Appendix A

Proof of the Nash-Aronson Estimate

Let A = div (AV), where A(x) is a symmetric matrix defined in JRm and satisfying the inequality 1/11 ::; A::; 1/21, with constants 1/11 1/2 > o. The solution of the Cauchy problem

: = div (AVv), vlt=o = f ,

will be denoted by etA f. Set

'l/J = 'l/J(x) = ~. x, ~ E JRm, u(·,t) = e-'¢eAte'¢f, 1

f E Cg"(JRm), f(x) 2: 0 , lIulip = (JRm lu(x, t)IP dx r . Obviously

Let us multiply this equation by U2S- 1(X, t)1]2(t) , where 8 2: 1, 1](t) is a smooth non-negative function defined in [0,00). Then

{ 1]2 au U2s- 1 dx = -.!:.. i { 1]2U2s dx _ ~ ( d1] 1]U2s dx = JRm at 28 dt JRm 8 JRm dt

= 1]2(t) { (-Vu . AVU2s- 1 + ~ . AVUU2s- 1 - U~ . AVU2s- 1 + JRm

+ ~ . ~U2s) dx =

= 1]2(t) { (_ 28 - IVUs. AVus _ 2(8 - 1) ~. AVuSu8 + ~. ~U2S) dx. JRm 82 8

Setting here 8 = 1, 1](t) == 1, we obtain the following energy estimates

~ :t lIull~ < 1/21~12I1ull~,

lllu(T)II~dT < te21121{12tllfll~·

The application of the inequality

-2us~ . AVus ::; 8u2s~ . ~ + S-IVus . AVu8 ,

(A.l)

Proof of the Nash-Aronson Estimate 537

yields

Let us assume that 7](0) = O. Then

r 7]2U2S dx + rT r 7]2 1V'us I2 dx dt ~ JRm Jo JRm

sup O$t$T

(A.2)

~ 4 ((VII + 1)v21~12s2 + sup 1 ~~ I) faT iRm 7]U2s dxdt.

We recall the well-known multiplicative inequality

2(m + 2) p= ,

m (A.3)

which implies that

(A.4)

Fix to > 0 and consider the intervals

k = 0,1,2, ... ,

together with smooth functions 7]k(t), k = 1,2, ... , such that 0 ~ 7]k ~ 1 on .10 =]0, tor,

7]k = 1 on .Jk, 7]k == 0 on jo \ .Jk-l ,

Set 7] = 7]k in (A.2). Then using the inequality (A.4) we get

IlusliLP(RmX.Jkl ~ 4 (cll~ls + t~~3k) IlusIILP(Rmx.Jk_,) =

=4(CII~ls+t~~3k) Ilul~~III+'Y , LP(Rmx.Jk_,)

k = 0,1, ....

(A.5)

538 Appendix A

Then (A.5) can be rewritten in the form

k = 1,2, ... ;

the constant C2 in this inequality and the constants C3, C4, C5 in the subsequent ones depend only on Co in (A.3) and 1/1, 1/2. Hence, we conclude by induction that

Therefore

Since <1>6+' = Ilullu(lRmx[O,to))' it follows from (A.l) that

Thereby we have proved the estimate

(A.6)

as well as a similar estimate for e'l/! etA e-'I/! f. These estimates have been obtained under the assumption that f :::: 0; nevertheless, they hold for any f E L2(JRm ),

since the operator eAt is positive. Therefore we also have the dual estimate

which, being combined with (A.6), yields

2 'Y = -.

m

(A.7)

(A.S)

Let K(x, y, t) be the fundamental solution for the operator ! -A. Then it

follows from (A.8) (for details see Remark 1 below) that

(A.9)

Minimizing the last exponent, i.e., setting ~ = ~, we finally obtain

Proof of the N ash-Aronson Estimate 539

Remark A.I. Let us consider more closely the transition from the estimate (A.8) to the estimate (A.9). First of all we verify that operator eAt for t > 0 is continuous from L2(JRm ) to Co(JRm ), where Co(JRm ) is the space of functions f such that f is continuous in JRm and f(x) ---- 0 as Ixl ---- 00. For a smooth matrix A(x) this result follows from the classical theory. Let Ae be a sequence of smooth matrices such that A e ---- A almost everywhere, v1I ::; 2Ae ::; 4v2I. For f E Cgo(JRm ) and ue = eA• t f we have (see Section 2.1)

Therefore the family ue(x, t) is compact in L2(JRm ) for any t > O. This family is also compact in Co(JRm ) because of the estimate (A.6). Passing to the limit as € ---- 0 we see that

Hence, by duality, we obtain

eAt : M ____ Co ,

where M is the space of all finite Borel measures on JRm . Here the operator eAt is weakly continuous, in particular, if J.Le, J.L E M, and

then eAtJ.Le ~ eAtJ.L in L2(JRm).

Moreover, the estimates (A.6), (A.7), (A.8) still hold if we take 1jJ(x) equal to ~Ixl, V~ E JRl). Therefore etAJ.Le ---- etAJ.L (in the sense of uniform convergence on compact sets of JRm ). These arguments allow us to consider the initial value f(x) = 8(x - y), instead of the initial values in Cgo(JRm ), and therefore, to infer the estimate (A. g) from (A.8).

Remark A.2. The above proof can be easily extended to the case of operators

of the form p ! - div (A'V) with a non-symmetrical matrix A

Appendix B

Weak Convergence in L1 and Weak Convergence of Measures

Let Q be a bounded domain in lR,m, and W," Wo E L1(Q). The sequence of functions w" is weakly convergent to Wo in L1(Q) if

lim r w"cpdx = r wocpdx, 'icp E LOO(Q) . ,,~oJQ JQ

A sequence of functions WE E U(Q) is said to be equipotentially integrable if for any integer h > 0 there is a real 8 > 0 such that

for any € > 0 and any measurable set A c Q with IAI :S 8 (IAI stands for the Lebesgue measure of A).

A function h(t) (t 2: 0) is said to be coercive if it is non-negative, non­decreasing, and satisfies the condition

lim C 1h(t) = 00 . t~oo

Recall the following

Criterion of Weak Compactness in £1(Q). The following statements are equivalent:

a) the sequence w" is weakly compact in L1 (Q);

b) the sequence WE is equipotentiaUy integrable;

c) there exists a coercive function h(t) such that

sup r h(lwEI) dx < 00 ; E JQ

Generalized Lebesgue's Theorem. Let the sequence WE be equipotentially integrable and convergent to Wo almost everywhere on Q as € --+ O. Then Wo E

U(Q) and w" --+ Wo in U(Q).

The proof of the above statements can be found in: Dunford & Schwartz [1], Ekeland & Temam [1], Natanson [1].

Weak Convergence in L1 and Weak Convergence of Measures 541

One of the useful implications of the condition c) is that for any w E £1 (Q), there exists a coercive function I.{J such that 1.{J(lwl) E £l(Q). Of course, this result is very simple and can be proved quite easily on the basis of the following observation: for any given numerical series

00

Lan<oo, an;:::O, n=1

there exists a sequence An --+ 00 such that

In what follows, we consider finite non-negative Borel measures f..l on IRm. If fIRm df..l = 1 then f..l is called probability measure.

Definition B.1. A sequence of probability measures f..le is weakly convergent to measure f..l, f..le ~ f..l, if

(B.l)

for any continuous bounded function I.{J defined on IRm (then f..l is a probability measure, too ).

The following simple result can be very helpful in some situations.

Proposition B.2. Let f..le and f..l be probability measures, and let the relation (B. 1) hold for any I.{J E Cij(IRm); then f..le ~ f..l.

The distribution of an m-dimensional random variable 7] is, by definition, a measure f..l given by

f..l(A) = P{7] E A}

for any Borel set A c IR m .

Let 7]e,7] be random variables in IRm with distributions f..le, f..l, respectively. If f..le ~ f..l, then 7]e is said to converge in distribution to 7],

D 7]e --+ 7] .

A detailed exposition of weak convergence of measures and its properties can be found in Billingly [1], as well as in many other manuals on the Theory of Probability.

Appendix C

A Property of Bounded Lipschitz Domains

Let Q be a domain in IR m

Definition C.l. Q is said to be a Lipschitz domain in an open ball B, if there exist a vector v E IRm , Ivl = 1, and a constant c < 1 such that

v·(q-p):=;clq-pl, VqEBnQ, VpEB\Q. (C.1)

Definition C.2. Let Q be a bounded domain in IRm; then Q is said to be a Lipschitz domain, if its boundary can be covered by open balls such that the intersection of Q with each ball is a Lipschitz domain in that ball.

It can be easily verified that Definition C.2 is equivalent to the usual definition of Lipschitz domains in terms of epigraphs (see Ekeland & Temam [1]).

Let I be the identity mapping from IRm to IRm.

Lemma C.3. Let Q be a bounded Lipschitz domain in IRm. Then there exists a sequence of diffeomorphisms ej : IRm --+ IRm such that

ej(Q) c Q , lim Ilej - Illc2(JRm) = o. J~OO

(C.2)

Proof. Let us fix a system of balls covering the boundary of Q and such that Q is of Lipschitz type in each ball. Let B be anyone of these balls. Consider a function cp such that

cp E Cg'(IRm), cp > 0 III B, cp == 0 outside B .

Then the mapping 1

h(x) = x - --: cp(x)v J

is infinitely differentiable and satisfies the inequality

Ih(x) - h(y)1 2: Ix - yl (1 -} max Icp/l)

Therefore, the mapping h: IRm --+ IRm is injective, if j is sufficiently large. Taking into account the smoothness of h, we see that h is a diffeomorphism. Since h = I outside B, therefore h(B) = B.

Let us establish the inclusion

h(Q n B) c Q. (C.3)

A Property of Bounded Lipschitz Domains 543

Assuming the contrary, we can find a vector q E Q \ B such that h( q) E JRm \ Q. Therefore h(q) E B\ Q, since h(B) C B. According to the definition of Lipschitz domains, we have

v· (q - h(q)) ~ clq - h(q)1 ,

or <p(q) ~ c<p(q), which is impossible, since c < 1, and q E B implies that o < <p(q). Thereby the inclusion (C.3) is proved.

Taking into account (C.3) and the fact that h coincides with the identity mapping outside B, we obtain the inclusion h(Q) C Q.

Now, let {Bkh=l, ... ,N denote the above system of balls covering the bound­ary of Q, and let hk be the respective diffeomorphisms constructed for the same sufficiently large j.

It is clear that, as j -+ 00, the diffeomorphisms

converge to I with respect to the norm in cn(JRm ), where n is an arbitrary positive integer.

It remains to verify that OJ(Q) C Q. If q E Q, then, because of the inclusions hk(Q) C Q, k = 1,2, ... , N, we have OJ(q) C Q. If q E 8Q, then there exists the smallest k E {I, 2, ... ,N} such that q E B k • Therefore, by virtue of (C.3), we have hk(q) E Q. Hence

Thereby Lemma C.3 is proved. o

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Subject Index

abstract energy criterion, 162 almost-periodic function, 238 auxiliary - equation, 228, 325 - periodic problem, 17, 89, 100 - variational problem, 440

Barry-Essen estimate, 78 Bloch representation for the fundamen­

tal solution, 79 bulk modulus, 369 BVo(Q) space, 507, 521

Caratheodory condition, 420 central limit theorem, 61, 78, 269, 319,

324 - on the average, 270 chess Lagrangian, 438 closeness theorem, 65, 283, 335 compensated compactness, 3, 138, 230,

373 condition of cubic symmetry, 39 conductivity threshold, 313 convergence - in distribution, 61, 541 - of arbitrary solutions, 14, 151, 181 convex function, 415 conjugate function, 416 complementary energy, 428 critical probability, 304, 316

density of states, 357 deviatoric trace, 391 Dirichlet problems of type I and type II,

424 disperse media, 86, 279, 407 dual - analogue of r-convergence, 167

- boundary value problems, 428 duality - formula, 428 - principle, 430 Dychne formula, 455

effect of compressibility, 126 effective - conductivity, 93, 299, 443 - homogeneous medium, 12 elasticity tensor, 368 ergodic algebra, 243, 247 equivalence principle, 509 Euler equation, 8, 18, 40 extension - condition, 271 - property, 113, 117, 265 extremal relation, 431

Friedrichs inequality, 2

G-convergence, 149, 160, 180 r-convergence, 162 - of Lagrangians, 461 - of integral functionals, 499

Hashin-Strikman bounds, 187 Hashin structure, 195, 403 Hill material, 379 homogenization rule, 14 homogenized - differential equation, 12 - Lagrangian, 438, 440 - matrix, 12

incompressible elasticity, 382 infinite - conducting cluster, 314 - superconducting cluster, 318

570

Lavrentiev phenomenon, 423 Lax-Milgram lemma, 7, 180 L1-closedness, 482 limit load, 503, 508

Marchenko-Khruslov problem, 124 micro-nonhomogeneous medium, 12

Nash-Aronson estimate, 63, 272, 536 non-defective Lagrangian, 510 null-Lagrangian, 214

optimality criterion, 192

phase interchange equality, 213, 237 Poincare inequality, 3, 153 polarization tensor, 108 P6lya-Schiffer theorem, 109, 189 principle of periodic localization, 155,

206 property of the mean value, 5

quasi-convex function, 490 quasi-periodic function, 226

random - Lagrangian, 438 - motion, 61 - set, 250

Subject Index

Rayleigh-Maxwell formula, 45 regular Lagrangian, 424 relaxation principle, 488 residual diffusion, 23, 36 resistence threshold, 318

spherical trace, 391 shear modulus, 369 Sobolev space, 1 Sobolev-Orlicz space, 425 stabilization criterion, 67, 282, 324 standard Lagrangian, 421 stationary random field, 222 stratified media, 15, 168, 209, 355, 377 Sturm-Liouville problem, 349 surface - limit load, 518 - of fluidity, 508

theorem on conjugate functionals, 422

variational - criterion of r-convergence, 497 - method, 202, 395 virtual mass tensor, 108

*-weak convergence, 4

Encyclopaedia of Mathematical Sciences Editor-in-Chief: R. V. Gamkrelidze

Partial Differential Equations

Volume 30: Yu. V.Egorov, M.A. Shubin (Eds.)

Partial Differential Equations I Foundations of the Classical Theory

1991. ISBN 3-540-52002-3

Volume 31: Yu. V.Egorov, M.A.Shubin (Eds.)

Partial Differential Equations II 1992. ISBN 3-540-52001-5

Volume 32: Yu. V.Egorov, M.A.Shubin (Eds.)

Partial Differential Equations III The Cauchy Problem. Qualitative Theory of Partial Differential Equations

With contrihutions hy S. G. Gindikin. V. A. Kondrat'ev. E. M. Landis, L. R. Volevich

Translated from the Russian hy M. Grinfeld

1991. ISBN 3-540-52003-1

Volume 33: Yu. V.Egorov, M.A.Shubin (Eds.)

Partial Differential Equations IV Microlocal Analysis and Hyperbolic Equations

Translated from the Russian hy P. C. Sinha 1993. ISBN 3-540-53363-X

Volume 63: Yu. V.Egorov, M.A. Shubin

Partial Differential Equations VI Elliptic and Parabolic Operators

1994. ISBN 3-540-5467/1-2

Springer

D.Revuz, M. Yor

Continuous Martingales and Brownian Motion 1991. (Grundlehren del' mathematischen Wissenschaften, Bd. 293) ISB~ 3-540-52167-4

M. Ledoux, M. Talagrand

Probability in Banach Spaces Isoperimetry and Processes

1991. (Ergebnisse del' Mathematik und ihrer Grenzgebiete, 3. Folge. A Series of Modern Surveys in Mathematics, Vol. 23) ISB~ 3-540-52013-9

Ya. G. Sinai

Probability Theory An Introductory Course

Translated from the Russian by D. Haughton

1992. (Springer Textbook) ISBN 3-540-53348-6

M. V. Fedoryuk

Asymptotic Analysis Linear Ordinary Differential Equations

Translated from the Russian by A. Rodick

1993. ISB~ 3-540-54810-6

Springer