only on the starting and end point of the curve y.

Remark 7.8

( 1 ) Let XI, ..., x, E C w ( U ) be local coordinates on the open set U c M . Then any 1-form w E R ( U ) has a unique representation

with f i E C w ( U ) . But note that 1-forms w E R ( M ) cannot i n a natural way be identified with n-tuples of differentiable functions on the whole manifold. That i s possible only i f M is parallelizable, cf. Remark 7.5.

(2) I f F : M -, W C N is a differentiable map, ( y l , ..., y,) local coordinates on W , and w = Cj"=, gjdyj E R ( W ) , then with F j := yj o F E C o o ( M ) we have

where F * ( g j ) := gj o F is the pullback of the function gj E Cw(W).

Similarly to T M we can also define the cotangent bundle

and endow it with the structure of a differentiable manifold. We leave the details to the reader.

7.4 Tensor fields of degree 2

For a finite dimensional real vector space V we denote V* its dual space and B2(V) the vector space of bilinear maps V x V -+ R. We remark that

Chapter 7 Differentiable manifolds

there is a natural bilinear map, the tensor product:

where

Furthermore, there is a direct sum decomposition

where syrn2(v) contains the symmetric bilinear forms and Alt2(v) the antisymmetric (or alternating) ones.

Definition 7.11

(1) A (O,2)-tensor (field) on a manifold M is a map a, which associates to each point a E M an element a, E B~(T,M) such that for arbitrary vector fields X , Y E O(M) the function

belongs to Cm(M). We denote e012(M) the vector space of all (0,2)- tensor fields.

(2) A (2,O)-tensor (field) on a manifold M is a map T, which associates to each point a E M an element ra E B2(T,*M) such that for l-forms W , 77 E R(M) the function

belongs to Cm(M). We denote 0210(M) the vector space of all (2,O)- tensor fields.

Remark 7.9

(1) As for vector fields and 1-forms we can define for tensor fields a multiplication by functions.

(2) Given 1-forms w , E R(M) we denote w@r] E O 0 y 2 ( ~ ) the (O,2)-tensor with

(w @ rl)a = W a @ rla

for all points a E M .

173


O n the other hand, using biduality T,M % (T,*M)* we define, given vector fields X , Y E O ( M ) , the (2,O)-tensor (field) X 18 Y E O 2 ? O ( ~ ) i n the analogous way. O n an open set U c M with local coordinates X I , ..., x , E C m ( U ) , we may write the restriction to U of any (O,2)-tensor a uniquely as

with functions f i j E C m ( U ) . In the same way we have local representations

for (2,O)-tensors 7 E Q 2 1 0 ( ~ ) . (3) There is a natural bijection between Q 0 1 2 ( ~ ) and the vector space of

all bilinear maps

which i n both arguments commute with the multiplication by functions, i.e. f a ( X , Y ) = a ( f X , Y ) = a ( X , f Y ) : One associates to a (0,Z)- tensor a E Q0,2(M) the map (cf. (7.1))

Often, (0,Z)-tensors are defined by providing the associated map 8 ( M ) x 8 ( M ) 4 C m ( M ) only, and i t remains to check the above compatibility with respect to multiplication with functions.

( 4 ) Given a diflerentiable map F : M + N one can, as for 1-forms, define pullback homomorphism

W e leave the details to the reader

Example 7.4

( 1 ) Using (0,2)-tensors we can define infinitesimal metrics on a manifold: A metric tensor on a differentiable manifold is a (0,2)-tensor field a E Q 0 ' 2 ( ~ ) , such that a , is symmetric, i.e., a, E s y r n 2 ( ~ , ~ ) , and

Chapter 7 Dafferentiable manifolds 175

positive definite for all points a E M. Such a metric tensor gives rise to "punctual" isomorphisms

with corresponding global isomorphism

6 : Q(M) -% R(M), X H a (X, ..).

Indeed, if XI, . . . , x, are local coordinates on U c M and

then we have

(2) Given a metric tensor a E 0012(M) there is a "dual" tensor u* E

~ '>O(M) given globally by

and with respect to local coordinates as in the previous point:

if

where

7.5 Differential forms

For an n-dimensional vector space V denote BP(V) the space of all p- linear maps VP --t R: indeed, there is a natural isomorphism BP(V) E

(V*)@p with the p-fold tensor product of the dual space V* with itself. The subspace AltP(V) c Bp(V) consists of all alternating plinear maps (p-forms) a: : VP -+ W, i.e. linear in each of the p arguments and satisfying:

in particular o(vl , ..., v,) = 0, if vi = vj for two distinct indices i and j . If we fix a basis ul , ..., u, of V then such a map is uniquely determined by the values a(ui , , ..., ui,), where 1 5 i l < ... < i, 5 n; in fact they can freely be


prescribed. As a consequence, the vector space Altp(V) of all such maps satisfies AltP(V) = 0 for p > n := dim V and dimAltP(V) = for

p 5 n (by definition one takes ~ l t ' ( ~ ) := R). Furthermore we note that for w E Altn(V) and a linear map f : V -+ V we have

There is a natural projection Alt : BP(V) + Altp(V), which associates to each plinear form v an alternating one: it is defined as

with the symmetric group Sp on p letters, such that Alt(7) = 7 for alternating 7. Using this projection operator we define the exterior or wedge product

(P + ~ l t ( a @ D), ( ~ , p ) ~ a A p : = - p! . q!

where the (p + 9)-linear form a @ P is given by

The wedge product is associative, anti-commutative: a A P = (-l)pqp A a and bilinear. Taking as above a basis ul , ..., un of V with the dual basis u;, ..., u i of V*, we obtain the basis

for the vector space AltP(V). If V is in particular an inner product space, the spaces AltP(V) are

so too: First of all the inner product a : V x V 4 R provides a natural isomorphism ~ l t l ( ~ ) = V* 2 V, hence a natural inner product a* on V*. Then the inner product on V* induces an inner product on (V*)@p BP(V) > Altp(V), the space of plinear maps on VP, namely


We leave it to the reader to check that, if in (7.2) we start with an ON-basis of V we obtain an ON-basis of Altp(V) c BP(V). In particular, there is a non-trivial alternating n-form, that can be regarded canonical up to a sign, namely w = e; A ... A e;, where e;, ..., e; E V* = AI~'(v) is the dual basis of an ON-basis el, ..., en of V. If u1, ..., U, is any basis of V, then that form is

This can be seen as follows: writing ui = xk Xikek and A := (&) we see ( ~ ( u i , uj)) = AAt and hence 1 det A1 = Jdet(o(ui, uj)). On the other hand e; = XkiuE gives

e; ... e: = det(A) u: A ... u:.

Let us now consider the corresponding construction for the tangent bundle of a differentiable manifold M. A dzflerential form of degree p on M is a map 77 which associates to each point a E M an alternating p-form va E AltP(TaM) such that for any p-tuple of vector fields XI, ..., Xp E O(M) the function

is differentiable. The vector space of all differential forms of degree p is denoted RP(M) (such that RO(M) = Cw(M),R1(M) = R(M)). From our above discussion of alternating forms on a fixed vector space we derive immediately that any 77 E QP(U) on an open set U with local coordinates X I , . . . , xn has a representation

with uniquely determined functions fil,.,.,ip E Cw(U). As for (0,2)-tensors, to determine a differential form of degree p it suffices to give a plinear alternating map

compatible in each argument Xi with the multiplication with functions:


We use that in order to define a linear differential operator of degree 1, called exterior derivative,

For rl E RP(M) we define dq : O(M)P+' -+ Cm(M) by

where xk denotes a variable to be omitted. (In order to assure that the operation indeed defines a differential form one needs to verify (7.5) for cr = dq!) The given formula is obviously "local", i.e. we can apply it for any open subset U c M and then we have (dw)lU = d(wlu). If (xl , ..., x,) are local coordinates on an open subset U c M with corresponding coordinate vector fields 6'1, ..., an, then, because of [ai, a,] = 0, the second term in the above formula equals zero and we get

From this explicit description with respect to local coordinates one easily derives the Leibniz rule

for a pform q as well as ddq = 0. For differential forms we can define a pullback as for 1-forms, it com-

mutes with both the wedge product and the exterior derivative.

Definition 7.12 A differentiable manifold M is called orientable, if there is a nowhere vanishing differential form, called volume fonn w E Rn(M) of degree n = dim M , i.e. w, # 0 for all a E M.

If the manifold M is orientable and we have chosen an orientation, i.e. a nowhere vanishing n-form w E Rn(M), we call local coordinates XI, ..., xn on U C M oriented, if wJv = fdxl /\ ... dx, with a function f > 0. Now we want to define the integral JK q of an n-forrr q E Rn(M) over a compact set K C M: if K c U is completely contained in an open subset U with


oriented local coordinates X I , ..., x, and wlv = gdxl A ... A dx,, we set

L := J,,,, g(x)dxl ... dx,.

If the intersection KO := K n supp(7) is contained in an open subset U as above, we take

and finally in the general case we cover K with finitely many open coordinate sets Ui C M and choose a partition g, gi subordinate to the open cover (U := M \ K, Ui) of M. We define

and leave it to the reader to check that this definition does not depend on the choices made.

Chapter 8

Riemannian manifolds and Lie groups

8.1 Riemannian manifolds

In this chapter we study differentiable manifolds with some additional structure: Riemannian manifolds, where one has the notions of length (of curves) and angles between curves meeting in a point, and Lie groups, manifolds admitting a differentiable group law.

Definition 8.1 A Riemannian manifold is a pair (M, o) with a differentiable manifold M and a metric tensor a E Q0t2(M), cf. Example 7.4, (1).

Often the metric tensor a is simply called a Riemannian metric or infinitesimal metric, but here we shall use the word metric exclusively for distance functions d : M x M -+ W.

Example 8.1

(1) M = Wn with a := EL, dxi 8 dxi. (2) Any submanifold N c M of a Riemannian manifold (M, a ) is again a

Riemannian manifold with the pullback of o E e0t2(M) with respect to the inclusion N -+ M as metric tensor.

(3) Any differentiable manifold M can be endowed with a metric tensor: take a locally finite cover (Uj)jE .J with charts (Uj, cpj) (remember that our manifolds are metric spaces and hence paracompact!) and choose a subordinate partition of unity (gj)jE .J, i.e. gj E C r ( U j ) c C r ( M ) , gj 2 0 and CjE gj = 1. Define aj E E012(M) by

n

ajlu, :=g j . y ~ ~ ( ~ d x i 8 d x i ) and ajlM\v, = O i=l


and finally

(4) Using the formula (7.3) with the inner product a, on V := T,M we can at every point a E M of a Riemannian manifold (M, a ) measure the length of a pform r],. In that way we can associate to any pform r] E OP(M) a continuous function 171 : M - R>o.

Riemannian manifolds as measure spaces. Every Riemannian manifold (M, a ) is in a natural way endowed with its Riemannian measure, the Bore1 measure p := p, given on an open subset U c M with local coordinates X I , .. . , xn as follows:

with the unique function f > 0, such that the n-form

is pointwise associated to the metric tensor a, i.e. its value w, E Altn(TaM) is associated to the inner product a, on T,M for all a E U , cf. Section 7.5. In fact, if

we have, according to formula (7.4),

Using the measure p on M and the dual metric tensor a* E Q2>O(M), cf. Example 7.4, (2), we are going to define a bilinear form

resp. a quadratic form

Gradient, divergence and Gauss' Theorem. The metric tensor a provides natural isomorphisms T,M 2 T,*M, cf. Example 7.4, (I), and we

Chapter 8 Riemannian manifolds and Lie groups

denote

the induced isomorphism on the level of vector fields and 1-forms. In particular we can define the gradient Vf E @(M) of a function f E Cw(M)

by

On the other hand, given a vector field X E O(M) we can define its divergence div(X) € Coo(M): if U is an open subset admitting a nowhere vanishing n-form, we take an n-form w E Rn(U) associated to the metric tensor a (for connected U it is unique up to sign!) and denote w x E Rn-'(U) the n - 1-form with

Then the function div(X)ltr E Cw(U) is defined by requiring

Explicitly, if U admits local coordinates XI, ..., x, and

is the determinant of the coefficient matrix gij E Cw(U) of the metric tensor a = Cgijdxi 63 dxj, the vector field X = Cy=l fiai, has divergence

Assume now that W c M is a relatively compact open subset with smooth boundary, i.e. each boundary point admits a neighborhood U together with a function f E Cw(U) with non-vanishing gradient, such that W n U :=

{x E U; f(x) < 0). Denote n : a W -+ T M the outer normal vector field of dW, i.e. n(x) = V f (x) / (V f (x) ( holds in U n a W for U as above. Since a W itself is a Riemannian manifold, there is a Riemannian measure v on dW, and the Gauss theorem holds:

for any vector field X defined in a neighborhood of W and the Riemannian measure p on M .

183


Riemannian manifolds as metric spaces. Any connected Riemannian manifold admits a natural (global) metric (or distance function) d : M x M - RLo: first of all one defines the length L(y) of a Coo-curve y :

[a, b] H M as the integral

with Ij.(t)l := Jw. Define now the metric d : M x M --, R associated to the metric tensor

a as follows: The distance d(a, b) between points a, b E M is the infimum of the lengths of piecewise Coo-curves (being the sum of the lengths of the Coo-pieces) joining a and b. Then d is in fact a metric on M , and the induced topology on M coincides with the original one.

A bijective map f : M + M is called an isometry if it preserves the metric: d(f (x), f (y)) = d(x, y) for all x, y E M. The set of all isometries Iso(M) forms a group, in fact one can show

Iso(M) = {f : M --, M diffeomorphism, f *(a) = a). (8.2)

So a map f : M -+ M is an isometry, if it is a diffeomorphism preserving the metric tensor: f * (a) = a, i.e. such that all its tangent maps are isometries of inner product spaces. From this it follows immediately that all the constructions based on the metric tensor as for example the Riemannian measure or later on the different curvatures are preserved by an isometry.

Between nearby points a, b there are always curves of minimal length. They can be characterized locally by a differential equation. For that we need the notion of a covariant derivative or also connection, which is a map analogous to the action of vector fields as derivations

but where CW(M) is replaced with O(M).

Definition 8.2 A covariant derivative or connection on a differentiable manifold is a map

satisfying the following conditions:

Chapter 8 Riemannian manifolds and Lie groups 185

(2) For fixed Y E O(M) the map X H VxY commutes with the multiplication with functions:

for all f E Cm(M). (3) For fixed X E O(M) the map Y H VxY satisfies the following Leibniz

rule:

for all f E CM(M).

As in the case of tangent vectors one shows that VxYlu E O(U) depends only on XI and Y 1 U, and that there are induced connections VU on every open subset U c M . Now, if XI , ..., Xn E O(U) provide a basis for the tangent space T,(M) at each point a E U (a "frame" in classical terminology), we can write

with unique functions rfj E Cm(U), the so called "Christoffel symbols", of the connection V (with respect to the vector fields XI, ..., X,).

On the other hand any choice of n3 functions l?fj E CM(U) provides a connection on the open subset U.

If M is a Riemannian manifold, then there is a natural choice of a connection V: it is determined by the requirement that it should be "torsion free", i.e. its "torsion tensor TV" vanishes:

for all vector fields X, Y E O(M) (note that the expression TV (X, Y) commutes in both arguments with the multiplication with functions (one says: it is "Cm(M)-bilinear"), though the three summands do not!) and compatible with the Riemannian metric, i.e. the Leibniz rule

should hold for all vector fields X, Y, Z E O(M).

Parallel transport of tangent vectors. On a manifold M there is no natural way to compare tangent spaces with different base points as for example in Rn, where we have canonical isomorphisms TaRn Rn 2 TbRn.


But given a connection on M, one can at least define a "parallel transport" of tangent vectors along regular curves y : Z -+ M , i.e. y is a Coo-curve with nowhere vanishing tangent vector .j.(t). First of all, a vector field over y is a differentiable map Z : Z -+ TM, t H Zt, with n o Z = y or more explicitly, Zt E T,(tlM for all t E Z. For example t - $(t) is such a vector field over y .

We want to show that we can use the given connection in order to "differentiate" vector fields Z over y with respect to t E Z. In fact, the value (VxY), depends only on the behavior of Y near a and Xa, since Xa = 0 implies (VxY), = 0, as one easily sees writing X = C fixi with respect to a "local frame" XI , ..., X, (vector fields providing bases of the tangent spaces at all points near a E M). So VxaY := (VxY), is well defined. Furthermore, writing Y = C giXi one obtains in the same way, that V+(tlY is determined already by t I+ Y,(t), i.e., the restriction of Y to y. Finally, since any vector field Z over y (y being a regular curve) locally extends to a vector field Y E O(U) with an open subset U c M (i.e.: Zt = Y,(t)), we see, that for a vector field Z over y, the expression

V+Z : Z 4 TM, t V+(t)Z

provides again a well defined vector field over y, the "(V-)derivative of Z". We shall call a vector field Z over y a "parallel vector field", if V+Z = 0.

Since the differential equation V+Z = 0 in local coordinates is an explicit linear differential equation of first order, we see that given a tangent vector X,(to) E T,(to) M for some point to E Z, there is a unique parallel vector field Z over y with Zt, = X,(t,). That fact allows, depending on a curve y : [O, 11 -+ M connecting points a and b in M , to define an isomorphism TaM r TbM: given Xa E T,M take the parallel vector field Z over y with Zo = Xa and map Xa to Xb := 21.

Geodesics. A geodesic is a regular curve y : Z -+ M , the tangent vectors of which constitute a parallel vector field: V + j = 0. In local coordinates that equation becomes an explicit (nonlinear) differential equation of second order in the components of y. As a consequence, we get that for any Xa E TaM, there is for suitable E > 0 a unique geodesic y : [-E, E] -+ M with y(0) = a and j(0) = X,. More generally, there is a differentiable map, called the exponential map


defined on a neighborhood U of 0 E TaM with the geodesic y := yxa starting with tangent vector X, at a and yxa being defined on [0, 11. Then Exp(tXa) = yXa(t), and for sufficiently small U, Exp is a diffeomorphism onto its (open) image Exp(U) C M near a: its tangent map

is the identity on TaM. (The name exponential map is in analogy to the theory of Lie groups, where a similar construction, cf. formula (8.4), plays an important role.) The connection with curves of minimal length is now as follows: a curve, which cannot be shortened by a slight perturbation with starting and end point kept fixed turns out to be a geodesic. Locally any two points can be connected by a geodesic. Indeed, every point a E M admits "convex" neighborhoods U, i.e., such that any two points b , c E U can be connected in U by a unique geodesic of length d(b, c). Basic for the global behavior of geodesics is the equivalence of the below five conditions on a connected Riemannian manifold M - if any one of them is satisfied, then all of them are satisfied and M is called a complete Riemannian manifold. They read as follows:

(1) M is a complete metric space with respect to its metric d. (2) There is a point a E M, such that for all r > 0 the open ball B,(a) :=

{x E M; d(a, x) < r ) with radius r > 0 is relatively compact in M. (3) For all points a E M and r > 0 the ball B,(a) with radius r > 0 is

relatively compact in M. (4) There is a point a E M , such that the exponential map Exp can be

extended to the entire tangent space T,M. (5) For all points a E M the exponential map Exp can be extended to the

entire tangent space TaM.

In fact in a complete Riemannian manifold two different points can always be connected by a (not necessarily unique) geodesic of minimal length; so in particular the exponential map is surjective.

Curvature. The curvature R := R~ of a connection V is a generalized (O,2)-tensor, as it is the torsion tensor T := T ~ . The latter was defined in (8.3) by giving a Coo(M)-bilinear map

i.e. compatible in both arguments with the multiplication with Cm- functions: T(fX, Y) = fT(X, Y) = T(X, f Y). As a consequence of that


we can understand it as a family (Ta)aEM of R-bilinear maps

such that for X, Y E O(M) pointwise evaluation provides a new vector field T(X,Y) E O(M). For the curvature tensor R := R~ we use the same strategy, but on a higher level. The map

(where the target consists of all Cm(M)-linear endomorphisms of the vector space O(M)) associates to any pair (X, Y) of vector fields a Cm(M)-linear map R(X, Y) : O(M) - O(M), Z H R(X, Y)Z: it is defined as

with the commutator [Vx,Vy] = VxVy - VYVX of the R-linear maps Ox, Vy : O(M) + O(M). Since R(X, Y)Z is Coo(M)-linear in each of its three arguments (check this!), there is again the pointwise point of view: R is induced by a family (Ra)aEM of R-bilinear maps

such that for X, Y, Z E O(M) pointwise evaluation provides a vector field R(X, Y)Z E O(M).

Roughly speaking, the map Ra(Xa, Ya) : TaM -, TaM measures the difference in the results of parallel translation of a given vector Za E TaM first along an integral curve of X and then along an integral curve of Y and the analogous procedure with Y first and then X. Indeed R~ = 0 holds in a neighborhood of a point a E M for the canonical connection V of a Riemannian manifold M if and only if a admits an open neighborhood U C

M with local coordinates X I , ..., x,, such that the coefficients gij E Cm(U) of the metric tensor alu = gijdxi @ dxj are constants.

Obviously R(X, Y) is alternating in the arguments X, Y, i.e.

and skew symmetric:

it satisfies a sort of Jacobi identity, called the first Bianchi identity:


as well as

a ( R ( X , Y ) Z , W ) = a ( R ( Z , W ) X , Y ) .

We can derive from the curvature tensor R an "ordinary" tensor Ric E e0y2(M), called the Ricci (curvature) tensor of M :

with the endomorphism Ra(.., Xa)Ya : TaM - TaM of the tangent space T,M. Now the first Bianchi identity together with the fact that R ( X , Y ) is alternating in X , Y and has trace 0 at every point a E M (taking into account that Ra(Xa , Y,) : TaM - TaM is a skew symmetric endomorphism), implies that the Ricci tensor is symmetric

Ric(X, Y ) = Ric(Y, X ) .

Finally, for any point a E M, there is a unique endomorphism Fa : TaM - Ta M satisfying

for all tangent vectors X,, Ya E TaM. Its trace

defines a function R E C m ( M ) , the scalar curvature of the Riemannian manifold M .

8.2 Lie groups

Definition 8.3 A Lie group is a group G endowed with a compatible structure as differentiable manifold, i.e., such that the group operations

are differentiable maps.

Example 8.2

(1) Any real vector space V with the addition of vectors as group operation is a Lie group.

189


(2) The group of all invertible square matrices of size n

is a Lie group with the manifold structure induced from RnXn E Rn2 on the open subset GLn(R): it is obvious that matrix multiplication is differentiable, while for the inversion of a matrix we may apply the formula A-l = det(A)-lA* with the complementary matrix A* = ((-l)ifjdet(Aji)) of A, where Aji E is obtained from A by deleting the j-th row and the i-th column.

(3) Any subgroup G C GLn(R), which is a submanifold of GLn(R) is itself a Lie group. For example take the orthogonal group

(with the unit matrix I = In E GLn(R)) or the group Nn(R) of all upper triangular unipotent matrices, i.e. upper triangular matrices having all diagonal elements = 1. In a more abstract way we can describe Nn(R) using the flag, i.e., increasing sequence of subspaces, (V, = Ri x {O))i=o,...,n of subspaces of V := Rn. Then Nn(R) consists of all matrices having the flag subspaces Vi as invariant subspaces: A(&) c V, and inducing the identity on the successive quotients V,/K-l,i = 1 ,...., n.

(4) The Heisenberg group. If on V := Rn+2 we replace the above complete flag (V,)i=0,...,~+2 with the gap flag Vo C Vi C Vn+1 C Vn+2, the above description of Nn(R) leads to the Heisenberg group En c Nn+2(R): it consists of all upper triangular unipotent matrices

having outside the diagonal only nonzero elements in the first row and in the last column.

(5) Let M be a differentiable manifold and denote Diff(M) the group of all diffeomorphisms of M onto itself, endowed with the compact-open topology, i.e., a base of the topology is given by the finite intersections of sets W(U, K ) := {f E Diff(M); f (K) C U), where K c M is compact and U c M open. Insofar as we consider differentiable manifolds that


are metrizable, the compact-open topology is metrizable as well: the corresponding notion of convergence is that of uniform convergence on compact sets. Then any locally compact subgroup G c Diff(M) is in a natural way a Lie group and the natural action G x M -+ M is differentiable, cf. Definition 8.5.

The argument of the last point in Example 8.2 applies in particular to the group of all isometries of a Riemannian manifold M :

Theorem 8.1 Let M be a complete Riemannian manifold. Then the group Iso(M) of isometries, equipped with the compact-open topology, is a locally compact topological space.

Proof. Since Diff(M) is a metric space, it suffices to show that if a sequence qk E Iso(M), k E N, converges at some point a E M to some point b = limk,, qk(a), there is a subsequence that converges on compact subsets uniformly to an isometry q E Iso(M).

Observe that for every x E M , the sequence qk(x), k E N, is bounded: since r]k(a) -+ b, we have for x # a and all k sufficiently large,

Thus, since the balls on M are relatively compact and M is complete, the sequence qk(x) has a convergent subsequence in M. Furthermore, M admits a dense countable subset which we enumerate as x,, m E N. Con- sider subsequent extractions T, j E N, such that qk~(x,) converges to some y, E M as j -t co, and let j, E N be a monotone sequence such that d(qkjm(xm),ym) 5 2-, whenever j 2 j,. Then for every n E N,

d(qkjmm (xn) ~ n ) -+ 0. Let q(x,) := y, and note that q is an isometry on {x,),~~: for

m > n, d(ym, yn) = limd(qkj.(xrn), qkT(xn)) = d(xmr 5,). The metric space M being complete, it is easy to see that q extends by continuity to an isometry on M. If qk -+ 7 pointwise, but not uniformly on some compact set K , then there exists a sequence xk E K such that d(qk (xk), q(xk)) > 6 > 0. Now, K being compact, we may assume xk -+ x, whence

a contradiction, since the right hand side converges to 0. Finally note that r] not only preserves distances as a limit of distance-preserving maps, but also is bijective: injectivity is immediate, while surjectivity is seen as


follows: take a point z E M and zk := T ) ~ ' ( Z ) . Then we have d ( a , z k ) =

d(T)k(a), ~ ) k ( z k ) ) = d(T)k(a), Z) + d ( b , z ) , and thus almost all zk belong to the relatively compact ball B,(a) with r = d ( b , z ) + 1. Any limit zo of a convergent subsequence of (zk) then satisfies r](zo) = z.

Coro l lary 8.1 If M is a complete Riemannian manifold, then the group Iso(M) of all isometries i s a Lie group.

For g E G denote A, : G + G , x H gx resp. Q, : G -, G , x H xg the left resp. right translation by g, which obviously are diffeomorphisms of the underlying manifold (but of course no group homomorphisms!). In particular there are induced maps on the level of vector fields:

(A,), : @ ( G ) + O ( G ) , X H (A,) ,(X) = T A , 0 X 0 A,-%,

and in the analogous way for right translation.

Def in i t ion 8.4 A vector field X E O ( G ) on a Lie group G is called left resp. right invariant if (A,) ,(X) = X resp. ( e g ) * ( X ) = X for all g E G .

R e m a r k 8.1 W e denote g c O ( G ) the subspace of all left invariant vector fields on the Lie group G . I n fact, the evaluation at the neutral element e E G , i.e, the map

is an isomorphism of vector spaces, since the formula X , = T e ( A g ) ( X e ) for X E g shows that any tangent vector X e can uniquely be "embedded" into a left invariant vector field X , such that we may identify T e G with g. Furthermore i t follows that a Lie group is parallelizable: i f the values of the left invariant vector fields X I , ..., Xn E g at e E G form a base of T e X , they do as well at any point g E G .

Basic for the theory of Lie groups is that the commutator of left invariant vector fields again is left invariant, i.e. the subspace g c O ( G ) is closed with respect to the Lie bracket: [g, g] c g, and thus a Lie algebra, cf. Definition 7.9. As i t turns out, the Lie algebra g determines G locally, i.e., the group operations on any sufficiently small neighborhood U of the neutral element e E G . For a more precise statement we refer to Remark 8.2 below.

E x a m p l e 8.3

(1) If G = Rn with the addition of vectors as group multiplication, then g = Rdl + ... + Ran and consequently [ X , Y] = 0 for all X , Y E g.


(2) If G = GLn(R) c Rnxn, we can identify the corresponding Lie algebra

gIn(R) TI(G) with Rnxn: A matrix A E RnXn corresponds to the tangent vector XI = Ci,j(A)ijdij E TI(G), and it generates the left invariant vector field X with

With respect to these identifications we obtain the Lie bracket

Rnxn x RnXn 4 RnXn, (A, B) H [A, B] = AB - BA,

the commutator of the matrices A, B E RnXn. (3) The Lie algebra h ( R ) c gIn(R) of the group Nn(R) C GLn(R) consists

of all upper triangular matrices with only zeros on the diagonal. (4) The Lie algebra $, c gIn+2(R) of the Heisenberg group Wn consists of

the matrices

8.3 The exponential map

Let y := yx : Z + G be the integral curve of a left invariant vector field X E g with y(0) = e. Since X is left invariant, t I+ gy(t) is the integral curve of X through g E G (for t = 0). Now take g = y(s). Then t H y(s f t ) is as well the integral curve through g = y(s) (for t = 0). With other words y (s + t) = y (s) y (t) . As a consequence of that we see that y can be extended to the entire real line and that it is a group homomorphism from the additive group R to the Lie group G, a one parameter subgroup. In the whole we obtain a differentiable map

also called the exponential map of the Lie group G. But note that exp, in spite of the formal analogy with the exponential map Exp of a Riemannian manifold, need not be induced by the exponential map Exp corresponding to a suitable choice of a G-invariant metric tensor 0 E 0°12(G). Indeed,


such a tensor u is determined by the inner product ue on TeG, but for the latter there is in general no natural choice. Also here, the tangent map

To exp : To (TeG) Te G -+ TeG

is the identity on TeG, and so there are open neighborhoods U c TeG of 0 and V c G of e, such that

is a diffeomorphism.

Example 8.4

(1) For G = V, a vector space, we find the exponential map

exp = id" : V " ToV --+ V.

(2) For G = GLn(R) we get

and that fact also explains the name exponential map.

The exponential map satisfies exp(X) exp(Y) = exp(X + Y), whenever X, Y E TeG r g lie on the same line through the origin 0 E TeG. In the general case, since exp is a diffeomorphism near 0, it follows that, after possibly shrinking U, there is a Cw-map P : U x U -+ g such that

exp(X) exp(Y) = exp(P(X, Y)). (8.5)

That equality, together with some more information about the function P , is called the Baker-Campbell-Hausdorff formula. In fact, P is real analytic, so in a neighborhood of 0 E g TeG it admits a representation

as a sum of homogeneous polynomials

of degree k. There is even a more detailed description using the Lie bracket: the k-th polynomial Pk can be written as a linear combination of Lie monomials [[..[[Zl, Z2], Z3] ..., Zk-l], Zk] of degree k, where the factors Zi are either X or Y, with rational coefficients independent of the actual Lie group


G! There is a fairly complicated recursion formula for the polynomials Pk, we mention here only

and

An important consequence of the Baker-Campbell-Hausdorff formula is the fact, that the Lie algebra g of a Lie group G determines the "germ" of G near the neutral element; more precisely, the situation is as follows:

Remark 8.2 Lie algebras and Lie groups. For any (finite dimensional) real Lie algebra g there is a unique (up to isomorphism) simply connected Lie group G with Lie algebra (isomorphic to) g. Furthermore, given any homomorphism f : g -+ f j of Lie algebras and H a Lie group with Lie algebra f j , there i s a unique homomorphism cp : G + H with f = T&. If T,cp : g + f j even is an isomorphism, the group homomorphism G + H is a local diffeomorphism; i n particular, the kernel of cp is a discrete subgroup D c G. Hence, if H is connected, i t follows that H G/D.

Example 8.5

(1) Nilpotent Lie groups. A Lie group G of dimension n is called nilpotent if its Lie algebra is nilpotent, i.e. if the descending central series (gi)i>o of g, inductively defined by go = g and gi+l := [g,gi], satisfies gk = (0) for some k . Here we denote [g,gi] c g the subspace generated by the elements [X, Y] with X E g, Y E gi. In fact, for a nilpotent algebra the above sequence is necessarily strictly decreasing and hence g, = (0) for m := dimg, in particular all Lie monomials of degree > m vanish. Examples are N,(R) and all its Lie subgroups, in particular the Heisenberg group W, c Nn+2(R). In that situation the Baker-Campbell-Hausdorff map P is polynomial: P = CETl Pk with n := dimG, and hence defined everywhere: P : g x g + g. If in addition G is simply connected, the exponential map exp : g + G is a diffeomorphism. For example, for N,(R) the exponential map


has the inverse

Finally, an arbitrary simply connected nilpotent group is isomorphic to a closed subgroup of N,(R) resp. of the form exp(g) with a Lie subalgebra g c nn(R). In particular, for a simply connected nilpotent group G the exponential map exp : g -+ G provides a differentiable chart, and in that case one also speaks about exponential coordinates for the group G. For G =

W, the corresponding Lie algebra fj, C nn+2(R) can be thought of as Rn x Rn x R, where a triple (x, y, t) with x = (XI, ..., x,), y = (yl, ..., y,) corresponds to the matrix

The group law of W, then reads in exponential coordinates (x, y, t) E

t), = Rn x Rn x R as

(x, y, t) . (x', y',tl) = (x + XI, y + yl , t + t l + 2(x. y' - x'. y))

with the standard scalar product x . y := xlyl f ... + xnyn on Rn. In order to see that, use the Baker-Campbell-Hausdorff formula together with the fact that triple brackets are trivial in the Heisenberg algebra fj,. The left invariant coordinate vector fields associated with the exponential coordinates (x, y, t) are

Xi = a,, + 2yidt,

Y z = dy, - 2xidt,i = 1, ..., n,

and Ti = at.

(2) Carnot groups. A nilpotent Lie group G is called a Carnot group if its Lie algebra is stratified, i.e. endowed with a grading (or stratiification), a direct sum decomposition


such that [V,, V,] c V,+j, where we set Vk := (0) for k > p, and g, as a Lie algebra, is generated by Vl. For example, any "two step algebra", i.e., such that all triple Lie products [ [ X , Y ] , Z] vanish, is stratified: set V2 := g1 and take as Vl any complementary subspace. Since this applies in particular to the Heisenberg algebra $,, we see that the Heisenberg group Hn is a Carnot group with 1/1 = Rn x Rn x (0) and Vz = ((0,O)) x R. Trivially, any vector space V as a group is a Carnot group with Vl = V.

8.4 Lie group actions

Definition 8.5 A (differentiable) action of a Lie group G on a differentiable manifold M is a differentiable map

satisfying

For a point x E M the set

is called the orbit of the point x, the subgroup

of G is called the stabilizer or isotropy (sub)group of x.

Example 8.6 Taking M = G there are three natural actions of a (Lie) group on itself:

(1) the left translation (g, x) H gx = A, (x), (2) the right translation with the inverse (g, x) H xg-l = eg-I(%), and (3) the conjugation (g, x) H gxg-l = e,-1 (A, (x)) = A,(@,-1 (x)).

Definition 8.6 An action G x M + M of a (Lie) group is called

(1) transitive, if for any pair of points x, y E M there is a g E G with

gx = Y (2) free, if G, = {e) for all x E M.


We remark that an action is transitive, if there is only one orbit Gx = M; in that case M is diffeomorphic to the coset (or "homogeneous") space GIG, of G relative to the closed subgroup G, C G, via the orbit map GIG, -+

M, gG, t-, gx: any closed subgroup H C G is a Lie subgroup of G (i.e. in addition a submanifold of G) and the left coset space again a differentiable manifold with the G-action G x G/H + GIH, (g, xH) t-, gxH): if N L, U is a closed submanifold of an open subset U c G transversal to all cosets meeting it, i.e. such that TaG = Ta(aH) @ TaN for any a E N, the composition N -+ G -, G/H is a local homeomorphism onto an open subset of G/H, where we thus get in a natural way differentiable charts for G/H from those for the submanifold N. In general, M is the (set-theoretically) disjoint union of the G-orbits Gx. We have still bijective continuous maps GIG, -, Gx c M, and if we in addition know that Gx c M is locally closed in M, then the orbit map is also a homeomorphism (Gx being locally compact) resp. a diffeomorphism: the orbit Gx is even a submanifold in the open subset U c M, in which it is closed, hence a differentiable manifold. Induced actions. Given an action on M , there is an induced action

G x T M + TM, X , t-, Tg(X,),

where we identify g E G with the diffeomorphism M -+ M, x t-, gx. On the level of vector fields then, there is the action by conjugation:

G x O(M) + O(M), (g, X) t-, g,X := Tg o X o g-l,

and for 1-forms it is, up to "inversion", the pullback

In an analogous way we obtain natural actions

G x 0 ° * 2 ( ~ ) -+ 0 0 1 2 ( ~ ) , (g, a ) H ga,

where

as well as

G x Q'>~(M) + 0 2 1 0 ( ~ ) , (g, a ) I+ ga,

with


8.5 Integration

An important role for the analysis on Lie groups (or more generally, locally compact topological groups) is played by the Haar measure, the analogue of the Riemannian measure for Riemannian manifolds: there is a measure p on the Borel subsets of G, unique up to a positive scalar factor, such that p(K) < co for all compact sets K C M and p is left invariant, i.e. p(gA) = p(A) for all Borel sets A c G and elements g E G. The existence of the Haar measure is easy to see using n-forms. Take an element we E

Altn(TeG) and extend it to a left invariant n-form. Then

where G is endowed with the orientation defined by w .

Example 8.7

(1) For G = Rn the Lebesgue measure is a Haar measure. (2) For G = GLn(R) c Rnln we can take p = A / ( det I n with the deter-

minant det : GLn(R) + R and the n2-dimensional Lebesgue measure X := Xn2 on the open subset GLn(R) c Rnvn 2 IRn2.

(3) The next example shows that in general the Haar measure need not be right invariant. Since pg(A) := p(Ag) also is a Haar measure on G, we obtain pg = a(g)p with a group homomorphism a : G -+ R*. E.g. look at G := R x R*, regarded as the group of affine linear transformations of R (with a pair (b, a ) corresponding to the transformation t I+ at + b), i.e. the group law is

and the Haar measure is p = x-lX with the two-dimensional Lebesgue measure X = X2. A computation now shows that ~ ( y , x) = x. But if G is compact, we have a(g) = 1 for all g E G, since then p(G) < co and a(g)p(G) = p(Gg) = p(G). In that case we normalize p by requiring p(G) = 1.

(4) For a simply connected nilpotent Lie group G the Haar measure is both left and right invariant and coincides in the exponential coordinates exp : g -+ G with a Lebesgue measure for the vector space g (A Lebesgue measure on a vector space V is by definition a Haar measure of V as a Lie group). Namely, with respect to exponential coordinates,

199


the left translation by g := exp(Y) looks as follows

with the polynomial P : g x g -+ g of the Baker-Campbell-Hausdorff formula 8.5 and 8.6. Its tangent map at A E g is in a natural way a linear map g -, g, namely the linear term in the polynomial P(Y, X + A), regarded as a function of X E g. But

P(Y, X + A) = P(Y, A) + X + N(X) + terms of higher degree in X

with a linear map N : g -+ g satisfying N(gi) C gi+l for the members gi of the descending central series: The Baker-Campbell-Hausdorff formula presents N(X) as a linear combination of Lie monomials of degree at least two with one factor X and the remaining ones being Y or A. As a function of X it is linear and maps gi into gi+l. Hence the tangent map at A E g is of the form idg + N , with a nilpotent endomorphism N : g -+ g, and has in particular determinant 1. Since this holds for any A E g, we find that a Lebesgue measure on g is also a left (and right) Haar measure for the Lie group G = exp(g).

Now assume that the compact Lie group G acts differentiably on M. Then there is an averaging or projection operator

onto the subspace of Q ~ ~ ~ ( M ) ~ of G-invariant tensor fields, which is obtained pointwise as follows

where the integration is evaluated in the finite dimensional vector space B~(T,M). As an important example we mention that we can always assume that a differentiable manifold M with G-action is endowed with a G-invariant metric tensor: we know that there is some metric tensor T E e0t2(M) and then consider a := H(T). Since

and with T also all translates g r are metric tensors, it follows that so is a.

Chapter 8 Riemannian manzfolds and Lie groups


As in the previous chapter our main references are the books [76] and [70]. In particular Gauss' theorem for Riemannian manifolds is an easy consequence of Green's theorem as proved in [76], Appendix 6. The role of the natural distance function on a Riemannian manifold is discussed in 1761, Ch. IV, Prop. 3.5, while the fact that isometries of a Riemannian manifold actually are diffeomorphisms is shown in [76], Ch. IV, Thm 3.10, as well as in [70], Ch. I, Thm 11.1. The different characterizations of complete Riemannian manifolds are presented in in [76], Ch. IV, section 4, and 1701 Ch. I, section 10.

A locally compact subgroup of the group Diff (M) of all diffeomorphisms of a differentiable manifold is a Lie group - cf. [94], p. 208 and p. 212. The Baker-Campbell-Hausdorff formula (8.5) is derived in [70], Ch. I., while for the relationship between Lie groups and their Lie algebras, cf. 8.2, we refer to [70], Ch. 11, Thm 1.11. Nilpotent groups and algebras are the subject of Ch. 111, section 2 also in [70]. The theory of the Haar measure is presented in 1641, Ch. XI.

201

Chapter 9

Sobolev spaces on manifolds and subelliptic problems

In this chapter we consider Sobolev spaces over Riemannian manifolds with rich symmetries and, specially, over Lie groups. In particular, the concentration compactness argument can be employed to prove energy minimizers in these spaces, with dislocations defined by actions of the isometry group.

9.1 Sobolev inequality on periodic manifolds

Let X, a be an N-dimensional complete Riemannian manifold with the metric tensor a E e0y2. One says that a Riemannian manifold X is periodic (or cocompact) if there is an open set V c X, such that

A compact manifold and a manifold whose isometry group is transitive (in particular, any Lie group equipped with a left shift invariant Riemannian metric) are obvious examples of a cocompact manifold.

We will study the Sobolev space H1(X) with the principal term of the form (8.1). In order to keep the notations closer to the Euclidean case, we will in what follows denote the dual metric tensor a* E O2y0 (Example 7.4, (2)) as a scalar product (., .) E @'lo on the cotangent bundle, and identify its arguments as 1-forms. We shall also write la[' := (a , a) for cr E R1.

Theorem 9.1 Let X be a complete periodic R iemannian N-manifold. Let p E [2, cm) for N = 1,2 and p E [2,2*] for N > 2. There exists a

203


C > 0, dependent on X and p such that for every u E C r ( X )

Proof. We give the proof for a non-compact X . The proof in the compact case follows immediately from Corollary 2.4 and partition of unity subordinated to the charts. 1. Let Ui, i = 1 , . . . , m, be a covering of the compact set by coordinate charts. Without loss of generality we can assume that they are trace domains, and thus by Corollary 2.4 we have with some Ci > 0

Let R := Urn=l Ui and C = max{C1, . . . , C,). Adding the inequalities for Ui, we obtain

2. Let V2, V3 be as defined in Appendix A with Vl = R and G = Iso(X). Repeating the preceding argument with a covering of V2 by finitely many sets qiV2, qi E Iso(X), noting that the Ui qiV2 c V3, and applying an arbitrary isometry q E Iso(X), we have

Let J c Iso(X) be a set given by Lemma A.l and let us add the inequalities in (9.3) over J. Then

Chapter 9 Sobolev spaces o n manzfolds 205

Problem 9.1 Prove that if R C X is a bounded set, then the set { ~ l n :

u E C r (X), JX (lduI2 + u2) 5 1) is relatively compact in LP(R, p), 1 5 p < 2*. Hint: use the correspondent result in the Euclidean space, draw the consequence for a coordinate chart, consider a covering of R by coordinate charts and a partition of unity subordinated to this covering.

The Sobolev space is H1(X) as the closure of Cr(X) in the norm

9.2 "Magnetic" Sobolev space

Let X be a complete Riemannian manifold and let a: be a differential 1-form on X . We consider a space HA(X) defined as a closure of C r ( X , @) in the norm

The quadratic form (9.6) plays in physics a role of an energy functional for a quantum system with an external potential magnetic field, which is represented by an exact differential Zform ,B = da. The form a E R1(X), called the magnetic potential, is determined by the magnetic field ,B up to an arbitrary closed form, and it is easy to see that (9.6) is invariant under the gauge transformations (a:, u) H (a+dp, ewu) with an arbitrary smooth

cp.

Lemma 9.1 (Diamagnetic inequality) Let a E R1(X). For every u E C,"(X, @) and at every point where u # 0,

Proof. Let v, w be respectively the real and the imaginary part of u. The relation (9.7) follows from the following chain of identities due to bilinearity


and the chain rule, considered at any point where u # 0:

Jdu - iuaI2 - (d(lu))l2

= lduI2 + I~1~1a1~ - 2v(a, dw) + 2(a, dv)

- ) u ) - ~ + w2a)dwl2 + 2vw(dv, dw))

= ( u ) - ~ { ) v ~ w - wdvI2 + 2 1 ~ 1 ~ ~ ~ . ( W ~ V - V ~ W ) + I u ( ~ ( c ~ / ~ ) 2

= ) U [ - ~ Iwdv - vdw + lu12~l 2 0.

Corollary 9.1 The following inequality holds:

Consequently, the space HA (X) is continuously imbedded into H 1 (X) .

Proof. Let T, : C r ( X ) -+ C r ( X ) , T,u := (u2 +c2); - E . For every function of the form T,u, (9.8) follows from (9.7). To obtain (9.8) for u one passes to the limit E -+ 0 and follows the argument in the proof of Proposition 2.2.

As a consequence of corollary (9.1), we have the "magnetic" Sobolev inequality when X is periodic:

p E [2,2*] for N > 2, p 2 2 for N = 1,2.

9.3 Magnetic shifts and D-convergence

Let X be a complete Riemannian manifold and let G be a subgroup of Iso(X), closed in the compact-open topology. Using the pullback action of Iso(X), R2 (X) -+ R2 (X), one calls the magnetic field /3 E R2(X) G-periodic if q/3 = ,B for all q E G or, in terms of the magnetic potential a E R1, d(qa - a ) = 0. We require a somewhat stronger condition, noting that if X is simply connected then the form qa-a is a differential of a function. That is, we assume that there exists a CM-function $,(.) : X -+ C, uniformly continuous in q with respect to the compact-open topology, such that

Chapter 9 Sobolev spaces on manifolds 207

From (9.10) follows that the magnetic field P = da is Iso(X)-periodic. In particular, we have dqid = 0, so that, since X is connected, qid is constant. Since $id is defined by (9.10) up to a constant, we assume that

The function +, defines the following set of transformations, known as magnetic shifts:

In particular, if X = IRN with the standard scalar product and G = IRN, with the actions QX := x + 77, every periodic (here, constant) magnetic field P: dp = 0, 77P = ,B, corresponds to the magnetic potential of the form a = (Ax, dx), where A is a constant alternating matrix, and the magnetic shifts corresponding to the field a use +, = Av. x.

Lemma 9.2 The set D G , ~ , up to the extension by continuity, is a group of unitary operators on HA (X).

Proof. It suffices to prove that

gv-1 = g,l (9.13)

and

9v-1 = 9; (9.14)

for every 77 E Iso(X). To prove (9.13), note that from (9.10) and (9.11) it follows immediately that

Then solving the equation gvu = v, one has v = e-iqqOv-luoq-l = ei*q-l uo 77-l.

In order to prove (9.14), consider the following calculations, taking into account that 77 E G, (9.15) and (9.13):


Lemma 9.3 Let X be a complete Riemannian manifold and let a E

n1 (X) . The space (HA(X), D G , ~ ) is a dislocation space.

Proof. By Lemma 9.2 the elements of DG are unitary operators. Thus it suffices by Proposition 3.1 to show that if qk E G, gqk f' 0, then g,, has a strongly convergent subsequence. Moreover, it suffices to verify the elementwise convergence on a dense subset C r (X).

Assume that g,, f\ 0. Then there exist u, v E C r (X) and a renumbered subsequence of qk, such that (g,,u, v) ft 0, so that qk(suppu) n suppv # 0. Let xk E suppu be such that qkxk E suppv. Since suppu is compact, a renumbered subsequence of xk converges to some x E suppu. Since suppv is compact and qk are isometries, a renumbered subsequence of qkx converges, and therefore, by Theorem 8.1, qk converges to some E I uniformly on compact sets. Then g,,v converges for any v E C r ( X ) .

Lemma 9.4 Let G be a closed subgroup of Iso(X) and assume that X is a complete G-periodic Riemannian N-manifold, that is, for some open bounded set V G X ,

Let r E (2,2*) and let ub E HA(X) be a bounded sequence. Then

Proof. The proof of necessity is repetitive of that in Lemma 3.3. To

prove sufficiency, assume that uk D%" 0. Let fi = V and let V2, fi and the countable set J c G be as in Lemma A.1. From (9.4) we have

By adding terms in (9.17) over q E J, we obtain

Chapter 9 Sobolev spaces on manzfolds 209

for an appropriately chosen "near-supremum" sequence qk E J . It remains t o note that b y compactness o f imbedding in t h e local Sobolev inequality (9.4), gVkuk + 0 in L T ( X , p ) , so that the assertion o f the lemma follows from (9.18).

Remark 9.1

( a ) Theorem 3.1 for the dislocation space ( H k ( X ) , DG,a) holds with (3.9) implying that whenever m # n, the sequence of isometries qk :=

) - '̂ ' i s discrete, that is, for every x E X q k ( x ) has no bounded Vk OVk subsequence. Indeed, i f some point x E X the sequence q k ( x ) had a bounded subsequence, by Theorem 8.1 qk would have a subsequence convergent to some 77 E G i n compact-open topology. O n this renumbered subsequence, with u E C r ( X ) \ { 0 ) , we have g (,) *g (,,) u - gVu # 0 ,

V k ' l k

which contradicts (3.9). Furthermore, i f X is Iso(X)-periodic, then (3.11) yields to interpretation of Lemma 9.4.

(b) If X is Iso(X)-periodic, the natural counterparts of Lemma 3.4 and Remark 3.4 hold true for the integral J F ( u ) d p , resp. J F ( x , u ) d p when a = 0 and for J F(Iul)dp, resp. J F ( x , 1ul)dp for a # 0 .

( c ) If X is Iso(X)-periodic, the minimum in

is attained. ( d ) If X is G-periodic with respect to a subgroup G of I s o ( X ) , then the

minimum i n (9.19) is attained even i f the condition u E H ~ ( x ) is replaced by u E H i ( R ) , provided that R c X is a G-flask set. If Sl i s a locally trace set ( in particular, a R E Ck,) such that for every sequence qk E G, there exists a q E G such that, up to a set of measure zero,

l imin f q k R c R , (9.20)

then R is a G-flask set. ( e ) Assume that I s o ( X ) has an infinite compact subgroup T such that when-

ever T # id and qk E G is discrete, the sequence q k l o T oqk is discrete. Let H&(x) be a subspace of H 1 ( X ) consisting of functions u such that u 0 T = u for all T E T . Then H $ ( x ) i s compactly imbedded into LP(X , p ) , 2 < p < 2*.

( f ) There are natural counterparts of Theorem 6.5 and Theorem 6.6.


The group DIs,(x),, generalizes the group DRN of Euclidean shifts, and similarly one can regard the conformal group of X as a generalization of the group of translations and dilations DN,R. This issue will be addressed below for the case of subelliptic operators on Carnot group.

9.4 Subelliptic mollifiers and Sobolev spaces on Carnot groups

The energy functional SwN I V U ~ ~ can be considered as a particular case of SRN ELl IXiuI2 where Xi are vector fields on RN, but if the sum is sparse, the quadratic form no longer defines a space that admits local compact imbeddings into LP-spaces: consider for example a sequence of the form c~k(x) := '$(XI, . . . , x N - ~ ) ' $ ~ ( x N ) where '$ E C r ( R N - l ) , '$k E C r ( R ) , '$I, --\ 0 and ll'$k\\2 = 1, in relation to SRN laiu12. Nonetheless, there exist collections of less than N vector fields on IRN that are not sparse in the sense above. In particular, this is the case when the set of subsequent commutators of a given collection of vector fields spans at every point the whole tangent space. Energy forms and correspondent differential operators defined by such collections are called subelliptic. In this chapter we consider subelliptic energies with rich homogeneity properties, namely invariant energy functionals on RN endowed with a Carnot group structure.

Let G be a connected and simply connected Lie group associated with a nilpotent Lie algebra 0, generated, as a Lie algebra, by a subspace Vl C g, and endowed with a stratification g = Vl $ .... @V, such that [V,, V,] c V,+j. We set the convention Vk := (0) for k > p, and let Yl, . . . , Y, be a basis for Vl. We may choose the basis { K k ) of g by setting Y,1 = K, m l = m, and selecting the basis Y,k, i = 1, . . . , mk, for every Vk, k = 1, . . . , p from the vectors Y, ,,..., i, := [Y,, , [Y,,, [. . . , [K,-, , Kk]]].

Let us fix on G exponential coordinates, which allows to use the same notations for an element Y of g, the left invariant vector field on G defined by Y and the first order differential operator Yu = u H du(Y) associated with this vector field. In these notations an element of 77 E G is represented by a point y E RN, y = {y.. 23 1 i = 1 , . . . , m j , j = 1,. . . , p ) ,

Cy mj = N: 77 = exp(C yi jxj) . Heisenberg group Wn in exponential coordinates (xl, . . . , x,, yl, . . . , y,, z) has a stratified basis consisting of = axi + 2yidz, i = 1 , . . . ,n, Y,+n,l = ayi - 2xidz, i = 1 , . . . , n, and Y12 = 8,.

Using exponential coordinates we define anisotropic dilations St : G -+

Chapter 9 Sobolev spaces on manijolds 211

G, t > 0, as the mapping y i j I+ t j y i j . Note that the Jacobian of bt in the exponential coordinates is tQ, where Q := CT=l jmj is called the homogeneous dimension. For example, the homogeneous dimension of IRN is N, the homogeneous dimension of the N = 2n + 1-dimensional Heisenberg group IHI, is Q = 1 2n + 2 . 1 = 2n + 2 = N + 1. We recall (Example 8.7) that the left and the right shift invariant Haar measure on Carnot groups coinsides with the Lebesgue measure. We endow the group G with a left- invariant metric tensor by fixing its value at the origin as an inner product on g where the basis {Xj) is orthonormal, and extending it to all points of G by the pullback action of the left shifts on Q'I~(G) (see Section 7.9).

Definition 9.1 Hilbert spaces V112(G) and H1(G) are completions of C r ( G ) in the following respective norms:

and

In particular, when G = W,,

H1 (G) is trivially imbedded into L2(G) and the subsequent argument shows that V1y2(G) is imbedded into L ~ ( G ) whenever Q > 2 (when Q = 1,2, then, with necessity, G E RQ with vector additions as the group law). Note that V112(G)-norm and LP(G)-norms are invariant with respect to left group shifts

For Q > 2 we define also the group of dilation actions


and its subgroup b2 = {h j E bR, j E Z), and observe that it preserves the Vly2(G)-norm as well as the L2*o-norm with 2*Q = a. Repeating the argument of Chapter 2, we can see that any function u E V112(G) has weak derivatives Y,u E L2 (G) . Definition 9.2 A convolution of u E Cw(G) and cp E C,"(G) is the following function:

Lemma 9.5 Let u,v E C,"(G), let 52 c G and let 01 > {qC-',q E R, C E suppv). Then

Proof. The estimate follows from the following chain of inequalities that employ, in particular, the Cauchy inequality and the invariance of the Haar measure.

2

Problem 9.2 Show that for u E Cw(G), v E C,"(G),

and

Consider now the following transformation

Chapter 9 Sobolev spaces on manifolds

and note that SG Itu = SG U. Let

Problem 9.3

(a) Show that if the vector field Y is left invariant with respect to the group shifts, then the vector field yR is right invariant.

(b) Show that

Y(u*v) = u * Y v and (YU) * v = - U * Y ~ V , U E c"(G),v E c ~ ( G ) . (9.30)

Lemma 9.6 (D. Jerison [72], Lemma 9.1) Let G be a Carnot group with the left invariant vector fields, identified via the exponential map with the stratification basis Y,., i = 1,. . . , mj, j = 1,. . . ,p , defined above. For every i, j , there exist differential operators Dijk, k = 1,. . . , m, such that for every

E CO='(G), cp E C,"(G)J

and there exist differential operators ~ ( ~ 1 , k = 1 , . . . , m, such that for any

u E CO0(G), cp E C,"(G)J

Proof. 1. Due to (9.30) it suffices to prove the following identities:

and

Since the vectors Kj are identified, via the exponential map, with the left- invariant vector fields on G that coincide at the origin with aij = a,,, , they

213


admit the coordinate representation

with Pij,kl(0) = 0. Since G is nilpotent, by the Baker-Campbell-Hausdorff formula, P i j , k l are polynomials. By the definitions of dilations bt and of the exponential map, the vector fields Xj are dilation-homogeneous of degree j in the sense that Y,j (uobt) = tj(Y,ju) oSt, and thus, Pij,kl are homogeneous polynomials of degree 1 - j when 1 > j and, with necessity, Pij,kl = 0 when 1 I j . For the similar homogeneity reason, PijZkl are independent of ykt,l, whenever 1' > 1. Thus,

In particular, we have

and, from JGY,j(vw) = 0, v,w E Cr (Q) ,

where Y,; denotes the adjoint operator for Yij with respect to the inner product of L2(G). Since, i11 the exponential coordinates on G, inversion of the element corresponds to inversion of the flow and thus to y w -y, the vector fields xr allow the following representation:

2. Let prove (9.33). It is easy to show by induction, starting with k = p and decreasing k to 1, that

with some polynomials qjkil. Indeed, when k = p, (9.39) holds trivially due to (9.37), and, assuming that (9.39) holds for k 2 ko, it will follow for k = ko - 1 from (9.38). From here follows, due to (9.36) that


with some polynomials Qjki l . Moreover, since Yij, j > 1, are subsequent commutators of Y,, for each i, 1 there exist differential operators A:) such that

Substituting (9.41) into (9.40), we get (9.33). 3. The relation (9.34) follows from

Indeed, when k = p + 1, no terms are left in the second sum. Let us prove (9.42) by induction. For k = 1 we have, noting that Ci,j j = Q ,

a t I tp = -Qt - 1 6 - - j - Q - l a . . zJ 2 3 ~ 0 St-' i f

= C tj-'&jIt(-jyijp). i j

This verifies (9.42), k = 1, with D f ) = 0 and Dijl : p I-+ jyiip. For the induction step for (9.42) from k to k + 1, it suffices to express the terms of the form t k - l d i k ~ t ( ~ i k k ( P ) in the form of the right hand side of (9.42), step k + 1. Due to (9.38) and since the polynomials Pik,lj are homogeneous of degree j - k and vanish at zero, we have

Since Y,. are iterated commutators of Yk and [xR,yR] = [ R X R , RYR] =

R [ X , Y ] R = [ X , YIR, the vector fields are linear combinations of operators of the form Y,:. . . qy. Thus, and since tj-'q:q:. . . qyItl j l =

~Fltq:. . . qyljl, the term t k - l q f ~ t ( ~ i k k p ) is of the form of the first term in (9.42) and therefore (9.34) is verified.


Let cp E C," (G; [O, 11) satisfy SG cp = 1. We define the mollification operator

Mt : L$,(G) -, CrC(G) by

Mtu := u * Itcp7 where Itcp := t-Qcp o &-I , t E (0,l). (9.43)

Lemma 9.7 If u E C," (G), then

Proof. Using the change of variables H c7 we get

which due to absolute continuity of u converges to zero as t + 0.

Lemma 9.8 Let R c R1 C G be two open bounded sets. There exists a C > 0 and to > 0 such that for every u E C r ( G ) and t E (0, to),

Proof. By the second assertion of Lemma 9.6, there exist $i E Cr(f l ) , i = 1,. . . , m such that

Let to > 0 be such that for t E (t, to), i = 1,. . . ,m , {qCP1 : 77 E

a,< E bt(supp$i)) C 01. Thus, by Lemma 9.7 and (9.46), Mtu =

u + Jot ~ , u d s = u + Jot xEl Y , * MSGids. Then, using the Cauchy


inequality, and a t a later step, Lemma 9.5 and (9.27), we have

9.5 Compactness of subelliptic Sobolev imbeddings

Lemma 9.9 Let R G ill c G be two open bounded sets. If there is a C > 0 such that uk E C r ( G ) is such that

uklR has a subsequence convergent in L2(R).

Proof. Let t > 0 be sufficiently small so that {rl[-', q E R, < E

Gt(suppcp)) c R1. Then for every E 0, u E C r ( G ) ,

IMtu(~)l 5 I ~ ( C ) I u ( q ~ t ~ - l ) l d ~ 5 I l~ l l rn l l~lli ,n, , SUPP'P

and

IY,,jMtul(~) = I U * (Y, ,~I~V)I (V) 1 C(t ) I l~ l l~ ,n , .

Then by ArzelbAscoli theorem (Theorem 1.1), the sequence Mtuklnl is compact in C(R1).


By (9.45) there exists a sequence t , -+ 0 such that

Let k j , ~ be an extraction such that M t , ~ k ~ , ~ is a convergent subsequence in L2(R). In particular we may choose it so that

and let for every m choose a subsequent extraction kj,, of kj,,-l such that

Then, by the triangle inequality,

This implies that ~ k ~ , ~ is a Cauchy sequence, and therefore, a convergent sequence, in L ~ ( R ) .

9.6 Subelliptic Friedrichs and Poincark inequalities

Theorem 9.2 Let R c G be an open bounded set. There exists a C > 0, such that for every u E CF(R),

Proof. Consider the inequality

which is true at the origin (Y,*yi = 1, y: = 0) and extends by by continuity to some open set V 3 0. With u E CF(V), using at the last step (9.51),

Chapter 9 Sobolev spaces on manifolds

we have

Then (9.50) with R = V follows by adding the inequalities for i = 1,. . . , m. Since any bounded set is contained in r](StV) with some 77 E G and t > 0, we have(9.50) for all open bounded 0.

Corollary 9.2 Let B be a unit ball i n the space V1>2(G). For any open bounded set V c G, the set Bv = {ulv : u E B) is compactly imbedded i n LP(V), P E [2,26).

Proof. Let x E C r ( G ) , xlv = 1 and let W C G be an open set containing suppx. Then by (9.50), the set {xu : u € B ) is bounded in H1 (W), and thus, by Lemma 9.9, it is compact in L2(V). The statement for p > 2 follows, by the Holder inequality, from the statement for p = 2.

Lemma 9.10 Let R c G be a connected open set and let u E L ~ , ( R ) . If for every cp E C r ( R ) and every i = 1 , . . . , m,

then u is a constant function.

If u E Cto,, the relation (9.52) is equivalent to q u = 0.

Proof. Since every Y E g is a polynomial of {Y,), from (9.52) follows that Ja uY*cp = 0 for any Y E g. Let U G R be an open bounded connected set, let v E U, and consider, in exponential coordinates at 7, the curve t H ~t = t y . Let cp E C r ( U ) . Since the set U is open, there is an E > 0 such that whenever lyl < E and t E [0, 11, V;'U c R. Then

which implies that U(~,'J) = u(J) almost everywhere in U, and therefore, u is constant almost everywhere in u (see the proof of Lemma 2.1 for details). Since U is arbitrary and R is connected, u is constant on R.

219


Theorem 9.3 Let R c 01 G G be two bounded domains and let 11, E

L2(R) be such that Jn 11, # 0. Then there exist a C > 0, such that for every u E C?(fil),

Proof. Assume that (9.53) is false. Then there exists a sequence uk E

C,M(R1) such that the left hand side of (9.53) converges to zero, while the right hand side of (9.53) equals 1. Note that by Riedrichs inequality, with

2 2 2 < an appropriate x E CW(Rl), ~ l ~ ~ \ ~ = 1, one has Lljn u 5 h, x u -

CJnl ELl I Y , U ~ ~ + CJnu2, SO {uk) is bounded in H1(R1). Then by Lemma 9.9, ukln has a renumbered subsequence convergent in L ~ ( R ) . Let u be the limit of ukln in ~ ~ ( 0 ) . Since we have S n 1 ~ u k I 2 4 0, due to the Cauchy inequality we also have, for any cp E CT(R), pXuk -' 0, and thus thus Ja uycp = 0, i = 1,. . . , m. By Lemma 9.10 u is constant in R. Moreover,

Thus u = 0 in R. On the other hand, Jn u2 = limJn u i = 1, a contradiction.

Lemma 9.11 There exist a C > 0, such that for every t E [O,1] and

E CF'(G),

Proof. Let U C G be an open set containing the origin and consider the sets Vz, V3 and V4 produced by (A.l) from Vl = U. From Theorem 9.3 with R = V2, R1 = V3 and 11, = 1 we have, by rescaling (9.53),

If we denote as & the set produced by (A.l) with Vl = Z for any given

Chapter 9 Sobolev spaces on manzfolds

Z c G, then for any set W c G, ht(W2) = (6tW)z . Indeed,

Then from Lemma A. l with X = G, Vl( t ) = 6tV1, Vz(t) = htVz, t > 0, follows that there is a subset J( t ) of G such that the sets r]StVz, r] E J( t ) , cover G, and the multiplicity of the covering of G by r]btVs, r] E J( t ) , is not greater than ISt(V4) l/l6tVl I = IV4 l/lVll, a finite number independent of t . Thus, adding the relations (9.55) over all r] E J( t ) such that r]btVz intersects b t h , we get that the sum in left hand side will be greater or equal to JG u2. In the right hand side we note a uniform (both in t and r ] ) multiplicity of

covering and subadditivity of ( J ~ ~ ~ ~ ~ \ u l ) and (9.54) follows.

9.7 Subelliptic Sobolev inequality

The analog of Sobolev inequality on a Carnot group of homogeneous dimension Q > 2 is continuous imbedding of D112(G) into L';, where 2; = 6. Theorem 9.4 There exists a S > 0 such that for every u E C,"(G), Q > 2,

Consequently, the space D112(G) is continuously imbedded into L'; (G).

Proof. Let u E C,"(G) and let x E c,"((+, 4); [O, 41) be such that ~ ( s ) =

s for s E [I, 21, ( ~ ' 1 5 2. We set x j ( t ) = 2-jx(2jt), j E Z and apply (9.54) to functions xj(luI) E C r , taking into account that, since derivation by Y, follows the chain rule, IY,xj (u ) 1 5 21Y,ul. Then

221


Taking into account the bounds of u on the respective sets of integration, we have

and, if we substitute t j = 2-(26/Q)jt, take the sum over j E Z, and note that each of the intervals [2j-', 2j+'] overlaps with the rest of them not more than four times, we get

The inequality (9.56) then follows from setting t = X(JG lul2;)6 with X > 0 large enough and collecting similar terms.

Note that from (9.56) easily follows, by multiplying u with a smooth cut-off function, that for two open sets U G W, W G G,

whenever u E C r ( G ) . By the Holder inequality this extends to

9.8 Concentration compactness on Carnot groups due to shifts

Proposition 9.1 Let G be a Carnot group. The pair (H1 (G), DG), where

DG ={go : u ++ u o q , q E G), (9.58)

is a dislocation space.

Proof. First, observe that g,, 0 in H1(G) if and only if qk has no bounded subsequence. Indeed, if, on a renumbered subsequence, qk -+ Q,

then for every u E C?(G) \ {0), (U o qk, u o q) -+ Ilu o q1I2 = llu112 > 0 and


D,, f\ 0. If, conversely, for any compact set K there is a jK E N such that vj @ K for j > jK, then for every u, v E C r ( G ) , (U 0 r]k, v) = 0 for all sufficiently large k, since the set { r ] E G : qsuppv n suppu # 0) is compact.

The group DG consists of unitary operators, so Proposition 3.1 applies. Relation (3.7) follows then from the observation above.

Let Go be a closed subgroup of G and assume that there exists a neighborhood of zero V @ G, such that

The group DG, is obviously also a dislocation set on H1(G). In particular, the Heisenberg group Wn has such a subgroup WE, consisting of points whose canonic coordinates (x, y, z ) take integer values.

Lemma 9.12 Let Go be a subgroup of a Carnot group G satisfying (9.59) and let uk be a bounded sequence in H1 (G) and let p E (2,25). Then

Proof. The proof of necessity is repetitious of the proof for Lemma 3.3 In order to proof sufficiency, consider (9.57) with where U = r]V2, W = r]V3, r ] E Go, with Vz, V3 given by (A.l) with X = G and V1 = V. We have

Let J c Go be as given by Lemma A.1. Then adding the terms in (9.61) over r] E J we obtain

where r]k E J is any sequence satisfying

Since uk 0 ~ 1 , ~ - 0, by Lemma 9.9, uk or] i l -+ 0 in L2(V2), and thus, due to the Holder inequality, in LP(V2). By (9.62) this implies uk -+ 0 in LP(G).


Remark 9.2 The following statements have immediate analogs in Chapters 4 4 .

(a) Theorem 3.1 for the dislocation space ( H 1 ( G ) , DG,), where Go is a subgroup of the Carnot group G , holds, with (3.9) implying that whenever

rn # n, the sequence @ ) - l q ~ ) has no bounded subsequence. Furthermore, i f Go satisfies (9.59), then (3.11) yields to the interpretation of Lemma 9.12.

(b) If (9.59) is satisfied, the natural counterparts of Lemma 3.4 and Re- mark 3.4 hold true for the integral SG F(u ) , resp. SG F(q , u ) .

(c) Let b E Lw(G) and assume that either b is Go-periodic and (9.59) is satisfied or b, := limlql,, b(q) 5 b(q), q E G. Then the minimum in

p E (2 , 25) , is attained. (d) Assume that G has an infinite compact subgroup T such that whenever

T # e and qk E G is discrete, the sequence q i l o T o qk is discrete. Let H$(G) be a subspace of H1(G) consisting of functions u such that U O T = u for all T E T . Then H$(G) is compactly imbedded into LP(G), 2 < p < 2;.

(e) There are natural counterparts of Theorem 6.5 and Theorem 6.6.

9.9 Concentration compactness on Carnot groups due to dilations

In this section we consider v ' > ~ ( G ) equipped with the product group D G , ~ = dR x DG or with its subgroup D G , ~ = dz x DG, where the group of anisotropic dilation actions d R is defined by (9.24) and d z = { h j E dR,j E

Z}. Note that dtq = qdt It is easy to see that the elements of D G , ~ extend by continuity to unitary operators on D112(G), and to isometries on ~~6 ( G ) .

Lemma 9.13 Let u E D112(G)\{O). The sequence h,,uoqk, qk E G , sk E

EX, k E W, converges weakly to zero if and only i f l;fskl + 1qkl CO.

The proof is repetitive of Lemma 5.1 and can be omitted.

Proposition 9.2 The group D G , ~ is a dislocation group on D1y2(G).

This of course implies that D G , ~ is a dislocation group as well.

Chapter 9 Sobolev spaces on manzfolds 225

Proof. Since D N , ~ is a group of unitary operators, by Proposition 3.1, it suffices to prove that

gk E DN,w, gk f\ 0 * gk has a strongly convergent subsequence. (9.64)

By Lemma 9.13, if gk f\ 0, then the corresponding parameter sequence (sk, qk) is bounded and has convergent (renumbered) subsequences qk +

E G, s k -+ SO E R. Let gou = h8,u 0 70. Then gku - gou for u E C e However, since the operators gk, go, are unitary, llgkull = llull = llgOull and therefore gk -+ gou. By density this extends to all u E V1>2(G).

Lemma 9.14 If uk is a bounded sequence in V1s2(G), then

Proof. Assume without loss of generality that uk E C r ( G ) .

The implication uk D2R 0 + uk D3Z 0 is trivial and the argument im-

plication lluk112;2 4 0 J uk D3R 0 is repetitive of that in Lemma 5.3.

Let us prove the implication uk Dsz L Z * 0 * uk + 0. Assume uk D2Z 0.

Let x E C?((:, 4), [O, 3]), such that ~ ( t ) = t whenever t E [I, 21 and I x ' I I 2.From (9.57) with U = V2, W = V3 provided by (A.l) with any fixed open set Vl ,

from which follows, if we take into account that ~ ( t ) ~ ; 5 c t 2 ,

Let J c G be as given by Lemma A.1. Adding the above inequalities over q E J and taking into account that ~ ( t ) ~ 5 clt12;, SO that by (9.56)

we obtain


Let qk E J be such that

="z Since uk 0, uk o r l ~ l --\ 0 in D1>2(RN) and due to Lemma 9.2,

J "lk V2


Let

xj (t) = 2jX(2-jt)), j E Z.

Since for any sequence jr E Z, hjkuk D2z 0, we have also, with arbitrary

j k E Z,



Adding the inequalities (9.67) over j E Z and taking into account that the sets 2j-I 5 lukl 5 2j+' cover IR with uniformly finite multiplicity, we obtain

Let jk be such that


and note that the right hand side converges to zero due to (9.66). Then from (9.68) follows that u k --, 0 in L~; , which proves the lemma.

Remark 9.3 The following statements have immediate analogs in Chap- ters 5 and 6 with the proofs that require but trivial modifications.

(a) Theorem 3.1 for the dislocation space (D112(G), D G , ~ ) holds i n the form similar to Theorem 5.1 i f we replace N i n all the exponents with Q; replace 0 E RN with e E G; read 2-j . +y and 2j(- - y ) as ~-'(62-j .) and d2j 7. respectively and interpret (3.9) as I jim) - jp) 1 +

(m)-17)p)l --, co whenever rn # n. I6,c) Vk

(b) The natural generalizations of Lemma 5.2, Lemma 5.5 and Lemma 5.6 hold true.

(c) The natural analog of Theorem 5.2 holds true, provided that N i n the condition (5.19) is replaced by Q. The minimizer in the case F ( s ) =

1 . ~ 1 ~ ~ when G i s a Carnot group of rank two is found in [60], Theorem 2 -Qp 1.1, and is a scalar multiple of ((1 + x2 )2 + 16y ) , where x , y are

the exponential coordinates corresponding, respectively, to the strata Vl and Vz.

(d) There are natural generalizations of Theorem 6.7 and Theorem 6.8.


Sobolev inequalities on compact Riemannian manifolds follow immediately from those on bounded domains. On paracompact manifolds, Sobolev imbeddings into LP(X, w) with some weight w are immediate, but if the manifold sprouts "tentacles" where the local Sobolev constant goes to infinity, there is no imbedding with weight w = 1. Global Sobolev inequality is essentially determined by existence of lower bound for scalar curvature. We refer the reader to the book of E. Hebey [67] for details. Periodic manifolds provide an immediate elementary example of Sobolev inequality with a constant weight.

For the general matters concerning the magnetic Schrodinger equation, see J. Avron, I. Herbst and B. Simon ([B]) Existence of minima in (9.19) for the case of the constant magnetic field on R3 was proved by M. Esteban and P.-L. Lions in [50]. The generalization of their work to periodic manifolds with a periodic magnetic field, outlined in Section 8.3 is due to 11151, which builds upon the paper on the non-magnetic problem 1541. The concentration compactness argument in this case becomes an obvious analogy


of that in the Euclidean case, once it is shown that magnetic shifts (or actions of isometries in absence of magnetic field) are dislocations, that (3.9) in this case amounts to discreteness of the sequence of isometries, and that in case of periodic manifold, DI,,(M)-weak convergence is LP-convergence. The paper [55] gives constructions and classification of invariant Rieman- nian metrics for arbitrary noncompact differentiable manifolds equipped with a proper transformation group. Typical existence results in literature for minimization problems of the type (9.19) in literature involve a potential term J Vlu12 that provides a variational penalty a t infinity that overrides the contribution of the magnetic field (see for example the paper of K. Kurata [77] and references therein).

Subelliptic differential operators on RN associated with quadratic form a(u,u) = JCi Ixi(u)12 were introduced by L. Hormander ([71]), who proved that if subsequent commutators of Xi span the whole tangent space, then the operators are hypoelliptic, i.e. the form a on functions with compact support dominates their L2-norm. A necessary and sufficient condition of hypoellipticity was given by C. Fefferman and D.H. Phong [53] in terms of geometric optics (or the Carnot-Caratheodory metric, where the balls are defined as a union of integral curves y([O, 11) of unit vectors from the span of Xi). Subelliptic operators on Lie groups and related Sobolev spaces were first studied by G.B. Folland [57]) and G.B. Folland and E. Stein, [58]. For general Sobolev inequality for subelliptic quadratic forms on Carnot groups we refer to the book of N. Varopoulos ([118]), where the proof of Sobolev inequalities is based on estimates of the subelliptic heat kernel (cf. D. Jerison and A. SBnchez-Calle [74]; the textbook [82] gives a detailed presentation of this approach to Sobolev inequalities in the Euclidean case); the proof of the subelliptic Sobolev inequality (9.56) in this chapter follows in fact a much more general construction for axiomatic Sobolev spaces (see P. Koskela and P. Halasz, [63], for metric spaces, or M. Biroli and U. Mosco [19] for quasi- metric spaces that cover the case of quantum fi-actals, for a textbook on axiomatic Sobolev spaces see L. Ambrosio and P. Tilli, [5]), where the central assumption is the scaled Poincark inequality. On the Carnot groups, due to their shift and dilation invariance, the scaled Poincark inequality follows from the Poincar6 inequality in a ball, which is in turn, a consequence of compactness in the Sobolev imbedding, which in turn requires an appropriate construction of mollifiers, Lemma 9.6 (see also the construction of subelliptic mollifiers in L. Capogna, D. Danielli and N. Garofalo [28]). This chain of arguments for the case of Carnot group was provided by D. Jeri- son in [72], and followed by the proof by D. Jerison and A. SBnchez-Calle,

Chapter 9 Sobolev spaces o n manzfolds 229

[73], who used operators on Carnot groups as local approximations for the general operators of Fefferman-Phong type. A proof of compactness in the Sobolev imbeddings in the setting of Fefferman-Phong operators, was done by D. Danielli, [41].

The concentration compactness argument in generalization of problems of the type considered in Chapters 3-6, has been applied to the case of the Heisenberg group by S. Biagini [18] and numerous authors afterwards. This chapter does not present actual existence results, which become a trivial analogy of the results in the Euclidian case, once Theorem 3.1 is analytically interpreted for this case. To this ends we show that the Lie group shifts and anisotropic dilations are dislocations, and give the analytic interpretation of the separation relation (3.9) as that the corresponding sequence of group shifts is discrete, and of D-weak convergence as L P - convergence in Lemma 9.12 and Lemma 9.14. A detailed implementation of the concentration compactness argument in the subcritical case is given in the paper of I. Schindler and K. Tintarev [107]. For the critical case we refer to G. Citti ([37]) and N. Garofalo and D. Vassilev ([60]). A tentative framework of concentration compactness for on topological measure spaces equipped with a transformation group and an invariant Dirichlet form is found in the paper of M. Biroli, I. Schindler and K. Tintarev [20].

Chapter 10

F'urt her applications

This chapter contains several further examples of concentration compactness argument.

10.1 Dilations on the sphere and Yamabe problem

Let SN, N > 2, be the standard sphere, that is, a unit sphere in RNfl with a metric induced by the Euclidean metric on IRNS1. We consider SN imbedded in RN+l = RN x R with its center placed at the point (0 , l ) . Then one defines the stereographic coordinates on SN by the map < : R~ -+ R N + ~ .

Let

Let

where lr(5) is the Riemannian measure on sN (and thus, in the stereo- N

graphic coordinates, dp(<) = (&) dx) and note that this norm is

equivalent to the ~ l ( S ~ ) - n o r m (9.5). An elementary computation shows that

231


so that A extends to an isometry 2)ll2(lRN) + H1(sN). Moreover,

Due to (10.3), (10.4), the Sobolev constant (2.31) can be represented as

In particular, substitution of the minimizer (5.35) into (10.1) yields that the minimizer in (10.5) is a constant function. It should be also noted that due to the isometry (10.1), the group of dislocations D N , ~ defines a group of dislocations {AgA-',g E D N , ~ ) on H1(SN), the unitary actions of translations and dilations in the stereographic coordinates.

The minimization problem (10.5) is known as a particular case of the minimization problem

where X is a compact Riemannian manifold with a metric g, R is its scalar curvature, and integrals are taken with respect to the Riemannian volume. The problem (10.6) is a variational formulation of the Yamabe problem ([123]): given a complete Riemannian manifold X, g and a function R :

X + R, find a function v > 0, such that the scalar curvature of the manifold X with a new metric u A g equals R. Furthermore, the quantities in (10.6) are invariant with respect to conformal transformations (see e.g. [7], p.126). In the case of prescribed constant curvature (R = const) the solution was given by a succession of papers of H. Yamabe [123], N. Trudinger [117] who completed the case of nonpositive scalar curvature, T. Aubin [6] (the case of positive scalar curvature when N 2 6 and M is not conformally flat) and R. Schoen [108] who, using his celebrated positive mass theorem, has proved existence in the remaining case.

10.2 Global compactness in spaces H ~ ( R ~ ) and lDrnv2(RN)

In this section we consider the Sobolev space Hm(RN), m, N E N, defined as completion of C r ( R N ) with respect to the norm (2.59). We equip Hm(RN) with the group of shifts DRN.

Chapter 10 Further applications 233

Remark 10.1 Obviously, the shifts are unitary operators on H " ( R ~ ) for all m. A literal repetition of the proofs of Lemma 3.1 and Lemma 3.2 shows that u ( - yk) --\ 0 if and only i f lykl i 0;) and that D w ~ is a dislocation group. Thus, Corollary 3.3 can be repeated verbatim, pending an analytic interpretation of DRrv -weak and D Z N - W ~ U ~ convergence in this case. The latter is, like in the case m = 1, equivalent to LP convergence, as stated below.

Let 2; = for N > 2 m and let 2; = +0;) otherwise.

Proposition 10.1 If ur, is a bounded sequence i n H m ( R N ) , m E N, and D N

q E (2,2;) , thenuk 0 (or.' u k D3N 0 ) i f and only i f uk i 0 i n L ' ( R ~ ) . Loo Moreover, i f 2 m > N , then u k !kN 0 + uk - 0 .

Proof. Necessity. Since u k -+ 0 in LP, for every sequence yk E R N , u k ( . - yk) 4 0 in LP. Therefore u k ( . - y k ) - 0 in Lp, and since u k is bounded in H m , u k ( . - y k ) --\ 0 in H m .

Suficiency. Assume that u k D* 0. By density we may assume that u k E C r ( R N ) . Let us fix an open bounded set B c RN. By compactness of Sobolev imbeddings (Theorem 2.10) for every yk E 7LN,

Since the sequence vk := Vm-I uk is bounded in H 1 ( R N ) , and for every

yk E 7ZN , (vk ( a - yk), p ) H ~ 4 0 for every p E C r ( R N ) . Therefore vk !kN 0 in H1 and thus vr, -+ 0 in L ~ ( R ~ ) , p E (p , 2;). This, together with (10.7), implies that u k -+ 0 in wrn-lJ'. Note that by (2.68) the space Wm-'>p

is continuously imbedded into L* if p ( m - 1) < N , into L9 for every q E (2 , m) if p ( m - 1 ) = N and into Cb(RN) n L2 if p ( m - 1 ) > N . Substitution of p = 2; into N-g-l) yields 2; and we have ur - 0 in L9, q E ( 2 , 2 h ) when p ( m - 1) 5 N , that is, 2 m I N . Furthermore, if 2 m > N ,

we have u k + 0 in L o 0 ( R N ) and for every q > 2, 1 ~ ~ 1 ' 5 ~ ~ ~ ~ l l+

0.

Remark 10.2 Lemma 3.4 and Remark 3.4 extend to the case m > 1 with 2* replaced with 2;. No substantial changes in the proof are needed.

Consider now the space V m t 2 ( R N ) , N > 2 m , defined in Section 2.11, and equip it with the group of unitary operators DFTZ := DRlv x 6,,~,


where

N - 2 m . dmYz := { h j : h j u = 2 7 3 ~ ( 2 j . ) , j E z). (10.8)

I t is easy t o see that operators from 6m,2, and thus, from DGtZ, preserve the L2:-norm.

Remark 10.3 An elementary reproduction of the argument in Lemma 5.1 and Lemma 5.2 yields that gk E DGYw, gk -\ 0 , gku =

( hSh )u ( . - yk) i f and only if lskl+ lykl + oo and the group DG,w (and thus DF,z) is a dislocation group on V m ~ 2 ( R N ) . This and the subsequent lemma yield natural analogs of Theorem 5.1, Proposition 5.1 and Lemma 5.4.

Lemma 10.1 If uk is a bounded sequence in N > 2m, then D ~ Z

uk 0 , if and only i f u k --, 0 in L2>.

Proof. Necessity. Since uk + 0 in L2> then for every sequence gk E

DG,z, gkuk --, 0 in L2>. Therefore u k ( . - yk) - 0 in L2;, and since uk is bounded in V ~ ~ ~ ( I W ~ ) , gkuk - 0 in Vm12(RN) . Sufficiency. Without loss o f generality assume that uk E C r ( R N ) . Let vk := V m - l u k . Then

and therefore by Lemma 5.3, vk -+ 0 in L2* ( R N ) , or, in other words, uk -t 0 in Dm-l12;. By (2.66), the space Dm-l12; is continuously imbedded into L2;, which proves the lemma.

There are immediate analogs o f Lemma 5.5 and Lemma 5.6 for the case m > 1. W e also have an analog o f Theorem 5.2, and, in particular, the following

Proposition 10.2 The minimum in

is attained. Moreover, eve y minimizing sequence for (10.10) has a renumbered subsequence such that, with suitable gk E DF,z, the sequence gkuk converges to a minimizer.

Proof. Let uk be a minimizing sequence for (10.10), namely, 11uk112; = 1

and -+ sLm). Without loss o f generality we may assume that, on

Chapter 10 Further applications

a renumbered subsequence,

DJ,z Indeed, if uk 0, then by Lemma 10.1, uk -' 0 in L~;, which contradicts the assumptions on uk. Thus, renaming gkuk with suitable gk E DF,Z as uk, we have (10.11). From Proposition 1.6 we have

and, by Theorem 3.2, on a renumbered subsequence if necessary,

The last two inequalities may hold simultaneously only if ( ( ~ ( ( 2 ; takes values 0 or 1. Since w # 0, IIw112:, = 1. Then (10.12) provides

~ l l : , , ~ 5 sLrn), which implies that w i s a minimizer.

When m = 2, a minimizer for (10.10), with the norm specified as (2.65), is

N-4

where c x ~ = ( N ( N - 4)(N2 - 4)) and

From a simple calculation

one can easily deduce that a multiple of w2 solves the correspondent Euler- Lagrange equation

For the further details see the paper E. Hebey and D. Robert, [69].

235


10.3 Minimizer in the Nash inequality

In this section we prove existence of the minimizer for the Nash inequality (2.35).

Theorem 10.1 There exists a minimizer in the problem

Moreover, there exists a sequence uk E H1(RN) such that lluklll = llukllz =

1, I J V U ~ J J ; -+ CN and, for every such sequence there are some yk E RN such that uk(. + yk) converges to a minimizer in H1(RN) n L ~ ( R ~ ) .

Proof. I f w e s e t f o r e v e r y u ~ H ~ ( R ~ ) 2 N

S = l lu l l~ ,v(x) := sTu(sx),

then we will have llvlll = 1 1 ~ 1 1 ~ = 1 and

CN = inf v€H1(WN):IIvII~=II~llz=1

(10.16)

Let uk E C r (RN) be a minimizing sequence for (10.16), that is, lluk[ll =

IIukll2 = 1 and JWlv IVukI2 + CN. Let yk E RN be such that uk(. + yk) -\ w # 0. (Note that if there

is no such yk and w, then uk sN 0, and, by Lemma 3.3, uk + 0 in LP, p E (2,2*). Since lluklll = 1, from the Holder inequality follows that uk + 0 in LP, p E (1,2*), and in particular, in L2, which contradicts the constraint value llukllz = 1.)

Let now vk = uk-w(--yk) andlet t = IIwII1, T = IIwII$. Then from the Brhzis-Lieb Lemma (Theorem 3.2) follows, on a renumbered subsequence,

On the other hand, by Proposition 1.6 and (2.35),

Consequently,

1 2 (1 - T)l+2/N(1 - t)-4/N + T1+2/Nt-4/N 1

which is false, unless t, 7 E {O,l). Since w # 0, we have with necessity t = T = 1 and therefore w is a minimizer. Moreover, this also implies


that uk(. + yk) -+ w in L1 n L ~ . By the weak lower continuity of norm, J J V W ~ ~ 5 CN, which due to the definition of C, yields J IVwI2 = CN and thus uk(- + yk) 4 w in H1 n L1.

A rearrangement argument yields that the minimum in (10.16) is attained on a decreasing radial function, and it is shown in [29] that this minimizer is unique up to translations and dilations.

10.4 A minimization problem with nonlocal term

Let N E N , N > 2 , P E (O,N), CIE (-,-),

and consider

where 6 satisfies

Note that when CI is in the interval above, we have 19 E (0,l).

Lemma 10.2 Let ql > f i > q2 2 1. There exists C > 0 such that whenever u E Laql (RN) n L ~ ~ Z ( R ~ ) ,

Proof. Changing the integration variables (x, y) to (x, z) = (x, x - y) we represent @ (u, u) as (u, u) + @2 (u, u), where

and


Let p satisfy $ + 1 = $, which implies p > and = < :. Then from the Holder inequality we obtain

Then, since 1 + 1 = 2, by the Young inequality for convolutions (1.6), P 91

The same argument applies to Q2: the only modification is that the choice of 92 < f i yields p < in the relation $ + 1 := 2, which assures

92

that ~zl-p'fi is integrable in the exterior of the ball. Consequently,

Note that one can choose qi in (10.19) so that aqi E (2,2*) and therefore, by to the Sobolev imbedding, @(u, u) is bounded on bounded sets of H1(RN) and is continuous at zero in H1 (IRN).

Problem 10.1 Verify that @(u, u) is continuous in H1(IRN).

In particular, @(u, u) is bounded whenever llVu112 = 1 and llul12 = 1. Since both @(u, u) and the product IIvullg lluIIi-' are preserved under the transformation u H t ~ ( t . ) , we have the inequality

and the constant 6 in (10.18) is positive.

Lemma 10.3 Let uk be a bounded sequence in H1(RN). If uk D ~ N 0, then @(uk, uk) 4 0.

Proof. By Lemma 3.3, uk 4 0 in L ~ ( R ~ ) for any r E (2,2*). By the assumptions on a! one can choose ql, 92 from Lemma 10.2 so that aql , a92 E (2,2*). Then @(uk, uk) + 0 by (10.19).

Chapter 10 f ir ther applications 239

Lemma 10.4 Let u k be a bounded sequence i n H1(IRN) and let y?) t zN , w ( ~ ) E H 1 ( R N ) , be as i n Corollary 9.3. Then

Proof. By continuity of @ and Lemma 10.3 it suffices to prove (10.21) M when u k = E n = , w ( ~ ) ( . - y?)), A4 E N, and w ( ~ ) E CP(IWN). Then

(n) a (m) a

~ ( u k , u k ) = c J J I ~ ( ~ ' ( x - Y ~ ) I I W ( ~ ) ( Y - ~ k ) I dxdy

m,n=l, ..., M la: - Y I P

+xJJ ~ ~ ( n ) ( ~ ) l a l ~ ( m ) ( ~ - ~ i ~ ) (n) a

la: - Y I P + Y k ) I dxdy.

m#n

To prove the lemma it suffices to show now that the second sum converges to zero. Indeed, whenever m # n,

(n) a (m) a l w ' n ' ( x - ~ k ) I l w ( m ) ( ~ ~ k ) I dxdy

la: - YIP

=JS ~ W ( ~ ) ( Z ) I " I W ( ~ ) ( ~ + y?) - yim))la dxdy + 0 ,

Ix - YIP

since 1 y?) - yim) ( 4 oo by (3.38).

Corollary 10.1 Let u k u i n H1 ( R N ) . Then, on a renumbered subsequence,

Proof. Relation (10.21) applied to the sequence u k - u - 0 gives

Substitution of the right hand side of (10.23) into (10.21), once one takes into account that w(' ) = w limuk = u, gives (10.22).

Theorem 10.2 The infimum i n (10.18) is attained.


Proof. The problem (10.18) is invariant with respect to the transformation u I+ ut := t*u(t.), t > 0. In particular, for every u E H1(R) one can choose t > 0 such that llut 112 = 1. Therefore,

and every minimizer for (10.24) is a minimizer for (10.18). Let uk be a minimizing sequence for (10.24), namely, @(uk.uk) = 1, IJuk 112 = 1 and IlVukllg -f ~ 3 . Let yk E ZN be such that uk(. + yk) -, w # 0. (If there

is no such yk and w, then uk %N 0, and by Lemma 10.4 @(uk,uk) - 0, a contradiction.) Let vk := uk - w(. - yk) and let t = I I w [I ; , 7 = @(w, w). Then from Proposition 1.6 and Corollary 10.1 respectively follows that, on a renumbered subsequence,

On the other hand, by Proposition 1.6

which is true, given that w # 0, only if t = T = 1, which by the standard argument yields that w is a minimizer.

Observe that for the minimizing sequence uk selected in the proof, uk(. + yk) -, w in L2, and by weak lower semicontinuity, IVw12 5 K.$ . Then by definition of K., S IVwI2 = K.; = lim S ( ~ u ~ ( . + y k ) ( ~ and thus uk(.+yk) -' w in H ~ ( R ~ ) .

10.5 Concentration compactness with topological charge

This section presents a slightly modified version of the "splitting lemma" from the papers [15]), ([16] of V. Benci, D. Fortunato, P. D'Avena, and, in the second paper, L. Pisani. We focus here on the issue of conservation of a topological index under weak convergence, rather than on the actual problem (Derrick model) studied in [15], [16].

Let R c RN, N E N, be an open bounded set. We recall that there exists a unique function

u E ~ ( n ; RN), r] E RN \ u(dR) H deg(u, R, r ] ) E Z, (10.26)


(called topological degree) such that

deg(u, a, Y) = deg(u, %, Y) + deg(u, O27 Y) (10.28)

whenever R1, R2 are disjoint open subsets of R and y @ u(R \ (01 U fl2));

deg(h(t, .), R, ~ ( t ) ) is independent of t E [0, 11 (10.29)

whenever h E C([O, 11 x a; lRN), 77 E C([O, 11; RN), and ~ ( t ) $! h(t, do ) , t E [0, 11.

It is not required that any of the sets R, 01, 0 2 is nonempty, which immediately implies that deg(u, 0, y) = 0 for every u and y. We will not need here the constructive definition of the degree of smooth maps that is normally involved in computations (for additional details see, for instance, the book of K. Deimling [42]).

Let m, N E W, 2m > N > 1, and consider the Hilbert space Hm(RN; EXNf l ) equipped with the dislocation group DZlv. The variables in lRN+ l = R x RN will be denoted as ( = (Jo,<), = ( ( I , . . . , (N). Let

and let

K, := {x E RN : u0(x) > I ) . (10.31)

Since 2m > N, Hm(RN;lRNf l ) is imbedded in c ( R N ; RN+l), for any bounded set B c lRN the imbedding (of restrictions to B) into C(B; IWN+l) is compact (cf. (2.62)) and, by (2.63), for every u E Hm(RN; lRN+') the open set K, is bounded. Moreover, since x E dK, + uo = 1, 6 does not attain the zero value on dK,, the topological degree for (6, K,, 0) is defined.

Definition 10.1 The function car : A 4 Z,

is called topological charge.

Proposition 10.3 For every u E A there exists an E > 0 such that whenever v E C ( R ~ ; R ~ + ' ) , (Iv - u((, 5 E , one has v E A and car(v) =

car(.).


Proof. Note that infZEwnr lu(x) - (1,0)1 > 0, so no v with Ilv - u ( ( , sufficiently small attains the value (1,0).

Assume now that there is a sequence vk E A such that vk + u uniformly in RN and car(vk) # car(u). Then for all k sufficiently large,

Indeed, assume, without loss of generality, that there exists a sequence xk E K, \%, that is, uo(xk) > 1 and V ~ , ~ ( X ~ ) < 1, such that .iik(xk) = 0. Consider a renumbered convergent subsequence (the set K , is bounded) xk + xo E R N . Then uo(x0) = 1 and 6(xo) = limGk(xo) = 0, which contradicts the assumption u E A.

Observe now that for k sufficiently large it follows from (10.29) that deg(vk, K,, fl K,, 0) = deg(6, K,, f l K,, 0). Then (10.33) and (10.28) imply that

car(vk) = deg(.Uk, K V k , 0) = deg(ck, Ku, fl Ku, 0)

= deg(6, K,, n K,, 0 ) = deg(fi, K,, 0) = car(u),

a contradiction.

Problem 10.2 Prove that A is an open set and that the topological charge is constant on connected components of A.

The weak convergence in Hm(IRN) does not preserve the topological charge. Indeed, if w E c ~ ( I R ~ ; I R ~ + ~ ) n A, car(,) # 0, and yk E R N , lykl -+

oo, then w(. - yk) --\ 0, while car(w( - y k ) ) = car(w) # car(0) = 0. Moreover, a weak limit of a sequence in A does not necessarily remain in A. Sequences whose weak limits remain in A arise in [15] and [16] where the functional carries a variational penalty that prevents minimizing sequences from approaching the exceptional value (1,O).

Proposition 10.4 Let 2m > N > 1. Let uk satisfying the assertion of Theorem 3.1 for the dislocation space ( H m ( R N ; RN+l D Z ~ ) (in the sense

( e ) of Remark 10.1, and applied componentwise) with g i ' : u ++ u(. - yk ), y f ) E IRN and let

Assume that there exist an E > 0 such that


for all k E N. Then w ( ~ ) E A, ! E N, and there exists an integer L such that whenever e > L, ~ a r ( w ( ~ ) ) = 0, and

L

car (ug ) = car ( ~ ( ~ 1 ) . (10.36) e= 1

Proof. Since H ~ ( I R ~ ) is compactly imbedded into C(R) for any bounded set R c IRN and since the sets Re := {x E IRN, wf) > $ 1 are bounded, from (10.34) follows that

which yields w ( ~ ) E A for all e. Moreover, from (3.10) and (2.62) follows that there exists a L E N such that for all 4 > L, I ( ~ ( ~ ) l l , < 1, and therefore, c a ~ ( w ( ~ ) ) = 0. Let K ( ~ ) := {wf) > 1) (which is a non-empty set only if el L) and let

- Note that the sets K(e) + yf) are disjoint for all k sufficiently large, and pass to a renumbered subsequence so that they are disjoint for every k. Let us show that for all k sufficiently large

Indeed, assume, without loss of generality, that there is a sequence xk E lhfN such that

(4 (0 iik(xk) = 0, uk,o(xk) > 1, and w,, (xk - yk ) < l,! 5 L. (10.40)

From U ~ , ~ ( X ~ ) > 1 follows, due to Theorem 3.1 and (10.37), that xk - y p ) has a bounded subsequence for some lo 5 L. Then, on a renumbered subsequence, xk - yp) --, z E lhfN and from (10.40) follows that w@o)(t) = 0, w p ) ( z ) 5 1 and w p ) ( r ) 5 1. Therefore w(~o)(z) = (1,6) which contradicts ~ ( ~ 0 ) E A. This verifies (10.39). From (10.39), (10.28), disjointness of


- Ke and (10.37) follows that for all k sufficiently large,

-

= deg(w(e), Kt, 0) = car (w(~)) ) ,

which proves the proposition.


Dilations on the sphere defined via stereographic projection can be used to study problems with critical exponent there, see e.g. C. Bandle, R. Benguria [13]. The geometric background of the Yamabe problem is elaborated in the book of T. Aubin, [7]. The recent survey of A. Chang [35] has references to solution of Yamabe problem for the solved case, when the prescribed curvature is constant and to the current research for the case of variable prescribed curvature.

Concentration compactness in Sobolev spaces of higher derivatives and existence proofs for constrained minimizers is due to P.-L. Lions, [86] (subcritical case) and [88](critical case). A multi-peak expansion in V2v2(lRN) equipped with DT,w has been used by E. Hebey and D. Robert ([69]), who show uniqueness (modulo translations and dilations) of the positive solution to (10.14). Concentration involving terms with dilated solution w2 arises, in particular, in the semilinear elliptic problem involving a fourth order Paneitz type operator - a higher-order analog of Yamabe problem. Like in the case of Yamabe problem, the stereographic projection induces an isomorphism between critical problem in v2121RN and one on H2(SN).

P.-L. Lions included existence of the minimizer for the Nash inequality among problems that allow the concentration compactness argument, even if the existence is immediate from the rearrangement argument. The common reason to provide a concentration compactness proof nonetheless is that, unlike the rearrangement argument, it can be extended to the case of variable coefficients. The nonlocal problem in Section 4 has been studied by Lions in [89], V3.

Additivity of a topological index in "multi-peak" decompositions is a natural conjecture, but the papers [15]) of V. Benci, D. Fortunato,


P. D'Avena, and ([16] of the same authors with L. Pisani, are the only ones known to us that implement this extension of the concentration compactness. Proposition 10.4 is a model statement in the setting of their work.

Appendix A

Covering lemma

Let X be a locally compact metric space endowed with a differentiable action of a group G, and a Radon (i.e. bounded on compact sets) measure p that is invariant with respect to G: p o q = p, q E G. One says that the action of G on X is properly discontinuous, if for every pair of compact sets K, L c X we have K n qL # 0 for at most finitely many 77 E G. Given an open set Vl c X, we define the sets Vn by induction:

Vn+1 := U q K , n E N . (A.1) q€G:qVnnvnZO

Obviously, the sets Vn are open and, since id E G, Vn c Vn+l.

Lemma A. l Assume that there exists an open, relatively compact set Vl C X , such that p(Vl) > 0 , and that the corresponding set V4 is relatively compact i n X . Let Xo c X be a closed set and assume that, wenever X o # X , that the action of G on X is properly discontinuous. Assume furthermore that there exists a set Jo c G such that the collection of sets { q V ~ ) v E ~ o covers X O . Then there is a subset J of Jo such that the sets in the collection {qv~) , ,~ J are mutually disjoint, while the collection {?7V2}qE J

covers X o . Moreover, the multiplicity of the open cover {77V3}qE J is uniformly bounded. I f Xo = X , the multiplicity of the covering does not exceed

p(V4)/p(V1).

Proof. The second assertion of the lemma follows from the first one by the following argument. Let first X = Xo and consider a point x E CV3, C E

G. The multiplicity there will not exceed the number of q E J, such that qV3nCV3 # 0. This number is not larger than the number of q E J such that qVl C CV4. Since these sets are disjoint, the multiplicity of the covering at x does not exceed p(V4)/p(Vl). Note that p(V1) > 0 by assumption, and p(V4) < 00 since p is Radon measure and V4 is by assumption relatively

247


compact. Let now Xo # X. Then the action of G is properly discontinuous and therefore every point x E X has a neighborhood U c X intersecting only finitely many qV3, q E G. After a shrinking of U we may assume x E ql/3 for all those q and it follows that there are no more of them than elements in the finite set {q E G; qV3 n 7 3 # 0).

Now let us construct the subset J c Jo: We assume first that (qV1),, Jo

is locally finite. By induction we define subsets Jk = Ak UBk C Jo such that the number

of elements in Ak equals k,

and qV1 n <I4 = 0 for any rl E Ak, < E Jk, q # <. Furthermore Ak c Ak+1 for all k, while Bk 3 Bk+1 with np=oBk = 0. Since the cover {qVl)l)EJo was locally finite, the latter implies that any relatively compact set K c Xo is contained in UVEAk r]V2 for sufficiently large k. Finally take J := UFO Ak. Begin with A. := 0, Bo := Jo. Write Jo as a sequence Jo = {ql, 72, .. .). Now

assume Ak, Bk have already been constructed. Let mk := min{m : vrn E Bk). Set Ak+l := Ak U { q m r ) and let Bk+l := {q E Bk;qVl n qrn,Vl = 0). Since all sets qVl, q E Bk \ Bk+1 are contained in qrnkV2, the families (qVz),E~k+l and (qVl),EBk+l together constitute a cover of Xo.

If Xo # X , the covering ( Q V ~ ) , ~ ~ is already locally finite, Vl being relatively compact. Otherwise, X being paracompact, there is a locally finite refinement (U,),, j of ( ~ V I ) , ~ J, with a subset j c Jo and U, c r]Vl. But then ( ~ V I ) , ~ j is also locally finite: If qVl n cVl # 0, then r]Vl c CV2 and consequently, U, c <h. So there will be only finitely many r] E j such that qVl intersects <VI. •

Appendix B

Rearrangement inequalities

A measurable function u : ElN -t R is called vanishing at infinity if for every X > 0,

For every such function u there exists a symmetric-decreasing rearrangement u*, a radial symmetric nonincreasing function such that for every X > 0

I(. E ItN : Ju(.)I > X)J = 1{5 E ItN : u*(x) > A)) .

Equivalently, for every monotone function F : [O, oo) -+ R

If u, v E L ~ ( w ~ ) , then

If u E D ~ * ~ ( I w ~ ) then u* E D ' ? ~ ( I R ~ ) and

Relations (B. I) , (B.2) and(B .3) trivially extend to partial rearrangements in n < N variables u*(., x,+l,. . . , xN). Complete proofs, further statements and references can be found in [75] and [82].

249

Appendix C

Maximum principle

Proposition C.l (Weak maximum principle) Let R c JRN be an open set and let X > 0. Let u E H 1 ( R ) . If for every nonnegative v E Hh ( R )

S , ( V U . V v + Xuv) t 0 ,

then

u 2 - sup u- an

(C.2)

We recall that we use the notation u- := (-u)+.

Proof. Set M = supan u- 2 0. If M = CQ, the statement is immediate. If M < CQ, set v = ( M -/- u ) - . Taking into account Proposition 2.2 and noting that v E HA(R), we have from (C. l )

This implies v = 0 and (C.2) follows.

Proposition C.2 (Strong maximum principle) Let R c JRN be an open set and let X 2 0. Assume that u E H 1 ( R ) satisfies (C.l). Then if for some open ball B G R,

inf u = inf u 5 0 , B R (C.4)

then u is constant in B

See [62] for a proof.

251

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Index

ArzelA-Ascoli theorem, 7

Baker-Campbell-Hausdorff formula, 194

Banach-Alaoglu theorem, 15 BrBzis-Lieb lemma, 74

modified, 73, 74, 117, 118, 134, 209, 224, 227, 233

Cauchy-Schwarz inequality, 8 compactness

of sets in C ( X ) , 7 of Sobolev imbedding

basic lemma, 39 basic lemma, Carnot group,

217 bounded domain, 49, 50 for 49, 219 for trace on the boundary, 52 subspace of symmetric

functions, 77, 78, 91, 209, 224

unbounded domain, 90 conformal Laplacian, 231 connection, 184, 185 convergence

D-weak, 60 in ( v ' , ~ ( R ~ ) , DN,z), 110 in (H(R?), DN-i,z), 127

in (H'(BN), DZN), 71 in (HI (G), DG), 223 in (Hm(BN), D=N), 233 in (H;(~),DG,,), 208 in (12, DZN), 62 in ( V ' ~ ~ ( R ~ \ Bn),Dn,~) , 132 in (D1,'(G), DG,R), 225 in ( Z ) ~ ~ ~ ( I W ~ ) , D;;tVZ), 234

of operator sequences strong (elementwise), 17 weak, 17

weak, 13 uniform boundedness

principle, 14 convolution, 6

on Lie group, 212 cotangent space, 170 covariant derivative, 184 critical exponent

for H ' ( w ~ ) , 44 for H m , 233 for trace on the boundary, 54 on Carnot group, 212

critical sequence, 144 curvature, 189

scalar, 189, 232

deformation lemma, 142 degree, 241 derivative

Frkchet, 21 linear, 20

261


weak, 30 of u+, u-, 33 of higher order, 57

diamagnetic inequality, 205 differential, 171 differential form, 177

space RP(M), 177 volume form, 178

dilations, 109, 126, 131, 234 on Carnot group, 211 on standard sphere, 232

dislocation space (H(R?), D N - i , ~ ) , 126 (H1(RN), DRN), 70 (H1(RN), DzN), 70 (Hm(RN), DRN), 233 (vl" ( R ~ ) , D N , R ) , 109 (v l " ( R ~ ) , D N , z ) , 109 (l2,DzN), 61 PlS2 (G), DG,R), 224 ( V " ~ ( I W ~ \ R n ) , D n , ~ ) , 131 (H'(G), DG), 222 (H:(x),DG,,), 207 ( v " ~ ~ (RN), DE,), 234 definition, 61

dislocations definition, 60 group of, 61

divergence, 183

equicontinuous set, 7 exponential map

Exp, on Riemannian manifold, 186 exp versus Exp, 194 exp, Lie group, 193 product formula, 194

extension domain, 37 exterior derivative, 178

flask set (G, T)-flask set, H ' ( Iw~) , 83

examples, 87 geometric condition, 85, 86

Dn,z-flask set, 134 geometric condition, 135

flask subspace, 69

F'riedrichs inequality, 37 on Carnot group, 218 geometric condition, 38, 39

Frobenius theorem, 169

Gauss formula, 183 geodesic, 186

Holder inequality, 4 generalized, 4

Haar measure, 199 on nilpotent Lie group, 199

Hahn-Banach theorem, 14 Hardy inequality, 45

with singularity on a hyperplane, 46

Hardy-Sobolev-Maz'ya inequality, 136

Heisenberg group, 190

integral curve, 168 isometry group

Iso(M), definition, 184 is a Lie group, 192 local compactness, 191

Lagrange multipliers, 20 Lie algebra, 170, 195 Lie bracket, 169 Lie group, 189, 195

action, 197 Carnot group, 196 nilpotent, 195

magnetic shifts, 207 manifold, 161, 162

co-compact, 79, 203 imbedded, 162 orientable, 178 periodic, 203 Riemannian, 181

complete, 187 metric tensor, 174 minimizer

critical scalar field equation Bliss-Talenti solution, 120

BrBzis-Nirenberg problem, 121

oscillatory nonlinearity, 119 with small potential term, 125

critical semilinear Neumann problem, 130

Hardy-Sobolev-Maz'ya inequality, 136

Nash inequality, 236 problem with nonlocal nonlinear

term, 239 Schrodinger equation with

magnetic field, 209 semilinear elliptic problem

of higher order, 234 singular subcritical problem, 133

on D,,z-flask set, 134 subcritical scalar field equation

on !RN, 75 on a flask domain, 89 comparison condition, 95, 96 flask with Friedrichs

inequality, 93 penalty at infinity, 98 penalty at infinity in the

average, 102 reverse penalty condition, 105 with discrete symmetry, 100

subelliptic semilinear problem critical, 227 subcritical, 224

mollifiers, 39 on Carnot group, 216

Mountain Pass Theorem, 144

Nash inequality, 46 in parametric form, 43, 220

operator adjoint, 17 imbedding, 19 isometric, 18 self-adjoint, 18 unitary, 18

orthogonal, 8 orthogonal complement, 10

orthogonal projection, 10 orthonormal basis, 11

Poincar6 inequality for trace on the boundary, 53 in the weak form, 41 on Carnot group, 220

Riemannian measure, 182 Riesz representation theorem, 11

Sobolev inequality critical, 44, 221 for trace on the boundary

critical, 54 subcritical, 54, 55

on periodic manifold, 204 on the half-space, 47 subcritical, 48

Sobolev space on Riemannian manifold, 205 subelliptic, 21 1 with external magnetic field, 205

solution of mountain pass type critical nonlinearity with penalty,

157 critical oscillatory nonlinearity, 155 on flask set, 149, 153 penalty at infinity, 150

solutions of mountain pass type on Carnot group, 224, 227

space of linear maps t ( V , W), 5 Banach, 2 dual, 5 Euclidean, 2 functional spaces:

C(X) , 2 CW(0) , 3 C?'(R), 3 cm ( 0 ) , 3 cEc(n), 3 Cloc(X), 3 H1(G) on Lie group, 211 H'(Q), 31 HA (x) , 205

263Index

Concentration Compactness

Hrn(S1), 57 H;(n), 31 H,"(R), 58 LP(R), 2 LP(R, w), 2 LP,,(fl), 3 'D1,'(IRN), 43 'D1"(G) on Lie group, 211 ~~s~ p N ) , 58

Hilbert, 8 intersection of Banach spaces, 3 normed, 1 separable, 13

stereographic map, 231

tangent bundle, 166 tangent map, 165 tangent vector, 164 tensor, 173

spaces @ 0 3 2 ( ~ ) and Q2,0(M), 173 topological charge, 241 trace domain, 35, 36

vector field, 166 space @(M), 167

weak convergence decomposition for (H'(G), DG), 224 for (HL (x) , D G , ~ ) , 209 conservation of topological charge,

242 for (H'(IRN), Das ) , 72 for ('D','(R~), D N , ~ ) , 113

with radial symmetry, 114 for (Vll'(G), DG,R), 227 general theorem, 62 uniqueness, 68 with isometric dislocations, 67

weak limit, 14

Yamabe problem, 232 Young inequality for convolutions, 6

264

tintarev k., fieseler k. h. - concentration compactness(2007)(264)

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