estimates near the boundary for solutions i

COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS, VOL. XII, 623-727 (1959)

Estimates Near the Boundary for Solutions of Elliptic Partial Differential Equations

Satisfying General Boundary Conditions. I * S. AGMON, A. DOUGLIS, L. NIRENBERG

Hebrew The University New York University of Maryland University

Introduction

The Schauder estimates [33, 341 for second order linear elliptic differential equations play an important role in the existence theory for linear, and in particular, for non-linear elliptic equations. These estimates are for solutions of a second order linear elliptic equation in a domain %, with Holder continuous coefficients1 and are of two kinds. The first kind, “interior estimates”, give bounds for the derivatives up to second order of the solution u, and their Holder continuity, in any compact subset of the domain, the bounds depending on 1.u.b. IuI and on the moduli and Holder continuity of the coefficients of the equation. The second kind, “estimates near the boundary”, apply to solutions of the Dirichlet problem. Assuming the existence of Holder continuous first and second derivatives of the solution on and near a smooth portion r of the boundary 9, one estimates these quantities in a suitable sub-domain of % abutting r, the bounds depending as before on 1.u.b. 1 ~ 1 etc., and on estimates for the boundary values of u.

Simplified derivations of the estimates of Schauder have been given by Douglis and Nirenberg [lo], Morrey [28], Miranda [25] and by Graves [la], where a comprehensive discussion of the estimates and existence theory is

* A major part of the work for this paper was done in 1955-1956 a t the Institute of Mathematical Sciences under U. S. Army Ordnance Contract DA-30-069-1253 and the National Science Foundation, Contract Nos. NSF G 1217 and NSF G 1973. During the later work, A. Douglis was supported by Contract A F 18 (600)-673, Air Force Office of Scientific Research at the University of Maryland. L. Nirenberg was a Sloan Fellow during the academic year

‘A function g in a set in Euclidean space is said to satisfy a Holder condition with ex- 1958-1959.

ponent a, 0 < a < 1, if

g is Holder continuous (exponent a) in a domain if it satisfied a Holder condition with exponent a in every compact subset of the domain.

623

624 S. AGMON, A. DOUGLIS AND L. NIRENBERG

presented. Jn [lo] the “interior estimates” were extended to general elliptic systems of arbitrary orders. In this paper we shall derive “estimates near the boundary” for elliptic equations of arbitrary order under general boundary conditions, not merely the Dirichlet boundary conditions.2 We have obtained these results for general elliptic systems, but for simplicity, we treat here in detail the theory of a single equation for one function. Systems will be treated in a forthcoming paper. We propose also to treat at some later time pointwise estimates near the boundary for overdetermined systems and boundary conditions.

In addition to the Schauder estimates we present analogous L, estimates, for p > 1, up to the boundary. Interior L, estimates can be obtained rather easily as has been noticed by several authors (KoSelev [ 191, Nirenberg [32], Browder [7]; also Greco [13], and KoBelev [lS] for estimates up to the boundary for second order elliptic equations) with the aid of the Calderon- Zygmund theorem (see [S] and Theorem 2 of [O]). These boundary estimates have been obtained by us only recently; previously we proved such estimates only for p = 2. L, estimates up to the boundary for elliptic boundary value problems have been announced earlier by Browder

Recently KoSelev established L, estimates up to the boundary for the Dirichlet problem for elliptic equations of arbitrary order. We use our L, estimates (see Section 11) in proving differentiability of solutions of nonlinear problems.

For the case p = 2 our result has now been derived by several people. Subsequent to our work Schechter [35] derived similar estimates for overdetermined systems which agree with ours in the case of one operator. The L, estimates overlap also with unpublished inequalities of Aronszajn and Smith extending [4]. Agmon [ 2 ] , and independently Hormander, have solved a general coerciveness problem (extending the fundamental paper [4] by Aronszajn) involving L, estimates which include ours, for p = 2, as a special case. Some of his results have recently been extended by Agmon to L,, p > 1. Our method for deriving the L, estimates, even for f i = 2, differs considerably from the others and may be carried out fairly easily with the aid of a potential theoretic result, Theorem 3.3-once the explicit formulas for solutions of equations with constant coefficients, in a half-space, are known.

2Estimates near the boundary for other than Dirichlet boundary conditions, including cases where the coefficients of the boundary operators are discontinuous (“mixed” boundary conditions) have been obtained by C. Miranda [26] for second order equations.

3Since this paper went to the printers there has appeared a more complete announcement of Browder’s results which overlap considerable with ours. Estimates and existence theorems for elliptic boirizdary value problems, Proc. Nat. Acad. Sci. U.S.A., Vol. 45, 1959, pp. 366-372.

BOUNDARY ESTIMATES FOR ELLIPTIC EQUATIONS 625

We shall consider a linear differential operator L with complex coefficients operating on functions uix) defined in a domain 3 in (7zfl)-space: x = (x,, - * * , x,+,). L is assumed to be elliptic, i.e., if L’(x; D ) = L’(z; a/axl , - , a / a ~ ~ + , ) is the leading part of L (the part of highest order), then for every real vectoi S = (t,, - . * , En+,) # 0 and for every point x in 9, L’(x; 5) # 0. In case n > 1 it is easily seen that the order of L is even (Lopatinskii [21]). For this purpose set = (6, , - - . , t,), 5,+1 = t. For fixed real 5 # 0 consider the roots t of the polynomial L.’(x; t, t). Clearly L’(x; -5, -7) = 0, and since the set of real non-vanishing 5 vectors is connected for n > 1 it follows that there are as many roots t of L’(x; t, t) with positive imaginary parts as with negative. Hence the degree of L‘ is even, equal to 2m. In the sprtcial case of two variables, n = I, this is not necessarily true, as we see for instance with the Cauchy Riemann operator L = E S i a .

In this paper we shall consider only operators L of even order 2m. In the case of two variables we impose a’ condition on L :

For every point x in the closure 3 of 9 we assume the

CONDITION ON L. For every $air of linearly independent real vectors S, 8’ the polynomial in the variable t, L’(x; S+tZ), has exactly m roots with positive imaginary part.

We shall actually use this condition only at points x on the boundary of 9, and for 9, S’ the tangent and normal vectors, respectively, to $.

Because of ellipticity, however, the condition on L is invariant under de- formation and change of sign of the vectors, and so holds generally provided it is satisfied at a particular point for some pair of independent vectors,

We shall also assume that L is uniformly elliptic, i.e., that for some positive constant A , the “ellipticity constant”, the inequality

(1) A-1(8(*” 5 IL’(x; 3)l 5 A ( 5 p holds for every real vector S, with 1-31 = (2 ,$)%, and for every point x in the closure 9 of 9.

We shall consider solutions ~ ( x ) of (2) LU = F satisfying m boundary conditions on a portion r of the boundary 9. These con& t ions (3) Bjzt = CDj, j = l ; .* , rn , are expressed by differential operators B,(x; D ) , i = 1, * , m, defined for x in r. Each B, is of order m, and has Lomplex coefficients; denote its highest order part by S; , j = 1, * * *, m. The mj are non-negative integers, and may be greater than 2m, the order of m. We shall impose a certain algebraic


condition on the operators L’, Si ,and shall then call the Bj “complementing boundary conditions”. The algebraic condition is the following

COMPLEMENTING CONDITION. At any point x of rlet n denote the normal to s;d and S # 0 any real vector parallel to the boundary. W e reqzlire that the $01~- nomials, in z, x’;(x; Z+zn), j =’ 1, * * - , m, be linearly independent moddo the polynomial nr=l (z--ti(S)), where z$(8) are the roots of L’(x; E+tn) with positive imaginary parts.

Without imposing any conditions, except an appropriate degree of smoothness, on the lower order coefficients of L, B, we shall derive estimates up to the boundary for solutions of (2), (3) satisfying the conditions above, and also prove differentiability theorems for solutions near the boundary. In addition we show that these tonditions are necessary for such estimates to hold.

Hormander [15] has considered solutions of an equation with constant coefficients satisfying, on a plaqar portion of the boundary, conditions (3) with constant coefficients. He ‘characterized completely, for hypoelliptic operators L, these operators B, (with constant coefficients) for which solutions u of ( 2 ) , (3) with F and Qj in C” are necessarily themselves of class C”, and characterized for elliptic L those operators B, for which the solutions are analytic at the boundary, in case F and the are analytic. In case the operator L is elliptic, and L and the B j are homogeneous, his condition on the B, agrees exactly with the complementing boundary condition above. In the more general (non-homogeneous) case which he treats his condition also involves the lower order terms of the B, . In order to obtain our estimates near the boundary for arbitrary lower order terms, our condition is however, as remarked above, necessary as well as sufficient.

Certain simple elliptic operators such as the Cauchy-Riemann operators mentioned before under,‘ say, the condition that 9 e u = 0 on the boundary, seem to be outside the scope of our theory for single operators. Although this is not a linear condition over the complex numbers it seems worthwhile to have a thecg-y which includes this case. And in fact, as M. Schechter pointed out to us, if we break up the Cauchy-Riemann equation into real and imaginary parts (the way they are usually written) we find that they are covered by our theory for systems.

In deriving the Schauder estimates near the boundary we shall rely on the interior estimates given by Douglis, Nirenberg in [lo]. These were derived for equations with real coefficients, and the proofs do not extend directly to equations with complex coefficients. However, any elliptic equation (or system) with complex coefficients may, on separation into real and imaginary parts, be written as a system with real coefficients, and this system is also elliptic. Thus we may safely refer to the results of ClO].

BOUNDARY ESTIMATES FOR ELLIPTIC EQU.4TIONS 627

We mention that the boundary qperators for the Dirichlet problem

where n is the normal to the boundany, satisfy the “Complementing Con- dition” relative to any operator L for which the “Condition on L” holds. In [15] Hormander characterizes all those operators B, which satisfy the Com- plementing Condition relative to every operator satisfying the Condition on L.

This paper is primarily concerned with estimates. However, these sug- gest the possibility of new existence theorems. In particular, the fact that our estimates hold in the Dirichlet problem for any elliptic operator (for n = 1 we require the Condition on L ) wggests that it should be possible to derive existence theorems for such operators L, not merely for operators that are strongly elliptic, to which, up to now, the existence theory for the Dirich- let problem has been confined. Suppose for example that L has highly differentiable coefficients so that, in pkrticular, the formal adjoint L* is well defined. Let I I ~ t j l ~ , ~ , denote the square root of the sum of integrals of the squares of all derivatives of u up to order j , and denote by H1,L, (&5,Ls)

the completion of C” functions (with compact support) in 59 with respect to this norm, 1 1 l l j , L s . A direct (Hilbert space) approach to the Dirichlet problem would be: For given F in HO,Ls = L, , to find a function zt in S = H 2 m , L , n Hm,L, satisfying

Ltt = F , if one assumes, say, that the only solution in S of L* u = 0 is u = 0. From our estimates it follows easily that the range of L : S --f L, is closed. In trying to show that the range is dense we would be led, on supposing the contrary, to a function v in L, such that (Lu, v ) = 0 for all u in S. Here ( , ) is the L, scalar product in 9. Thus v is a “weak’solution” of L*v = 0. If we had a differentiability theorem at the boundary asserting that such a weak solution belongs in fact to S, we would know by uniqueness for L* that v = 0, hence that the range of L is dense. Thus the direct Hilbert space approach seems to require a strong differentiability theorem at the boundary for weak solutions.

Recently Agmon [3], using some results of this paper (in particular Section 2) has proved such a differentiability theorem at the boundary for weak solutions of a wide class of boundary value problems, and hence has obtained existence theorems for such problems-in particular for the Di- richlet problem. illso, M. Schechter 1361, using a different and elegant Hilbert space argument which does not require such a strong differentiability theorem, has independently proved existence theorems for a wide class of boundary problems, including also the Dirichlet problem. These proofs all make use of the L , estimates up to the boundary.

628 S. AGMON, A. DOUGLIS AND L. NIREiNBERG

Section 12 is concerned with straightforward applications of the estimates up to the boundary-chiefly to esistence thlzorems. In particular, using the above results of Agmon, Schechter we show, under optimal conditions on the coefficients, that uniqueness for the Dirichlet problem for L implies existence. We also describe (Theorem 12.5) the approach to existence theory via the continuity method (in a slightly more general form than that used by Schauder {33]) ;in this connection some open questions are formulated.

Before giving a detailed description of the results of the paper we wish to point out some of the other main items. The basic work is concerned with systems (2), (3) having constant coefficients and only highest order terms, and for a domain '3 which is a half-space. We obtain representations for solutions of such a system with the aid of explicitly constructed Poisson kernels. For two dimensions, n = 1, such kernels were constructed by Agmon [I]. In treating the general case we make use of an identity of F. John [17].

From these formulas we obtain, in the case of Dirichlet boundary data for a homogeneous operator with constant coefficients in a half-space, an extension of the maximum principle for second order operators. This asserts that if u(p ) is a solution of the homogeneous equation and is (for simplicity) of class C" in the closed half-space, and if u(P) = O(IPI"-l) for large IP1, then

1.u.b. IDm-luj 5 constant 1.u.b. lDm-lu/, %

(4)

provided the right side is finite. Here 1.u.b. refers also to all derivatives Dm-l of the order m-I. We have not generalized this to equations with variable coefficients (however, in Section 9 we prove a general result involving the Holder norms of the derivative's of order wz-1). Recently Miranda [37] obtained a general result of the nature of (4) for strongly elliptic equations with variable coefficients in two dimensions; he uses the results of Agmon [l]. Previously he had proved such a result for the biharmonic equation in two dimensions [24].

In Section 3 we present some general potential theoretic results concerning certain convolution operators taking functions of n variables into functions of n+l variables. Two of these are generalizations of results of Privaloff and M. Riesz. Except for Theorem 3.4 this section may be read without reference to the rest of the paper, and the results there, we feel, should prove useful in other problems.

We call attention also to Lemmas (9.1), (9.1)', (9.1)", and to Appendix 5 which is never referred to in the rest of the paper. Here we show that solutions having square integrable derivatives are in fact classical solutions


to which the interior estimates of [lo] apply. Other appendices contain proofs of some of the results in the text.

One of the main features of the Schauder estimates is their use in nonlinear problems. In Theorem 12.5 we prove a local perturbation theorem for solutions of nonlinear elliptic equatfons involving a parameter. It is here that one sees the strength of the estimates.

We have included (Chapter 111, and part of Chapter V) estimates tor equations in integral, or variational, form. Tbe remainder of the paper may be read independently.

Outline

Chapter I is concerned with the system with constant coefficients in the half-space, and potential theory.

Section 1. Weformulateinapreciseway theboundaryvalueproblem (2), (3) (with F = 0 ) in the half-space and do some preliminary algebra.

Section 2. We construct the Poisson kernels, and also prove the extended maximum principle, which is a special case of a more general result, Theorem 2.2.

Section 3. The potential theory. Section 4. A general representation formula is obtained for solutions of

( 2 ) , (3) (with F # 0) with the aid of the Poisson kernels and the fundamental solution for elliptic equations with constant coefficients.

Chapter I1 deals with the Schauder estimates. For I 2 I , = max (2m, m j ) we estimate the Holder continuity of derivatives up to order I of solutions of (2), (3) near the boundary in terms of 1.u.b. I%] and the given data F , Oj.

5. We introduce the basic norms and state some simple calculus lemmas.

6. With the aid of the potential theoretic results of Section 3 and the explicit formulas of Section 4 we prove the Schauder estimates for solutions of the equations with constant coefficients in a half-space considered in Chapter I. We also prove a general Liouville theorem for solutions of such equations.

Section 7. The Schauder estimates for equations with variable coefficients in general domains. Theorem 7.3.

Chapter 111. We prove Schauder estimates for equations in integral, or variational, form such as arise from regular variational problems. These involve estimating derivatives of lower order than I,.

Section

Section

630 S. AGMON, A. DOUGLIS AND L. NIRENBERC

Section Section 9. Equations with variable coefficients. Our main result, Theo-

rem 9.3, contains as a special case for the Dirichlet problem the result that if we have an estimate of all derivatives up to order k 2 ?n--1 on the boundary, and of their Holder continuity (as functions on the boundary), then we can estimate these functions and their Holder continuity in the entire domain.

Chapter I V is concerned with applications of the estimates and some comments. Section 10. We prove that the Condition on L and the Complementing

Condition are both necessary for our estimates (Schauder and L,) to hold,

Section 11. Differentiability at the boundary for solutions of nonlinear elliptic equations satisfying nonlinear boundary conditions. The first variation of the equation and the boundary condition is required to satisfy (locally a t the boundary) the Condition on L and the Complementing Condition.

Section 22. We first prove some obvious consequences of the Schauder estimates, such as the compactness of solutions and the finite dimensionality of the solutions of the homogeneous system. In Theorem 12.5 we describe the continuity method. We then prove Theorem 12.6, the perturbation theorem for nonlinear elliptic equations.

The rest of Section 12 is concerned mainly with the Dirichlet problem which we solve under mild hypotheses on the coefficients. For weakly positive elliptic equations we obtain a unique solution for the problem in case the Dirichlet data (13’/8n)j-~u = @, belong to Cm-j+u, j’ = 1, . a , m. The solution belongs to Cm-l+a in B. See Theorems 12.10, 22.11.

Section 13. We sketch a general estimate of Schauder type for a certain class of semilinear elliptic equations (see Nagumo [30]).

Chapter V deals with the L, estimates a t the boundary. Section 14. The constant coefficient case in the half-space. Section 15. The general case.

We wish to thank Dr. M. Schechter for a number of stimulating dis- cussions, and particularly Professor Hormander for his many helpful and constructive suggestions.

8. The constant coefficient case in a half-space.


CHAPTER I

EQUATIONS WITH CONSTANT COEFFICIENTS I N A HALF-SPACE AND POTENTIAL THEORY

1. Preliminaries

We shall consider functions defined in domains of (n+ 1)-st dimensional Euclidean space, E 2 1, with coordinates xl, - - ., xn+l and use the notation

a axiJ D . = -

Greek letters p, y , p, Y, will be used to &note vectors with integral components pi 2 0; however, E will simply denote a vector r" = (tl , * - , tn+*) with scalar components, and we shall write

5 1 nfl J n+1

= (B1, - * * ,

Sfi = 28, . . . EBn+l DB = @I . . . D b n + l , =; 2 B1 .

Thus a typical differentiation operation of order 1 will be written DB with For convenience, when there is no danger of confusion, we will

often denote such an operator by Dz. The letter cc will always denote a fixed positive number less than one. Much of the work will be concerned with functions defined in a half-

space given, say, by x,+~ 2 0. In that case, we shall write t = x,+~ and x = (xl , * * , x,), and shall denote the points by P = (2, t ) . Vectors with n + l components will be denoted by ( E , z), where [ = (tl , - * - , En). D, will denote differentiation with respect to t , and D, = (Dl , - - - , D,) differentiation with respect to x l , * * * , x, , DL or Di representing higher differentiations with respect to the xi. If P = (x, t ) , Q = (y, t), we introduce the Euclidean distance

= 1.

JP-QI = (l~-ylg+ ( t - ~ ) 2 ) ' ,

lx-yI2 = 2 (x i -y i )2 .

The scalar product of real vectors x, y will be denoted by x * y = 2 x, yi . Z = Z, will denote the hemisphere in x, t-space given by IxI2+t2 < y2 ,

t 2 0, and a = a, its planar boundary: 1x1 < Y , t = 0. A good part of the paper will deal with an elliptic operator

L (D) = L (Dz , D t ) with complex valued constant coefficients and having only terms of highest order. This will act on functions defined in a half-space t 2 0. The corresponding characteristic form L(E, z) is different from zero for all real nonvanishing (6, T). For real 5 # 0 consider the roots t of the polynomial L(6, z). If n 2 2, it follows easily that there are as many roots with positive imaginary parts as with negative imaginary parts. For, if z is a root for given E , then -t is a root for - E , and the assertion follows from


the fact that for n > 1 the set of real vectors t f 0 is connected. Thus in particular L is of even degree 2m. In the special case of two variables, n = 1, this is not necessarily true, the Cauchy-Riemann equations L(5, z) = [+it being a counter-example. As in the introduction, we shall impose the following

CONDITION ON L. L(5, z) i s of even order 2m, and baas, for each real 6 # 0, exactly m roots z with positive imaginary parts.

We shall also assume that L is uniformly elliptic, i.e., that for some constant A , the "ellipticity constant", the inequality

holds for all real (t, 7). (1.1) A-1(1512+22)m I IL(t, .)I 5 A(ItjZft2)"

A number of quantities can be estimated in terms of A and m. Writing 27n

LO, T) = 2 CJ5)zZ-d ' i 4 .

we see from (1 .1) that A-1 5 C, 5 A , and we find successively for i = 1, 2, * * , that for = 1, iC,(t)i 5 constant depending only on A and m. From this it follows easily that there is a constant C, depending only on A and m, such that the roots ~ ( t ) of L(5, t) for real 5, 151 = 1, satisfy

(1.2) IYm T(gl-1, I.(t)I 5 c. Denote byt t (E) ( T ; ( E ) ) , k = 1, * - - , m, the roots of L ( t , z) with positive

(negative) imaginary parts, and set

(1.3) 1 p=O

We observe that

(1.4) a+([, 2)'= (-l)rnM-(-& -2).

It is readily seen that the coefficients a:@') are analytic functions of E for real nonvanishing t and are homogeneous of degree $.

With M+(5, z) we associate the polynomials (in z) of degree j

@(t, 2) = c .,+(t)z? We claim, and this is the reason for their definition, that they satisfy

j

p=O j = 0, - - a , m-1. (1.5)

where y is a rectifiable Jordan contour in the complex plane enclosing all the roots ~,f( t ) in its interior; d,, is the Kronecker delta.

Proof: By deforming the contour to a large circle about the origin whose


radius goes to infinity, we see immediately, since the highest order term of .lf2-l-jtk is a0([)trn-l-3fk, that (1.6) holds for j 2 k . For j < k the polynomial M;j;-l-j tk differs from zk-j-lM+ by a polynomial Q of degree at most k-1, so that the left-hand side of (1.6) equals

Since the degree of Q = k - 1 is less than m- 1 we find, on deforming y again to a large circle, that this integral is zero.

We consider now certain boundary operators given by m differential operators with constant (complex) coefficients H,(D) of order m, with no lower order terms, j = 1, * * * , m. Our aim in this and the next section is to study the following

BOUNDARY VALUE PROBLEM. Let #,(x), j = 1, - . ,, m, be C" functions of compact sz$port in n-space. We seek a C" function u(x, t ) in the half-space t 2 0 satisfying

In order that this problem be well posed we shall, as in the introduction,

COMPLEMENTING CONDITION. For every fixed real E # 0 the polynomials

In other words, for fixed 5 # 0, let

impose the following algebraic condition on the operators L, B j :

Bj([, t ) ( i p i t) are linearly iadefiendent mod ill+([, t).

m B:(E, t) = 2 bjk(E)tlc-l = El,([, t) (mod M+)

k=l (1.8)

be the remainder when B,@, z) is divided by M+(E, t) (each considered as a polynomial in t). Then, the Complementing Condition means that

d(5) = det Jlbjk(E)j\ # 0 for real E # 0.

We note that d(E) is analytic for real E # 0, and we introduce the deterrni- nant constant

(1-9)

which is positive if the Complementing Condition is satisfied. Remark. Because of (1.4) the Complementing Condition is equivalent

to the condition that the B, are linearly independent mod M-([ , z) for every real 5 # 0.

In solving the boundary value problem (in Section 2) we shall make use of the following: For fixed real 5 # 0 there exist polynomials (in T) N j ( [ , t ) ,


j 1 1, - -, m, with coefficients which are analytic in E for t red # 0, such that

(1.10)

where y is a contour in the upper half of the complex plane enclosing all the roots of iW+(f, t) in its interior. Indeed, let llbjk((E)ll be the inverse matrix of the b,,(E). Then the polynomials, in z,

m

= 2 by' b j p d P a (in virtue of (1.6)) 4, P

= b P k b,, = 6 j k . P

We complete this section by examples showing that, if either the Con- dition on L or the Condition on the B, is violated, then there are solutions of the homogeneous system (1.7), in which 4, = 0, with bounded derivatives up to any given order I , but such that the Holder coefficient in the Holder condition (with any exponent u < 1) for derivatives of order I is as large as one pleases in any half-neighborhood of the boundary t = 0.

Suppose first that the Condition on L is satisfied but that the Comple- menting Condition on the B, is violated. Then there is a real unit vector t such that the polynomials B, ( E , t) (in t) are linearly dependent mod M+ (6, t). Hence, it follows easily that there exists a function v ( t ) + 0 satisfying the following differential equation and initial conditions:

M+(& --iD,)v = 0

[B,( f , --iD,)v],=, = 0

for t 2 0, for i = 1, - - - , m.

(1.12)

Since each zero of Af+(E, z) has a positive imaginary part, v and its derivatives tend to zero exponentially as t + + co. Obviously, given an integer I 2 0 there exists a positive number a such that D:v(a) -0: v ( 0 ) = c # 0. For ;1 2 1 define (1.13) vA(x, t ) = il-lefAE'"v(;1t).

Since L and B, are homogeneous we see that vA satisfies (1 .7 ) with dj = 0. Also, vA and its derivatives up to the order I are bounded uniformly for t 2 0, whereas


blows up of the order of ;Ia as il --f 00.

for 1.2 = 1). Define also in this case Suppose now that the condition on L is violated (which is possible only

where $1 , . * * , T$) are the ~(6) roots with positive imaginary part of L(E, T ) . Thereexistsin this caseaunitvector 5 = 5 1 such that r (5) 2 m f l . Since, for such 6, Ill+([, T ) is of degree 2 m + l , whereas the number of boundary operators is m, it follows that there exists a function v ( t ) $ 0 which satisfies (1.12). The functions vA(z, t ) defined by (1.13) again are not subject to the estimates in question.

2. Solution of the Boundary Value Problem in a Half-Space; The Poisson Kernels

We consider the problem (1.7) and assume that the Condition on L and In terms of the the Complementing Condition on the B j are satisfied.

polynomials N,, of (1.11) we introduce in the half-space t > 0 the

POISSON KERNELS. For mj 2 n,

Here, and throughout, the principal branch of the logarithm in the complex plane sIit along the negative real axis is taken, dm, is the area element on the unit sphere 161 = 1, and the p p are absolute constants given by

(2.3)

1 = - (2xi)"(m,-.)!

if m, 5 n,

Finally y is a Jordan contour in Y m t > 0 enclosing all the roots of M+(E, t) for all 161 = 1.

With the aid of the Poisson kernels we can write down the explicit solution of the boundary value problem (1.7).


THEOREM 2.1. The fzmct ion

% ( X I t ) = z: Kj(x-9, t)4j(Y)dY = 2% * 4, i s (2.3)

is a solut ion of problem (1.7). Here integration is over the full n-space, * denotes convolution. The

cjj , we recall, are C" functions of compact support. Our construction of the Poisson kernels, and their use in proving

Theorem 2.1, are based on an identity due to F. Jo'hn and used extensively in his book [17] (see p. 11). This asserts that a differentiable function c j with compact support in E, may be represented in terms of plane waves by the formula4

where q is any positive integer of the same parity as n, A , represents the Laplacean 05 , and dm, represents the area element on the unit sphere IEl = 1.

Before proving the theorem we apply the identities

valid for integral A, p, the CA, f i being appropriate constants, to represent the Poisson kernels K j in the form

(2-6) Ki(Z, t ) = dpf*)'2Kj,JX, t ) , q a = d ,K j ,a+2 .

Here q is a non-negative integer with the same parity as n and, for m, 2 n,

Kj.* =

(2.6)'

and, for mi < n,

'The formula in [17] is stated just for real valued f(z), and the real part of the right-hand side is taken unnecessakily. The apparently more general formula (2.4), valid for complex f (z) , is, however, an immediate consequence of John's.


p .( - l)l+W-"

2ni(mj+q) ! (%-?,aj- 1) ! K . = __!_.

3 3 q

( 2 . G ) "

________ log -1 dz * &+tz i I. N,(E, z)(x * t+tT)rn'+Q

. L1 [J:, M+(E, t) It is easily seen that K j , q and all its derivatives up to order mj+q-1

are continuous in the closed half-space t 2 0 (the same is true of derivatives of order mj+g if n. 2 2).

Proof of Theorem 2.1: Inspection shows that the Poisson kernelsK,(x, t ) , and the kernels K j , q , are analytic solutions of Lu = 0 for t > 0. Hence u(x , t ) is an analytic solution of the same equation in f > 0. Setting

(2.3)j uj = Kj * 6, it will suffice to show that uj belongs to C" in t 2 0 and that (2.7) B,(D)uj = dkj+j(x) for t = 0.

Consider any partial derivative of order s of uj(x, t ) . Choosing an integer q of the same parity as n. and such that q 2 s-mj+l we have, for t > 0,

D'%j = D" L I ~ + ~ ) ' ~ K ~ , ~ ( X - Y , t)cjj(y)dy i (2.8) = Ds j C j ( y ) d ~ + q ) ' 2 K j , q ( ~ - y , t )dy

= 1 D"j,,(s-y, t ) - d!"+"'"j(y)dy

after partial integration, since C j is C" with compact support. Since, as remarked above, DsKK,,,(z-y, t ) is continuous in the closed half-space t 2 0, it follows that Dsuj can be extended as a continuous function in the entire closed half-space t 2 0. As s was arbitrary we have proved that ztj E C" in t 2 0.

To verify (2.7) choose q so large that q 2 m,-m,+I, k = 1, - - , m. Using (2.8) we have

B,(D)Z&, 0 ) = j d:+q)'a+,(y) - B,(D)Kj,,(=y, 0)dy (2.9)

= j .4r+*)'a+j(x-y)Bk(D,> D,)K,,,(y, O)d!l,

after a change of variables.

B ~ ( D ) K ~ , ~ ( ~ , 0) = constant j do,

Assume first that K # i. Using (2.6)', (2.6)" we find that on t = 0

151=1


since, according to ( l . l O ) ,

Thus (2.7) is proved for k # i. Next suppose h = i. If mj 2 n we have, using (2.5), (2.6)' and (l.lO),

( 2 . lo),

where y,(y) is a homogeneous polynomial of degree q. Similarly if mi < n we find, using (2.6)" and ( l . l O ) ,

where again yU denotes a homogeneous polynomial of degree q.

(3.2) and rechanging variables, that From (2.9), (2.10)', (2.10)'' we find, after inserting the value of j3, from

Here we have used the fact that

SA!"+')/a+i(g) - y u ( x - ~ ) d ~ = J+,(y) * d ~ + q ) ' 2 ~ q ( ~ - ~ ) d ~ = 0

since yp is a polynomial of degree q and is therefore annihilated by By John's identity (2.4) the right side of (2.11) equals +,(x), and the

proof of the theorem is complete. Remark 1. It is clear from the above proof that, if + j belongs to

Cn-m5+2+s for s 2 - max m, and has compact support, j = 1, * , m, then %(z, t ) is a solution of the problem (1.7) of class C" in t 2 0.

Much stronger results will be proved in later sections. We shall prove a lemma concerning the smoothness of the Poisson

kernels. Using arguments similar to those employed by John in [l?] Chapter 3 it can in fact be shown that each kernel K,,,(x, t ) is analytic in t >= 0 except a t the origin.

It is convenient to introduce the constant (6.12) E = A + b + A - l + n + m + ~ ~ ~ , ,


where A is the ellipticity constant of (1 .1 ) ; b is a bound for the coefficients of the operators L, R j ; A is the determinant constant (1.9).

LEMMA 2.1. The kernels K j , a ( ~ , t ) are o f class C" in t 2 0, except at the origin, and satisfy

s 2 0.

Fairthemore, if s 2 m,+q+l, then DsKj , , is homogeneous of degree m,+q-s, artd the logaritltmic term in the inequality may be omitted. Here C(s, E ) is a constant depending only on s, q alzd E.

The lemma is proved in Appendix 1. From (2.6) we obtain the following similar statement for the Poisson kernels K, :

(2.13) IDSK,,,I 5 c (s, E ) (\x12+t2)(mf+Q-s)P (1 + lbl PI I ) 9

(2.13)' IDsKK,I 5 c(s, E ) (l+llog I P I I ) ( ~ ~ ~ ~ + t ~ ) ( c m t - n - s ) i 2 , s 2 0.

Furthermore, if s > m,-n, the kernel D S K j is homogeneous of degree mj-n-s, and the logarithmic term in (2.13)' may be omitted. Here c ( s , E ) depends only on s and E.

Because of the reproducing properties (2.7) of K j we can assert that (even for K = i ) (2.14) Bk(D)Kj(x, 0) = 0 for x # 0, j, k = 1, - - , m,

and hence, with the aid of (2.13)' and the theorem of the mean

(2.14)' IBk(D)Kj(x, t ) ( 5 constant - t(l+llog ~ P ~ ~ ) ~ P ~ m ~ - m ~ - ' + l ,

where the constant depends only on E . Because B,K,,,(x, 0) = 0 for x # 0, k # j, as established in the proof

of Theorem 2.1, we find in a similar way, with the aid of (2.13), that if s = n+q+wz,-mk 20, then

(2.15)

Furthermore, t

(2.15)' /!?,(D)K,(x, t)I s C ( E ) (/zIZ.t/2) l n - r l ) , 2 *

In Section 4 we shall make use of Remark 2. Let I , = max (2m, mj) and, for j = 1, - . , m, let dj (z ) be a

given function of class Cn+l@'j+z satisfying Dkr$,(z) -= O( (1 +log lxl)lx~--1---~ ), k = 0, . * * , l 0 - m j ,

for large Is]; let D?"j be a particular differential operator in the x variables of order lo-nzj, and let G,(X, t ) be defined for t > 0 by

a,@, t ) = 2 D>-jBj(D, , D,)K(x-y , f)Ci(y)dy. i

640 S . AGMON, A. DOUGLIS AND L. NIRElNBERG

Then Gj may be continued as a continuous function into the closed half- space t 2 0 and Gj(x, 0 ) = Dk"f$j(x).

we see easily that partial integration is valid, with no contribution from infinity, so that G j may be written in the form

Proof: Because of (2.13)' and because of our conditions on the

Gj = 2 J B,K,(x-y, t)DEp-%#i(y)8dy. i

Let [ ( y ) be a C" function with compact support which equals one in a sphere IyI < R and satisfies 151 5 1. We may write

where Gj = R,Wl+W,,

w1 = ?JK1(3"-Y. t)S'(P)D>"l$i(Y)dY,

w2 = g B,K,(x-!/, t ) ( l - c ( Y ) ) D ~ - " ~ ~ i ( Y ) d Y .

Using Remark 1 we see that B, W , may be extended as a continuous function in t 2 0 and, for 1x1 < *R,

B, Wl (x, 0) = Dy%& (x) *

For 1x1 < $R consider the function w2 . Since the integrand is nonzero only for IyI > R, in which region i ly l 5 Iz-yl 5 Qlyl, we see from (2.14)' and our assumptions on $j that

(1 + log ~ ~ ~ ) ~ ~ Y ~ ~ ~ - ~ j - ~ ~ - ~ ~ y ~ 2 ~ - f l - i - ~ ~ - ~ o + ~ ~ a ~ lw,(x, t)1 5 constant - t 1 I4 > R

5 constant t dY

which tends to zero as t -+ 0. Thus Gj(x, t ) is continuous at every point (x, 0) in 1x1 < *R and assumes the asserted value there. Since R was arbitrary the proof is complete.

We conclude with an interesting result based on (2.15), (2.15)'. The result will make use of the following norm for any no:n-negative integer j :

and the trivial inequality

t (2.16) oj2 $(y)dy 5 constant - 1.u.b. 141,

where the constant depends only on n, the dimension of the y-space.

THEOREM 2.2. Consider the solution u(z, t ) given by (2.3) and let 1 be an integer satisfying 1 2 m, , j = 1, - * - , m. Then , for any i = 1, - * - , m and any differentiation 0k-i in the x coordinates, the following inequality holds:


1-u.b. lo~--”fB,(D)~(~, t)I 5 C(1, E ) L: [9j]t-m, f

where the 1.u.b. i s taken over the fzdll half-space t 2 0. As a special case, for the Dirichlet boundary conditions, with

B, = Di-1 we obtain

AN EXTENDED MAXIMUM PRINCIPLE (WEAK FORM). The solwtion (2.3) of the Dirichlet problem

Lzd = 0, D:-lzl = $bj, t = 0 , j = l ; - - , m ,

t > 0,

satisfies

or, equivalently, (2.17)

Here the least upper bounds are taken with respect to all derivatives of order m-1 and on the left with respect to all x, t in the half-space, on the right with respect to all x.

Proof of Theorem 2.2: It suffices for any fixed j = 1, - * , m to prove that

Suppose first that j = i, then we have

and the result follows from (2.15)‘ with the aid of (2.16).

1.u.b. lDm-lzd(x, t)l 5 C ( E ) I: [$bJm+ I

1.u.b. JD-lzl(zJ t)I 5 C ( E ) 1.u.b. IDm-lzd(z, 0)l.

1.u.b. lDi-.-”rBi(D)Kj(x, t ) * dr(x)l I C(1, E)[94j]L-m,.

o;--”jBzKi(xJ t ) * (qx) = BiKi(Z, t ) 0 DZ--n< 9i J

Consider now j # i. According to (2.6) we may write

I(x, t ) = D~+””BiKj(x, t ) * $j(~) = rD~-m,H(dknts)’aK,,p(~-y, t )+j(y)dy

and let q be fixed satisfying q > l-mj-n. If 1--mj is even we may write this, after partial integration, as

I = DL-mf B id~“C~- ’~ t ) ’2Kr , , ( x -y , t ) d ~ - m * ) 1 2 ~ j ( y ) d y .

Applying (2.15) we find

from which, by (2.16), follows the inequality

(2.18) I I I a(lj E, q)IdiIt-m, - If finally j # i and 1-m, is odd, 1--m, = 2k+l , we write

I = 2 s o;---”aa B,d~+Q)’2-*-’Ki,*(2-y, t)D,rd;$,(y)dy. r


Appealing again to (2.15) and (2.16) we obtain the same estimate (2.18). This completes the proof of the theorem. With the aid of the results of Section 6 we may derive (confining

ourselves for simplicity to C" functions) AN EXTENDED MAXIMUM PRINCIPLE (STRONG FORM). Consider C'"

functions q35(x) with finite, j = 1, - * , wz. There exists a unique C" solution u of the Dirichlet problem

L u = 0, D:-'u = $,, t = 0 , i = l ; . - , m ,

t 2 0, (2.10)

satisfying (2.20) %(P) = O(lP1"-l) for large IPI. This solution u satisfies (2.21) 1.u.b. IDrn-lu(x, t)I 5 E(E) 2 [56JnZ-,. Thus any C" solution u(x, t ) of Lu = 0 in t 2 0 which i s O(lPi"-') for ]PI large, and whose derivatives Dm-lu(x, 0 ) are bounded, satisfies (2.22) ID"-'u((P)I 5 E(E) 1.u.b. IDm-lu(x, 0)l.

Proof: From the Theorem of Liouville type a t the end of Section 6 we see first that any two solutions of the problem differ by a polynomial v of degree m-1. However, the only such polynomial whose derivatives

, 7 = 1, - - + , m, vanish on t = 0 is v = 0, so that the uniqueness is established.

To prove the existence we may, without loss of generality, assume that D",,(O) = 0 for 0 5 K < m-j, j = 1, - -, m-1; this is equivalent to ad- ding a polynomial of degree m-2 to u. Let ((x) be a C" function defined for all x, vanishing for 1x1 2 1 and equal to unity for 1x1 5 &. For 1 > 0 let uA be the solution given by (2.3) of

5

Dj-1 '

LUA = 0, t z o D y u , = [(h)C$&), t = 0, j = 1, * - .) m.

2 "(k)C$&)lm-j I cM,

Setting 2, [C$j]m-j = M we see easily, because of our normalization of the 4, at the origin, that

j

where c is an absolute constant independent of A. Applying now the extended maximum principle in its weak form above we obtain the estimate

(Dm-luA(P)( 5 c C ( E ) M . Using the Schauder estimates of Section 6 one can see rather easily that this implies that as I - - + 0 the uA converge to a solution of our problem which satisfies the desired inequalities (2.20), (2.21).


We remark that uniqueness does not hold if condition (2.20) is omitted since ZL = t m is a solution of the homogeneous system (2.19).

3. Potential Theoretic Considerations

3.1. In this section we shall study certain integral transforms of functions f ( x ) , x = (xl , - - - , x,,), into functions ~ ( x , t ) , t 1 0. Let K be a kernel defined in the haIf-space t 2 0 and homogeneous of degree -n:

Here 1x1 = (zz;)%, P = (x, t ) , IPI = (IzI2+t2)%. Unless stated otherwise we shall assume throughout that S(z, t ) is continuous on the half-sphere t 2 0, Izj2+t2 = 1, and satisfies a uniform Holder condition at points P on the plane t = 0, i.e., if P = (x, 0) and Q = (y, t ) are unit vectors, then for some positive constants K and u’, u‘ I 1,

(3.2) p(p)-Q(Q)l KPQ“’, max lQl 5 K ;

here PQ represents the geodetic distance on the unit sphere from P to Q , Additional smoothness assumptions under various circumstances will be stated when needed.

In addition we make the basic assumption

(3.3)

where dw, represents element of surface area on the unit sphere 1x1 = 1. For 92 = 1 this assumption takes the form S ( x , 0) = -S)(--x, 0) .

We consider the transformation

(3.4) ‘u(x,t) = pW-% t ) f (Y)&> t > 0,

integration being over the entire n-dimensional y-space. Before describing our results we first write K as a sum:

= k’ ,+K, ,

and note some properties of these kernels. In virtue of the condition (3.2) and the inequality /elQ’ 5 C(u’) (sin O).’

for 0 5 (3 5 in, where the constant C ( d ) depends only on u’, we see that


5 K C (U')f"' (1x1 2+

so that ta'

(3.6) I q Z , t)l 5 C(+ ( I x I 2 + p ) (n+a'G *

It follows, by setting x = tv, that

where C, depends only on a, cc' and n. In addition we easily verify

where C, depends only on cc' and n. By condition (3.3) it is clear that we can write zd in the form

where by (3.7) we see that the last integral is convergent; in the first integral we mean the limit of the integral taken over large spheres about x with the radii going to infinity.

Suppose in addition to our assumptions above that K ( P ) has derivatives up to order 1 with respect to t which are continuous in t > 0 and bounded on \PI = 1. Setting

gj(q = J D:K(x, t )dx, t > 0 , i l l , we see readily that g,(t) = gj(l)t--j andgj+l(l) = -igj( 1). Hence if gj,(t) = 0 for some t > 0 and a certain j o > 0, then g,(t) = 0 for all t > 0 and i = 1, - - - , 1. Moreover we have the following result related to the assumption (3.3):

1 S j l Z . LEMMA 3.1. Condition (3.3) holds if and O d y if g j ( t ) = 0, t > 0,

To prove the lemma it suffices to consider the case j = 1. Consider the


decomposition (3 .5) of K. We know from (3 .7) that for t > 0, fK , ( x , t ) d x is absolutely convergent and, by the homogeneity, is constant. Hence its t derivative which equals J D,K,dx vanishes. On the other hand

Thus S D , K d x and J,+lQ(y, 0) d y vanish or differ from zero together, proving the lemma.

An immediate consequence is the following

COROLLARY. Let K (x, t ) be a homogeneous kernel of degree -n possessing continzlozts partial derivatives zlfl to order 1 in the half-space t 2 0 (the origin excepted). Suppose further that K i s a solution of a homogeneous partial differential eqzcation with constant coefficients of order I, containing the term D:K with a noitvanishing coefficient.

Indeed D f K is a linear combination of all the other 2-th order derivatives each of which contains at least one differentiation with respect to an x variable, and hence has zero integral with respect to that x variable. Thus f DIK ax = 0, and the corollary follows from the lemma.

We first derive a theorem which we call of Privaloff type. It is a simple extension of the classical inequalities of Holder, Korn, Lichtenstein, Giraud for integrals of the form (Cauchy principal value)

Then K satisfies (3 .3) .

3.2.

(3.4)' d x ) = jK,(z-% O)f(y)dy;

see Mihlin [23] and Bers 151. For 0 < u < u' we shall use the seminorms

(3 .10)

THEOREM 3.1. Under the assumfitions (3.2), (3.3) above, if f (x) i s in L, for some finite p 2 1 and [ f ] , < co for some fiositive u < u', then [uIa i s finite and

where C, depends only on a, u' and n.

shall show that

(3 .11) [ula 5 C , ~ [ f l a *

Proof: Since f E L, , the integral (3 .4) is absolutely convergent. We


from which the general result (3.11) follows immediately.

begin with the simpler case We shall consider separately the contributions of K , and K, to K , and

(1) Suppose K, = 0. By (3.7) we obtain immediately

Iu(x, t)-u(x', t)I

5 p&-% t ) ( f ( ! / ) - f ( ! / f ~ ' - - 4 ) 4 / 5 I u ] a ' I X - - X ' [ a 1 [ K , ( x - ! / J ' ) l dy

5 c, K [ f I a lZ--X'la> which is the desired estimate.

If t' 2 t and we set t' = ts we find I = 'u(2, t ) - U ( X J t ' )

= 1 ( K 2 ( z - q * t ) - -K2(x-! /J t ' )) f(?/)dy

= I (K2(-y, t)--K,(-YJ t ' ) ) f ( ' fY l )d9

by an obvious change of variable. Making now another change of variable in the second term being integrated we find, from the homogeneity of K , , that

I = Jk(--Y, 4 (f(x+Y)-f(-X+sY))dY, so that

1'1 5 1 IR2(-! /J t ) I I!/lady

5 c , K V ] a ( S - l ) a t a = CIK[flalt '--tlaJ

which again is the desired result-completing case (1) .

resentation (3.9), (2) Suppose K, = 0. We first have, assuming t' > t and using rep-

I%($, 1 ) - zt(x, t')l


where C' depends only on a.

in case K , = 0, Thus to conclude the proof of the theorem we have only to show that,

(3.12)

Set jx-x'j = d and let S be the sphere with center x' and radius 2d; let E be the exterior of S . Then, using (3.9), we may write

u(x, t ) - u(x', t )

= Is Kl (%-!A t ) ( f ( Y ) --f(4)dY

- Is K,(X'-Y, 4 ( ( f (Y) - f (x ' ) )aY

+ s, (m-Y' 4- fG@'-Yr t ) ) ( f ( Y ) - M ) &

+ (W)--f(X)) JE &(X'-Y1 W Y . Because of condition (3.3) the last integral vanishes. Let us denote the other three terms on the right by I,, 1 2 , I , . Clearly

the constant depending only on a and n. The same estimate holds for 11,l. To estimateI, we observe that d = Ix-d1 5 jx-yl, Ixf-yJ for y in E ,

so that by (3.8),

n

5 CIK[f],d.'J Ix-yla-a'-ndy I Y - 4 >d

5 constant - K[f],d"' da-a' = constant . tc[flaaU, where the constant depends only on a, a' and n. Note that the last integral converges because a < a'.

Combining these estimates for I , , I, , I , , we obtain the desired estimate (3.12), completing the proof of the theorem.

Note that our last argument yields the classical estimate for (3.4)'

(3.11)' [g], 5 constant - ~ [ f ] , , the constant depending only on a, CC' and n (this argument is, in fact, the usual argument.)

3.3. We turn now to an integral estimate, of Eesz type, for the function u defined by (3.4). We shall assume that f belongs to L,, 1 < f i < 00, and

648 S . AGMON, A. DOUGLIS, AND L. NIRENBERG

denote its L, norm by I f l L , . For fixed t > 0 we shall denote the L, norm of ~ ( x , t ) , regarded as a function of x, by I~tl~,,~. This result will not be used in any essential way in the paper, but is of interest.

THEOREM 3.2. Under the conditions (3.2), (3.3) above, a(x, t ) is in L, for each t and

t > 0,

where C depends only on u', p and n. We observe that for K = K , , i.e., K, = 0, the theorem is trivial. Ac-

cording to (3.7), K , has uniformly bounded L, norm for t = constant. Applying the well known result that convolution by a function in L, is a bounded mapping of L, into L, we obtain the desired estimate. Thus we need only consider K = K , .

Theorem 3.2 is a straightforward extension of recent results of Calderon and Zygmund in the important papers [S, 91. They proved such an estimate for the function g defined by (3.4)' under our conditions on K, (in fact they showed that condition (3.2) can be considerably relaxed). (We shall refer to Theorem 2 of [9] as the Calderon-Zygmund theorem.) Our proof, given in Appendix 2, is modelled after that in [9]: We prove the result (for K , ) for n = 1 by a simple reduction to the classical case of the Hilbert transform for which the result is due to M. Riesz (see [39]). It is then extended to higher dimensions folllowing [9].

We remark that for 9 = 2 a simple proof using only Fourier transforms may be given; this is similar to the argument in [8] p. 89, and requires only measurability and boundedness of 9.

The potential theoretic estimates used in deriving the L, bounds make use of the following seminorms for 1 < # < co and j a non-negative integer. For functions a(x , t ) define

I 4 L , , t 5 CfCCflL, 3

3.4.

)I/,* 1 4 f , L n = ( c 1 IDflWdxdt (3.13)

ISI=i t>O

For functions f ( x ) we define a seminorm l f l , - , ,D ,Lp in the following way. Suppose that f ( x ) is the boundary value of a function v(x , t ) in t > 0, f ( x ) = v ( x , 0 ) , with finite norm I v ~ , , ~ , , Then define (3.13)'

where the greatest lower bound is taken over all such functions v.

expressed in terms of the Fourier transform f(t) of f ( x ) . If we define We remark that for 9 = 2, j > 0, the norms (3.13)' can be conveniently

then it may be seen that there is a constant C depending only on j and n


such that

(3.13): C-llflj-l/z,L, 5 Ifli*-1,2,L, 5 Clfl3-1/2,L,. The norms (3.13), which were brought to our attention by Hormander (see for example Hormander, Lions ClS]) were used in the original draft of this paper. That draft, as far as integral estimates were concerned, dealt only with L, estimates 9 = 2 .

Inequality (3.13); suggests. the following CONJECTURE. For red a define f,(z) as the inverse Fourier transform of

/Sl5f(t). Now set

(3.131,

(3.13);

I f IL,,, L, = the L, n@Ym of f,-l/r.

C-’IfIj-l/D,L, 5 IfIi2.1!p,hp 5 cIfIi-l/v,L, .

Then there i s a constant C depending only on j > 0, n and ;b such that

We can now state our next main estimate (this surely holds under much weaker smoothness conditions on the kernel).

THEoREnf 3.3. Consider the transformation (3.4). Assume that the kernel K of (3.1) has continuous fiartial derivatives DK, DfK in t > 0 which are bounded by K on the hemisfihere IP] = 1, and that K satisfies (3.3). If f is in L, and has finite lf/l-l,D,Lp norm, then ~ ( x , t ) has finite norm, and, in fact,

I41,L, 5 c4fll-l/,,LD where C is a constant depending only on p and n.

Theorem 3.3 is proved in Appendix 3. Our original proof of Theorem 3.3 for ;b = 2 which was phrased in terms of the kernels (3.13), was based entirely on Fourier transforms, and required less smoothness of the kernel. The proof in the appendix makes constant use of the following simple consequence of the Calderon-Zygmund theorem.

LEMMA 3.2. Let G ( x , t ) be a measurable function defined in the half-space t > 0 a d satisfying

Here $2 i s a non-negative function defined on the hemisphere Sf : t > 0, IPl = 1 with Is+ 12‘dwp < K

for some r > 1 and some K .

Consider the function

650 S. AGMON. A. DOUGLIS AND L. NIRENBERG

%(x, t ) = f G ( x - y , t+s)v(y, s ) d ~ d s , s>o

where v (x , t ) i s a given function belonging to L , in t ;> 0, p 2 r/ (r- 1). Then u E L, in t > 0, and

with the constant depending only on r, K, # and n. Proof: Define a homogeneous kernel K ( P ) of degree -(n+l) in the

full P = x, t-space En+l as follows: K ( P ) = J2(P//P[)lPl-n-1 for t > 0, and K is odd in t. Then, for t > 0, we have

IUI0,Lg d constant * I&,L,

= S,,+,K(x-y. s+t)lv(?A s)l@2lds = u* (x, t ) ,

where we have extended v(y, s) as zero for s < 0. Now the Calderon-Zyg- mund theorem (Theorem 2 of (91) may be applied to the kernel K and it follows that

proving the lemma.

kernels G of degree -n-1 in the half-space t > 0 for which Lemma 3.2 will actually be used only for continuous homogeneous

G(x , t ) = O(1og l /t) as t -+ 0

uniformly for all points (z, t ) with 1x1 = 1. Clearly such kernels satisfy the conditions of the lemma for all r, and hence the estimate holds for all finite

3.5. We wish now to apply the preceding results to a kernel K of the p > 1.

form (see (2.6)’) (2.6)”)

(3.14) K(x, t ) = Dnaj+4+nK ,,G(xJ I ) .

By Lemma 2.1 K is homogeneous of degree -n and its first and second derivatives on I PI = 1 are bounded by a constant depending on E and q. Thus in order to apply Theorems 3.1, 3.2, 3.3 we need only verify that

f, K(z , O)dw, = 0. (3.15)

But this follows from the corollary to Lemma 3.1; for, K satisfies LK = 0 in t > 0.

Applying now Theorems 3.1, 3.2, 3.3 we obtain the following result on which almost all our estimates will be based (the constants depend only on the arguments shown) :

x1=1


THEOREM 3.4. Let u ( x , t ) = J K ( x - y , t ) f ( y )dy , with K a kernel of the form (3.14)

K(XJ t ) = m + g + n K f , * ( 2 , t) . (a) If f i s in L , for sowe finite f i > 1 and [ f ] , < 00 for some #ositive

u < 1, then [u], i s finite and

Lala S C(E, 4, [ f l a *

(b) If f belongs to L,, f i > 1, then so does u(x, t ) for every t, and

lUlL,,t 5 C ( E , 4, P)l f lL,

141,L9 s C(E? 4, !4lfll-l/Zl,L9 *

(c) If f belongs to L, for some finite 9 > 1 and lf/l.-l/,,L9 < co, then l z l i l , L 9 is finite and

3.6. In conclusion we mention a result of interest which is related to Theorem 3.1. We shall not make use of it, though it may be used, for instance, to give an alternative proof of a modification of Theorem 3.4(a); its proof is given in Appendix 4.

Let K(x, t ) be an infinitely differentiable kernel for t > 0 satisfying the following two conditions:

THEOREM 3 . 1 ~ .

(i) For every C" function f (x) with comfiact sufifiort the fzcnction

i s of class C" in t 20. (ii) There is an integer h 2 n s w h that for t > 0

(3.16) ID"(& t)I g C8(IZ12+t2)-(s+h)/2, s = 1, 2, . . .)

Assume that the s@@ort of f (x) is contained in some fixed sphere 1x1 5 R.

1 2 0,

where the constant C(L) i s inde9endent of f . Here summation i s over all derivatives of the orders shown.

where the C, form a n increasing sequence of constants.

Then, for 0 < u < 1,

2 TDZu]a 5 C ( I ) 2 [DZth-"f], 3

4. A Representation Formula in the Inhomogeneous Boundary Value Problem

In Section 2, with the aid of the Poisson kernels, we constructed solutions of (1;7)

L U = f ( X , t ) t > O (4.1) B,u = +Ax), t = 0 , j = l ; . - , m ,


in case f = 0. We propose now to construct solutions with f $ 0. Our aim here will not be to form solutions, although with our formulas such solutions can be easily written down, but rather to obtain a representation formula for a C" function u with compact support in t 2 0 in terms of Lu(x, t ) and B p ( q 0).

An essential tool is the fundamental solution T(P--P) , P = (x, t ) , P = (5, f) of the elliptic equation Lu = 0 with singularity at P = P . We refer here to the elegant discussion given by F. John in his book [17], pp. 69-70. There he constructs a fundamental solution having the form

(4.2)

where q(P) is a polynomial of degree 2m-n-1 for n f l even, 2m 1 rzfl, and q(P) is zero otherwise; y (Q) is an analytic function on I Q I = 1. From (4.2) it follows that

(4.3) IDsT(P)l 5 constant * jP(2m--n-1-s

holds for (a) s 2 0, in case n+ 1 is odd or nf 1 is even and greater than 2m, (b) s > 2m--n-1 if n + l is even and not greater than 2m. If n + l is even and 0 5 s 5 2m--n-l, then

(4.3)' IDsI'(P)l 5 constant IP12m--n-s-1 (l+llog IPII).

Inspection of the explicit formulas (in 1171) for the fundamental solution shows that the constants in (4.3), (4.3)' depend only on s, rn, n and the ellipticity constant A .

We consider now a C" solution of (4.1) with compact support in t 2 0 and seek a suitable representation. It will be convenient to extend f to the whole (rtf1)-space to be of class CN for N sufficiently large. This may be achieved, for instance, with a appropriate choice of constants 4, by setting

f N

(4.4) for t < 0

for t 2 0;

the 4, depend only on N . Having chosen some large N we now set

(4.5) v ( P ) = vjv(P) = /T(P-P)f , (P)dP,

integration being carried out over the full Z, €-space. The function v is of class CN+2m-1 in the full space (in fact it is in C" for t # 0) and satisfies Lv = f N . We set further

(4.6) B,v(x, 0) = Y&).


At this point it would be very convenient if we could assert that the following representation for u holds:

m

5=1 4 x 8 t ) = 45 , t ) + +&-Y> t ) (MY) -YAY) )dY .

However. the integrals on the right may not converge. As a substitute we shall obtain a corresponding representation for the derivatives of order lo of u, where lo = max (2m, mi). (Such a formula can also be obtained for some derivatives of lower order.)

THEOREM 4.1. If u i s a C" solution with compact sufport in t 1 0 satisfying (4. l ) , then the representation formulas

m r

(4.7)

hold for t > 0. Here v and w are defined by (4.5), (4.6), and f N i s a sufficiently smooth extension of f to the whole space.

Since u has compact support it is easy to derive from (4.7) a representation of zt itself.

In later sections (see corollaries to Theorem 6.1, Theorem 15.1) it is shown that the C" hypothesis in Theorem 4.1 may be replaced by appropriate finite differentiability of zc.

If we set u-v = w, q5j--lyj = B5wlt-o = w j , then it is to be proved that

D ~ O U ( C t ) = D ~ O V ( X , t )+ Z: J D ~ ~ K ~ ( ~ - Y , t ) ( + j ( y ) - ~ j ( y ) ) d ~ j=1

(4.8)

We have from (4.3), (4.3)': for large IPl, 1x1 and s 2 0

D"(P) = o(pp--n--l--s (1Slog PI))> D","'Oj(Z) = O((x(2--n-1-* (l+log Ixl)), (4.9)

where the logarithmic terms can be dropped if s > 21n-n-1. With the aid of (2.13)' it is seen that the integrals on the right of (4.8) are convergent.

The proof of Theorem 4.1 makes use of a simple uniqueness lemma.5

LEMMA 4.1. Let u be a function of class Clo in t 2 0 satisfying (4.10) Lu = 0, t 2 0,

(4.10)' B,u = 0, t = O , j = l ; . . , r n .

Assume that 21 and its derivatives zdp to the order lo are absolutely integrable on each $lane t = constant > 0, the integrals converging uniformly in t in every finite internal 0 < E 2 t 5 R. Assume also that u and its derivatives zi$ to the

&The assumptions here can be relaxed considerably. See end of Section 6.


orddr lo $assess uniformly bounded L, norms on the +lanes t = constant 2 0. Finally, suppose that

fjt,o lG(X, t ) I 2 d 2 d t < Then H = 0.

variables, Proof: Introducing the Fourier transform of u with respect to the x

&(x, t ) = ( Z Z ) - ~ / ~ e-i5’d.u(x, t)dz,

we see that, for t > 0, Zi is a continuous function in ( f , t) possessing continuous derivatives with respect to t up to the order lo . Furthermore, we see that

6 = (tl, * - ., tn), f

(4.11) t > 0.

Hence, it follows that, for a fixed E , G ( E , t ) is a linear combination of exponentials ellt (possibly multiplied by polynomials), where p is a root of L ( t , - ip) = 0. By our last assumption on u we find that Jj&(t, t) j2 dt < 00 for almost all 6 so that for these c5 the exponentials ,u have a negative real part. For these 5, h(5, t ) is a solution of the ordinary differential equation of order m : M+(t , - iD,)u = 0, see page 632. By continuity the same equation holds for all 5. If we show in addition that

(4.1 1)’ t = 0 , j = I , * . * , m ,

the result will follow. Since according to our Complementing Condition of page 633 the R j are linearly independent mod M+, we can infer from (4.11)’ that D:Zi(t, 0) = 0, 0 5 k 5 wz - 1, and hence that h(6, t ) = 0. But then

To establish (4.11)’ it will suffice to show that for every infinitely u(z, t ) = 0.

differentiable function p(6) of compact support we have

t = + O

But this follows readily from (4.10)’ and Parseval’s formula, since

i m i lim 1 y ( E ) B~ (6, f D,) ~ d t = lim (x) H u (x, t)dx t = + O t = + o s

n

Here the fact that L, norms are uniformly bounded is used when passing to the limit under the integral sign.


Proof of Theorem 4.1 : Consider the right-hand side of (4.8) for various derivatives Dzo. As is clear from the obvious compatibility relations which they satisfy for N sufficiently large, they are the derivatives of order /, of a C210f2 function g ( x , t ) . We easily see that

f > 0, Dlo--$m L g = 0,

and with the aid of Remark 2 (after Theorem 3.1) that

D2"jIljg = D>-ruj, t = 0, j = 1, . *, m. W'e intend to prove first that the square integrals on planes t = constant

2 0 of all derivatives of the form Dkg, I , 5 K 5 21, + 1, are uniformly bounded, next that all derivatives of the form D, D k g I , 5 k < 21, + 1, are square integrable in the upper half-space t > 0, and that the same derivatives are absolutely integrable on planes t = constant > 0, the convergence being uniform in any interval 0 < 8 5 t 5 R. The theorem then follows easily; for, from (4.9) the corresponding derivatives of w have this property and if h = ze, - g, we have

(1.12) Lh = 0,

D2-"'j B , h = 0, t > 0, t = 0,

with h satisfying the same integrability conditions as g. Lemma 4.1 may then be applied to the functions D?+l h and we conclude that D?+'h = 0. Thus, D2 h depends only on t . Since, however, D? h is square integrable on t = constant we also have D? h = 0. Hence all derivatives of h of the form Dzo D> h vanish. Operating on

(4.12)' DP-2m Lh = 0, t > 0,

with Dk-' D, we infer that 02" Dk-I h = 0. Operating next on (4.12)' with D $ J - ~ Dt we see that D:@+'" D)--$ h = 0. Operating in turn with D)+k D>-k, k = 1, 2, - * ., I,, we conclude in this way that D2lo h = 0 or that h is a polynomial. Since the derivatives D1o h are square integrable on t = constant, they vanish, i.e., (4.8) is proved.

Thus to conclude the proof of the theorem we show that for I , S I 5 21,f 1 each function

j D I K j ( x - Y, ~ ) w , ( Y ) ~ Y

has uniformly bounded L, norms on the planes t = constant. Also that for I , 5 1 5 21, the first derivatives of these functions are square integrable in the entire half-space t > 0, and finally that these last derivatives are absolutely integrable on t = constant > 0, the convergence being uniform in any interval 0 < E 2 t 5 R. Consider then a typical term


I,(%, t ) = J D"Kj(-y, t)q(?4)4/ for I2 I, - By (2.6)

I,(z, t ) = J D z d f + q ) / 2 K 3 , q ( ~ - ~ , t)co,($)dy.

If I - m, is even we may write this, after partial integration, in the form

(4.13) I j - - j Dzdp+q-i+m,)iaK j,J%-y, t)d:-j)'2Wg(y)dyt

q being chosen so that rt + q - I + m, is even and positive. By (2.13) and (4.9) there is no contribution from infinity; we see further that for I , 5 I S 21, + 1 any term Dl;-"j co, , and hence dt-'mj)/2 cog , is square integrable. It follows, furthermore, that the functions Dk-;-") wj(y) for I, 5 I S 21, have finite I l j - * , L 2 norms (see (3.13)'); for, D;--"'co,(y) is the boundary value of the function v,(y, t ) = Dk-j w,(y) t(t), where c(t) is a C" function of t in t 2 0 which equals one for t = 0 and vanishes for t > 1. The function vo then has square integrable first derivatives in the half-space t > 0.

Applying Theorem 3.4 (b) and (c ) for p = 2 we find for I, 2 I 5 21, + 1 that Ij has uniformly bounded L, norm on eacn plane I= constant, and that, for I, 5 I 5 21,, the first derivatives of I j are square integrable in the half-space t > 0.

Furthermore, writing

DIj = I gz+1 @+(l-2+9)/2 K . (% - qd(Z-mj)/2 3, Q y % ( Y ) 4 A

and noting that At-mj)'2 w,(y) is absolutely integrable and that, using (2.13),

IDz+l dp+q-z+mj)/a Kj,P(z) l 5 constant + (1x12 + t2)-n-l ,

we conclude that: DI, is absolutely integrable on every plane t = constant > 0, the convergence being uniform in any interval 0 < E 5 t 5 R.

In case I - m, is odd, I - m, = 2k + 1, we write

j ,a (z-Y~ t)Dgidtwj(~)d?/> (4.13)' I , == 2 D ZD If d:+P)/Z-k-1K

i

and use the same argument as above deriving the stated integrability properties for each member of the last sum. This completes the proof.

BOUNDARY ESTIMATES FOR ELLIPTIC EQUilTIONS 657

CHAPTER I1

THE SCHAUDER ESTIMATES

5. Notation

Let 9 be a domain in Ek (not necessarily bounded) and denote by 9 and 5 its boundary and closure, respectively. Cz(9) ( C 2 ( 3 ) ) will denote the space of functions possessing continuous derivatives up to order 12 0 in 9 (B). For f in C'(B) we define the pseudonorms,

vIz = v]? = 1.u.b. ID'ffl, where the least upper bound is taken over all derivatives of order 1 and over the domain B, and the norm

The subclass of those functions in Cz((a) whose derivatives of order 1 satisfy a uniform Holder condition of order a, 0 < a < 1, is denoted by C2+.(9). For f E Cz+'"(9) we define the pseudonorm

where the 1.u.b. is over P # Q in 9 and all derivatives of order I, and the norm

Iflz+a = Ifl,+Vlz+a Thus [/I, and I f l a are defined for every number a 2 0. We shall make use of the following lemma which is just a consequence

LEMMA 5.l.S Suppose 9 in Ek has the property that there i s a 6 > 0 such that every point P E % i s the extremity of a segment of length S lying entirely in 9. Then, given a 2 0, there are constants c = c(h, 6, a ) , d = d(h, 6, a ) depending only on the variables indicated, such that

v1rJ 5 cul:'a [fl:-"'"+dlflo 9 Ozbsa. If B i s the entire half-space t > 0, then we may take d = 0.

We shall make use of some other definitions. Let Z = ER : Iz(2+t2< R2, t 2 0, be a half-sphere in 2, t-space with z = (q , * - , xn). Let G = cR denote its planar boundary lzla < RB, t = 0. For the linear spaces CZ(Z), Cl(a), and CZ+'"(,Z), CZ-ta(c), 0 < OL < 1, we introduce the senlinorms

of the theorem of the mean.

Osee for instance the proofs of analogous results in Miranda [26] Section 33 and Douglis, Nirenberg [lo] Section 2.


v]v,l = v]i,z = 1.u.b. dpz lDz f (P) [ 5 RpfZlff for p any integer with p+l ;L 0. If $+Z < 0 we define = 0. Here the least upper bound is taken over all derivatives of order 1 and all points P in Z, d p denotes the distance from P to the spherical part of the boundary of Z, 1z(2+t2 = R2. Correspondingly we also define

ia

< - RP+z+alfl;+a ,

I f lp,z+a = l / l p , l + [ f l p , z + a - The norms Finally we set

for functions in C”(cr) are defined in the same way.

N

C.ul0.a = “Ula2 N

b l o , ~ = Iula - The new seminorms are subject to inequalities analogous to those of

Lemma 5.1, proved in a similar way with the aid of the theorem of the mean. LEMMA 5.2.6 Given p 2 0, a 2 b, there i s a constant = C(n, #, a )

(de$ending only on the variables indicated) such that

vi,, 5 ~[rti:% C~ILY+ [ ~ I ~ , ~ I .

6. The Schauder Estimates for Equations with Constant Coefficients

We consider, as in the previous sections, equations (4.1) with constant coefficients in the half-plane t 2 0, ie. ,

Lzl = 1, B,u = dj ,

t > 0, t = 0.

Set lo = m a (2m, mi) and let 1 be an integer 2 I, . Using the notation of Section 5 we shall assume throughout this section that u(x, t ) E and that f E Cz-2m+a in t 2 0, +* E Cz-mj+a on t = 0, for some positive ct < 1.

We derive first a basic estimate, for functions with compact support, from which the general inequalities will be deduced.

THEOREM 6.1. I n addition to the $receding conditions, assume that u has compact support. T h e n u E Cz+a and


where the constant depends only on I , a and the characteristic constant E.

and /, and to the plane t = 0 for the q5i(x).

obtained in Section 4, and on Theorem 3.4(a).

by (Z-Zo)-fold differentiation of (4.7),

The various norms in (6.2) refer clearly to the half-space t 2 0 for zd

Proof: The proof of (6.2) is based on the representation of u(x, t ) ,

Assume first that zl is of class C". According to Theorem 4.1 we have,

m

j = l D'u = D'v + z I j ,

(6.3) 1, = J D1K,(z-% t)(MY)--Wj(Y))dY = D'Kj L (+j-vj),

where z, = V , and yr are defined by (4.5), (4.6), with N sufficiently large. We see easily that for any fixed I for N sufficiently large the extended function f = f N (this enters into the definition of v ) of (4.4) satisfies

I, 5 k 5 1, where the norm on the left is taken over the whole x, t-space and that on the right over the half-space t > 0. The constant depends only on n and N (chosen sufficiently large).

V ~ l k - Z m + a 5 constant . Eflt-Zm+a 1

The function ZI = V , given by (4.5) satisfies the inequality

~ v l ~ + a 5 constant ' C#Nl~-2m+a 9

where the constant depends only on I , u and E. This inequality is by now well known (see for instance Bers [5]), and may be derived in fact from Theorem 3.1 for the case t = 0 (the Holder-Giraud case, see page 647). Thus, applying the previous inequality, we have

[ ~ ] l + a 5 constant * [ f ] t- tm+a , both norms here being taken in the half-space t > 0. From (4.6) we infer that (6.4) [y3]l-m,+a 5 constant [v],,, S constant * [#]z-2m+a , I , S k 5 1.

In order to apply Theorem 3.4(a) we use the representations (4.13). (4.13)' of the I , :

(6.5) j. a

for I - -m, even,

I , = p , 4 ( n + P - l + m d / 2 K * 4 ~ - w / 2 ( 9 5 j - v j )

(6.5)' I j = 2 D1Di5("+Q'/2-k-1K,,Q * D , L I ~ c o ~ , t

for 1--m, = 2 k f l . Here q > I--m,--12. We may now apply Theorem 3.4(a) to each term in If. Using (6.4) we find

660 S. AGMON, A. DOUGLIS A N D L. NIRENBERG

V j l a 5 c(E* a)C+rY~l l -m,+U 5 constant - ([4jlI-m,+a+ [flz-2m+a).

(6.6)

From (6.3) we find LDgu], 5 [D'v], + 2 [ I j I a , and, by (6.4), (6.6), m

[DZ.UI, 5 constant - (rfiz-2m+a + t: [4j11-m,+a~l j-1

which is equivalent to (6.2). Suppose now that u is merely of class CZo+", not Coo, in t 2 0. In order to

carry through the proof it suffices to have the representation (6.3) for 1 = 1, , i.e., (4.7). By differentiation we then obtain (6.3) for the value of 1 employed, and may proceed as above. But the representation (4.7) for functions a(z, t ) of compact support and of class CZo+OT in t 2 0 is easily obtained by suitably approximating u by C" functions u, and applying inequality (6.2) for 1 = lo to the u, . (In Section 8 such an approximation procedure is described for a more elaborate situation.)

This completes the proof of the theorem; incidentally we have proved

COROLLARY 1 . T h e representation (4.7) holds for functions u of class Clota, 0 < a < 1, and comfiact support in t 2 0.

Theorem 6.1 contains the basic estimates used to establish the Schauder estimates for equations with variable coefficients. Before treating the general equation we shall need a preliminary consequence of Theorem 6.1 for the special equations (6.1). The notation is that of Section 5.

THEOREM 6.2. Let u(x, t ) be a bounded solution of (6.1) of class C I O + ~ , 0 < u < 1, in the half-s$here Z = .C, : [xI2+t2 < h?, t >= 0. Assume that for fixed 1 1 lo ,

K = V l a m , 1-2n+a + 2 r4~1m,, I-m,+a I

i s finite. T h e n $6 is of class C1+u and

where the constant depends only o n 1, a and E . Proof: Because of the homogeneity of the norms in (6.7) we may assume

that R = 1. (a) We treat first the case that u e C1+Or in Z. We may then assume that [ulZ+" is finite. Otherwise we first apply (6.7) to u in ZR-e and then let E -+ 0 obtaining (6.7) for Z.

Let PI Q be two points in Z with 41P-Q[ 5 d, , ds and such that

N

Let [(x, t ) be a non-negative C" function which is identically one in


Izi2+t2 5 (lPl+&p)2, and vanishes for (x12ft2 2 (IPI+@p)2, so that [ ( P ) = [(Q) = 1. The function 5 can be so chosen that (6.9) ID"l I C ( k n)d;lc, k 2 0,

where C(R, n) depends only on k and n. We may apply Theorem 6.1 to the function v = [u to obtain the inequality (6.10) [v l t+a 5 constant * ( [ ~ ~ 1 2 - 2 m + a + 2 [ ~ j v ( % , 0)1l - -m,+a)*

I Now

L P r n L v = CDz-2mLu+Zk,0 coefficients - Dk[ * D1-kzr. Since Dk[ vanishes except for the points in Z whose distance from its spherical boundary is between i d p and i d p we find fairly easily with the aid of (6.9) and the theorem of the mean, that for k > 0

N

[Dk5 - D 1 - k ~ ] a 5 constant dp(l+a)l~Il-k+l, where the constant depends only on u, k and n. Furthermore

Thus we have N

[Lv] l - zm+a 5 constant * dp('+aJ ( [ L U I ~ ~ , z-zm+a+ i4J Similarly we find

N

CBjV(x, O)]l-m,+a 2 constant * d~(E+a'([+jlmj, l-mj+a+IUli).

Here the constants depend only on 1, u and n.

equalities (6.10), (6.8), the inequality Since A 2 dlp+OL[vIlfa we find, from the preceding paragraph

we find

and in-

N

1- %[ziIt+a 5 constant (CL2412m.l--2m+a + 2 [+$]ma,, l - m , + a + I ~ I z ) . N

By Lemma 5.2, the term constant * lull on the right may be estimated by ~ [ u ] ~ + ~ + constant * [u], , which on insertion in the preceding yields the desired inequality (6.7). (b) Suppose now merely that u E Clota and that 1 > I,. By case (a) we know that (6.7) holds for I = 2, . It follows that for Z' = ZR-, ,Z" = ZR-28 we have

(6.11) [%I& 5 constant ( [ / I & ~ + ~ + 2 C+iiTo,,+a+ [~I I ' )~ where the constant depends on E. We propose to show with the aid of (6.11) that u E Clfo. We shallcarry out only the first step, the proof that u E Czo+l+a ; the succeeding steps are similar (induction may be used). To prove that zc E CZo+lfa apply (6.11) to a tangential difference quotient uh of u,

N N

h = (h, , - * - , k), lhl < E ,


u(X+h, t ) - ~ ( x , t ) Ihl

uh =

We find that [u*]& is bounded by a constant independent of I t . It follows that u has derivatives of class Ca of the form D,D1ou. Equation (6.1) being valid, Db+lu also belongs to class Ca, so that in fact U U E C ~ O + ~ + " ; and so on for higher derivatives.

This completes the proof of Theorem 6.2. We conclude this section with an extension of Theorem 6.1. THEOREM 6.3. Let u(x, t ) be a solution of (6.1) of class Czofa , 0 < u < 1,

in the half-space t 2 0. Assume that V]2-2m+a and C3 [+ j ] l -m,+a , for fixed 1 2 1, , are finite and that

M,, = @ R-(l+") max lzcl R-+w Z R

i s finite. T h e n u is of class Pa and

[uI c+a 5 constant . ( V3 &-2rn+a + 2 [4fI Z-m,+a + M , ) where the constant depends only on I, u and E.

to u in ,Z = Z., , R 2 lO(lPI+IQI), we find Proof: Let P, Q be arbitrary points in t 2 0. If we apply Theorem 6.2

S constant * Rc+a(V]z-2rnia

+ 2 [4j]2-m,+a+R-(2+a) max Iul)* j X R

Letting R + co through a suitable sequence, and noting that d,/R -+ 1,

We note here as an immediate corollary a we obtain the desired result.

THEOREM OF LIOUVILLE TYPE. Let U ( X , t ) be a SOlUtiOrt O f (6.1) O f C h S S Clef", 0 < u < 1, in the upper half-space t 2 0. Assume that the functions f , bj are polynomials of degrees k, k+2mn--iPn,, respectively, and that 1imR+- m+a) max Iul= 0. Then u itself i s a polynomial of degree k+2m. -

ZR

7. Equations with Variable Coefficients

The transition from equations with constant coefficients to equations with variable coefficients is routine (see [lo]) but tedious and we shall not carry out every detail. We first consider such equations, as before, in the


( B + 1)-dimensional half-space x, t, t 2 0. The point (z, t ) will be denoted by P. Our equations with variable coefficients have the form

L ( P ; D)zc(P) = F ( P ) , t Z 0 D ) a ( P ) = @j(X)> t = 0 , j = l ; . . , m.

(7.1)

Here L and B, are differential operators of orders 2m and m, respectively; the coefficients of B, are independent of t. More explicitly, using the notation of page 631,

L = 2 aa(P)DJ,

B, = 2 bj ,y(x)Dy. IBI s29n

IYI S h j

We write L = L'+L'', B, = B;+ B;' , where L' and R; are the principal parts of L and B, , i.e., the parts of highest order. We shall make the following assumptions:

(i) Condition on L. We assume first that L is uniformly elliptic, that is, that there exists a constant A > 0 such that the characteristic form associated with L' satisfies (1)

A-'(ltl"z")" 5 IL'(P; t, .)I 5 A ( I t la++)" for all real 6 = ( E , , - - - , .$,J and real t, and all points P in the half-space t 1 0. In addition, in the case of two variables (n = 1) we assume that for each P on t = 0 and real 5 # 0 exactly half of the roots z of the polynomial L' (P; 6, z) lie in the upper half of the complex z-plane. Furthermore all coefficients of L are bounded in absolute value by a constant b.

(ii) Comfilementing Condition of the bowdary ofierators relative to L. The coefficients of the B j are bounded in absolute value by the constant b. Furthermore, for every fixed P* = (z*, 0) the system with constant coefficients consisting of the elliptic operator L'(P*, D) and the boundary operators B;(P*, D), j = 1, * - , m, satisfies the Complementing Condition of page 633 uniformly for all P*, That is, if A,* denotes the "determinant constant" (1.9) of the system, then

A,, 2Ll> 0

for some fixed constant A . For fixed 1 2 I, = max (2m, mj) ,

0 < u < 1, we shall assume that F and the coefficients of L belong to Cz-2m+a and have I Il-zm+c norms bounded by k , while @, and the coefficients of Uj belong to Cz-"i+" and have their I / l-m,+a norms bounded by k . (In Theorem 7.2 we consider solutions in a half-sphere Z: = ZR and there, of course, the coefficients of the operators L , B j are only required to be defined in Z and to have finite norms there.)

(iii) The coefficients are smooth.


In treating equations in integral (or variational) form we shall assume

In extending the theorems of Section 6 we use the notation of Section 6. In the following, constants depending only on A , d, b, k , I, n, a will be

denoted by C, , C 2 , - - * . THEOREM 7.1. Let u(P) be of class C2+a in t 2 0 and be a solzction of the

boundary value problem (7.1) in the half-space t 2 0. Assume (i)-(iii) and assume that / Z C / ~ + ~ is finite.

a slightly altered form of (iii).

Then

(7 .2 ) I4z+a 5 Cl(rFl1-2,+, + c r@~lz-,+a+I~lo)*

14 l+ar 5 C2(I~lzm,z-2nz+a + c I@ilm,, r-m,,a+I~lo).

THEOREM 7.2. Lei! u(P) be a solution of (7.1) of class Czo+a in the hemisphere .Z = .ZR , R 5 1, and assume (i)-(iii). Then u belongs to C1fu in 2 and

N N

(7.3) We shall first prove Theorem 7.2 using Theorem 7.1. In the proof we

shall assume that we already know that u belongs to Cz+a in 2, for this fact may be derived from the estimate (7.3) for I = I , , as in step (b) of the proof of Theorem 6.2, by taking difference quotients. In the proofs of both Theorems 7.1 and 7.2 we make use of Lemmas 5.1, 5.2 in the following form: For every E > 0 there are constants C, , el depending onIy on E, I , n, cc such that: 1) for u in the half-space t 2 0 we have

(7.4) I 4 1 s E"lz+a+Cll4O

(7.5) IuIz 5 ~ [ ~ I t + a + Q l I ~ i o .

2 ) in Z = ZR we have N N N

Proof of Theorem 7.2.: The reduction of the theorem to Theorem 7.1 is just the same as the reduction of Theorem 6.2 to Theorem 6.1, and we merely sketch it. Again we may assume that Iull+a is finite; otherwise we may first derive (7.3) for u in ZR-e and then let E + 0, obtaining (7.3) in ZR , We may also suppose that [ZC]~+~ > #ull+a ; for otherwise, (7.3) follows easily from (7.5) with E = 4. Thus there are two points P, Q in Z with 41P-QI 5 d p , d Q , and a particular derivative D z u such that

rv

N N

Let ( be the non-negative function defined in the proof of Theorem 6.2. Applying Theorem 7.1 to the function ZI = 5.u we obtain the inequality

[vIt+a 5 Cl(C~~II--2m+a + c I & + > 0~lz-m,+a+-l40~.

Following the argument of part (a) of the proof of Theorem 6.2 and using hypothesis (iii) we can establish the inequalities

BOUNDARY ESTIMATES FOR ELLIPTIC EQUATIONS 665 N

The remainder of the proof of the theorem follows that of part (a) of the proof of Theorem 6.2 with the aid of (7.5).

We turn now to Theorem 7.1. Its proof involves replacing the coefficients of L' and EI (the leading parts of L and B,) by their fixed values at some point, and taking all remaining terms in the equations (7.1) to the right-hand sides. Then we apply Theorem 6.2 and use (7.4) to estimate these extra terms on the right.

Proof of Theorem 7.1: As in the proof of Theorem 7.2 we may suppose that [ ~ . l ] ~ + ~ > &ulz+a so that there are two points P o , Q, where we may take Po = (0, - . - , 0, t o ) , Q = (y, t), t 2 to > 0, and a particular derivative D'zc such that

Now let 1 be a positive constant which will be determined later as a function of A , A , b, k , I, n, a. We distinguish a number of cases according to the relative positions of Po and Q,

(a) lP,-Ql 21. Then we have A 5 2 A - " [ u ] z

which, combined with (7.6) and (7.4), yields (7 .2) . (b) IP,-QJ < 1 and to 2 21. Then u is a solution of (7.1) in the sphere

with center at Po and radius 21. Since Q is in the concentric sphere with radius 1 the desired estimate for A, and hence for , follows from the interior estimates of Schauder type that were established by Dough and Nirenberg [lo].

The final case is (c) IPo-QI < A, to < 21. Consider the system of equations (7.1) in

.Z = and write it in the form

L'(Po ; D)u(z, t ) = @'(Po ; D)-L)u(x, t ) + F ( x , t )

BI(0; D)u = (Bi(0; D)-Bj)zl+@j(z) = $hi(%), f f (2, t ) , tzo

t = 0,

where 0 is the origin, the functions f , 4, being defined by these identities. Since P, , Q lie well in the interior of .Z = we may apply Theorem 6.3 to obtain

A Z f a A 5 C4([LI&,l--2m+a + I: rAl;,, ~-m,+a+IuIo)* 5

We shall estimate the terms on the right. Clearly


VIfm, z-2m+a 5 C'Ifm, z-zm+a

+ [ ( ~ ' ( p o ; D)--L'(P; D))u(P)I&z-am+a+ [" '~I&,l-~m+a . The first term on the right is not greater than C5Az+a[F]z-am+a. Since the coefficients of L belong to Cz-2m+a we see that the coefficients of L'(Po ; D)-L'(P; D) are bounded by constant - k 6 la (see condition (iii)), and a straightforward analysis shows that

"(L'(P9; D) - L ' ( R ~ ) ) ~ ( ~ ) l : m , z - z m + a

p"ZL]:m,z-~m+a 5 c~~.z+nluIz .

5 C5 (An [u] z+a+ INI 1 ) .

Similarly we find, with the aid of the theorem of the mean, that

Combining these inequalities we obtain

[/I&, z-zm+a 5 c5 C3-1 Z-Zm+a + ~ 5 ~ " C . u I , + , f ( ~ 5 f ~ 6 ) I ~ ~ l Z *

A similar argument yields

[+,I", , z-m,+a 5 C, [@,I l-m,+a

+ C, A" Cul *+a +C,I uI 1 ; inserting into the above inequality for A" we obtain

A s C,{ CFIt-zm+a + 2 [@iIz-m,+u) + Cgj*a/uIz+a

+ C,I 24 I z + c, P-" Iul 0 *

C,A" = t.

IuIz+a 5 4Cs(CFIZ-zm+, + 2 [@iIz-m,+a) +C,Iu.Iz -

j

We now fix A by setting

From (7.6) and the last inequality it follows that

By (7.4) we have C&l, 5 and the desired result (7.2) follows.

This completes the proof of Theorem 7.1, which, however, is not fully satisfactory in that we assume the boundedness of u, in contrast to the assumptions in Theorem 6.3. It should be clear, however, that our method of proof would yield an estimate should we allow the solution to grow with a certain speed at infinity, provided we made certain assumptions on the behaviour of the lower order coefficients near infinity. This would require a more careful use of Lemmas 5.1, 5.2.

We shall establish finally the Schauder estimates for a general domain 3, which may be unboundea; the result, even for a half-space as considered in Theorem 7.1, will be a sharper form of the theorem.

In the domain 9 (in (n+l)-space) with boundary 5b and closure 3

BOUNDARY ESTIMATES FOR ELLIPTIC EQU.4TIONS 667

we consider a bounded solution of the elliptic equation

LU = F, which satisfies certain boundary conditions

B,u = Qj, j = l ; . - ,m , on a portion I' of the boundary @ of 9; r may be the entire boundary. As before, L and Bj are differential operators of orders 2m and m, ; we set I , = max (Zm, w j ) and let 1 1 I, , 0 < cc < 1, be fixed. Consider a (possibly unbounded) subdomain 8 of 3 with the property that !,!l n $ lies in the interior of r (regarding these as sets in the n-dimensional boundary of 3). Under certain smoothness hypotheses on the boundary of 'B and under conditions analogous to (i)-(iii) we shall, roughly, establish the estimate

where the norms for the @jj will be defined in a fairly obvious way, and where the constant depends on the domains 91, 5D and on certain parameters (as in Theorem 7.1), but is independent of u.

We now describe our assumptions. We shall assume first of all that the portion r of the boundary is of class Cz+a. More precisely we shall assume the following concerning l?l and 9. There is a positive number d such that each point P in '2l within a distance d of has a neighborhood U p with the properties: (a) up n $I C I', (b) U p contains the sphere about P with radius &d, (c) the set up n 5 can be mapped in a one-to-one way onto the closure of a hemisphere ZR(p), R ( P ) 5 1, in (nf1)-space, with up n @ mapping onto the flat part of the hemisphere, by a mapping T p which, together with its inverse, is of class Pa; in fact each component of the mapping, and its inverse, i s assumed to have finite I I l + a norm (where it i s defined) bounded by a constant K independent of P.

Our assumptions preclude the boundary of the domain follotving back on itself or pinching into a narrow bottleneck.

Concerning the equation and the boundary conditions we shall assume that under each such mapping T , these go into a system (7.1) in satisfying the hypotheses (i)-(iii)-with the relevant constants A , A , etc. independent of P. In addition we assume the coefficients of L and B j to have finite norms I and I I l - - m j + a in '3, respectively, and L to be uniformly elliptic in 9, i.e., for some constant A (as in condition (i)) we have

A - y q 2 m 5 IL'(P, E)l 5 A1812"

for real E = (El, * * . , tn+J. After the transformation T, the functions 0, are defined on the flat part a, of the boundary of L'B(p, . We set

(7.7)

668 S. AGMON, A. DOUGLIS, AND L. NIRISNBERG

We assume finally that u is bounded and is of class Czofa in sbfr . We now state our main estimate, which includes the results of Theorem 7.1 and essentially of Theorem 7 .2 .

THEOREM 7.3. T h e solution u i s of class C2+" :in B a.nd

(7 .8 ) I4E" 5 ~(IFI?--2m+a + c I@jlCrn,+"f14?)> where the consfant C depends only o n d , K, and the constants E , ci, k and 1. (Recall (2.12), E = A+b+d-l+n+m+Zm, .)

Proof: I t follows from the interior estimates of Douglis, Nirenberg [lo] that u E CZta in Q. Furthermore near the boundary we may perform one of the mappings T p described above, and so imagine that we are dealing with a solution of (7.1) in ZR(p) . From Theorem 7 . 2 , we may conclude that u E Cz+a at boundary points of 8 belonging to 55, and hence in all of B.

Consider now any fixed point P in 2. If its distance from 9 is greater than d , we may apply the interior Schauder estimates of [lo] and infer that the derivatives of u up to order I at P and the difference quotients

are bounded by the right-hand side of (7.8), with the constant depending only on d and E , cc, K , 1. If the distance of P to 9 is not greater than d , we may perform the mapping T , and consider the transformed equation, with boundary conditions, in Zx(p, . I t follows easily from our assumptions that the distance of the image of the sphere about P with radius $d from the curved boundary of L'R(p) is greater than a constant depending only on d , K , R. We may then apply Theorem 7 .2 in L'R(p) and we find that the derivatives of u up to order 1 at P and the difference quotients

are bounded by the right-hand side of (7.8), with the constant depending only on d, K , and A , - * . , c c .

Combining these results we see that IuIZfL and the difference quotients

lDzN(P)--Dzu(Q)I, p, IP-QI"

are bounded by the right-hand side of (7.8) except in the case that IP-QI > i d . luly (4/d)", and the proof of the theorem is complete.

replace the term

But in that case the difference qu.otient is bounded by

= 9 and REMARK 1. I n case sb is bounded and l' = $, we m a y take in (7.8) by the L, norm of u


This follows easily from a general inequality which may be stated as follows: For every E > 0 there is a constant C depending only on E and 5D such that

REMARK 2. In case 5D is bounded, r = a, and the solution u of class Clota i s unique, we may take '21 = b and omit the term lul? on the right of ( 7 . 8 ) . T h e constant in (7.8) then depends on the equation but i s still independent of u.

Proof: The remark is a consequence of the following assertion: If the solution is unique, then

(7.9)

where the constant depends only on the equation, i.e., is independent of u. Assume that (7.9) does not hold. Then there is a sequence of functions u, in C1+a with lu,lF = 1 such that the terms in the brackets of the right-hand side of (7.9) go to zero for the corresponding u, as n + oc). From (7.8) we see that the norms lun/l+a are uniformly bounded. It follows that a subsequence of the zi, converge in the norm I /? to a solution u in of the limit equation] which is homogeneous. By uniqueness u = 0 contradicting the fact that [@ = 1.

CHAPTER I11

SCHAUDER ESTIMATES FOR EQUATIONS I N INTEGRAL OR VARIATIONAL FORM

8. The Constant Coefficient Case

We consider again a solution of the differential equation Lu = F satisfying B,u = @, on the boundary. But now the equations are to be written in a special form and we assume less differentiability of the solution. The arguments will be much the same as in the preceding sections, and we will mainly describe the crucial modifications and additions that are necessary.

In this section we consider the constant coefficient case (4.1), or (6.1), in the half-space t 2 0, i.e.,

t > 0, Lu = f , (8.1) Bau = $,, t = 0.

670 S. AGMON, .4. DOUGLIS .4ND L. NIRENBERG

Let now 1 5 lo be a fixed integer which is not less than the maximum order of differentiation with respect to t that occurs in the B, . We shall suppose that (in the notation of page 631)

and write (8.2) f = XD*fjj > 4j = 2 o',+i,y )

(8.3) B, = b5, DL D~j-lvl . Here the summations are for IpI 5 max (0, 2m-I), Iyl S max (0, mi-l), and mi-1 5 IvI 5 m i , j = 1, * - ) m.

The functions f a , #J~, will be assumed to belong to C" in t 2 0 for fixed positive u < 1 if 1 < 2m, 1 < m, . If 1 2 2m we assume f = fo to be of class Ci-2m+u; if I 2 m, we assume 4, = +5,0 to be of class Cz-mj+u.

Moreover, we shall assume that the solution u is of class Cz+a in t 2 0. But then we have to explain in what sense (8.1) is satisfied. It is the following integral (weak) sense: For every C" function g ( x , t ) with compact support in t > 0 and for every C" C(x) with compact support, we require that

jJ.(., t P ( D , , D,)C(x, t )dxdt

(8.1)'

Our aim is to prove analogues of Theorems 6.1 and 6.2. For functions zt, f a , # j , y defined in Z = we shall make use of the

notation if 1 < 2m,

For i = 1, - *

THEOREM 8.1. that u, f b vanish outside

I-2m

K~ = 2 [ f l f Rzm+k+ [/lf2,+, R Z f a if 12 ?m. k=O - , m,

Let zd be a solution of (8.1) of class Cz+a in t 2 0. Assume outside the unit hemis9here Zl, while the #Ji,y vanish

the unit sphere ul. Then, with R = 1 above, m

5=0 [ M ] ~ + ~ S constant * 2 K ~ ,

the constant defiending only on u and E.

BOUNDARY ESTIMATES FOR ELLIPTIC EQUATIONS 67 1

THEOREM 8.2. Let u be a bounded solution of (8.1)' of class Cz+a in the is finite. Let P, Q be two points hemisphere 22 = Z R , and assume that 2;

in ZaShR. Then for any derivative D'u we have

where the constant depends only on a and E. Theorem 6.2 is an analogue of Theorem 2 of [lo]; Theorem 8.2 is an

even closer analogue of Theorem 2' of [lo]. Let us now prove Theorem 8.2 assuming first that Theorem 8.1 holds.

The inequality is invariant under uniform stretching of the independent variables so that we may assume R = 1. Our proof follows that of Theorem 6.2. We consider again the function C(x , t ) of page 661 and shall apply Theorem 8.1 to the function v = Cu. However, we must be careful to write Lv and B,v(x, 0) in a special form. Thus we write

Lv = CLu + 2 coefficients * DkT * D2m-ku k>O

= 2 coefficient . W'(f,Dfl-fl'T) 8,8' + 2 coefficient Dfl(D*u Oar) ,

where the last summation is over IflIf161+1&1 = 2m, 161 > 0, 181 < I , 181 5 max (0, 2m--I).

Similarly R, v = 2 coefficient - 0: (tji, 0 z - Y ' C )

+ 2 coefficient D:(D"u - D"), 7*Y'

where the last summation is over Iyi+lt/+!dl = mj , /6/ > 0, /e l < I, IyI 5 mj-z-

Recalling that the left-hand side of (8.4) is not greater than d y [ v l I f a , we apply Theorem 8.1 and follow the proof of Theorem 6.2 to obtain (8.4). This involves estimating a large number of terms. We shall indicate the estimate for a typical term occurring in the expression for Lv. For R = 1, so that 5 d, 5 1, we have

d p V p - 8 ' 5 1 , 5 constant - ( [ f a ] = dF-lPl+l@'l+ I f l o d;-1fl[+1@'1). Note that the power of d, on the right may be negative. It is in order to render this harmless that the assumption d p 2 f R is made.

The other terms are estimated in a similar way, and we shall regard the proof of Theorem 8.2 as complete.

We turn now to Theorem 8.1 which is proved, as was Theorem 6.1, with the aid of the representation formula given by Theorem 4.1. We must however modify this formula and extend it to solutions of (8.1)'.


We remark first that we may assume f a E 0 for Ij3I < 2m-1. For, if T represents the operator: integration with respect to t from (x, 00) to (x, t ) , then we may represent Oafa with 1j3I < 2m--I in the form

DPfa = D/?D2"-1-!81 2m-z-181 . t P- f a )

Furthermore, we clearly have

[7-2m-2-181 f a ] , + p - z - ' f l I fa la 5 constant * ( [ f /91 ,+ [ f a l a ) .

Thus we shall assume f a = 0 for I/? < 2m--1. Consider the representation formula (4.7), where the function ZJ is defined in (4.5) in terms of a "smooth" extension f N of the function f to the entire space. The particular way in which the f N was constructed is not too important. Let us now modify f N

by first constructing extensions fa ,N of the functions fa by, say, formu~as (4.4), and then defining f N = 2 Dfl fa,N ; here N is chosen sufficiently large. With this new definition of f N the representation (4.7) still holds (for C" data) with y,(x) := B,v(x, 0) and, for P = (x, t ) ,

( 8 . 5 ) v(P) = C DS 1 T ( P - F ) f b , N ( P ) d P ,

where summation is for I#?I = max (0,2 m-1). In case 1 < m, it will be convenient to express B, in the form

B, = b , , r , r D p a , where summation is for IyI = m,-Z, Iyl+l,ul = mj .

of a polynomial 1' of degree at most 1,--I-1 such that The new representation formula to be established asserts the existence

D'u(x, t ) + V = D'v(x, t )

where

I , = 2s Dzb, ,y , ,D%Kj(x-y , t )DPv(y, 0)dy

I j = 1 D'K,(x-y , t ) B , ~ ( y , 0)dy

if I < m , ,

if 1 1 m i .

Before proving (8.6) we observe that the integrals on the right converge provided that, in t 2 0, f a E C" if 1 < 2m, f E C1--2nr+a if 1 2 Zm, #,, E C" if I < m, , $j E Cz-mj+a if Z 2 ms . This may be seen with the aid of (8.5), (4.3), (4.3)' and (2.13)' just as in the proof of Theorem 4.1.

Theorem 8.1 follows from (8.6) in essentially the same way as Theorem 6.1 did from the representation (4.7), and the details of the argument may be left to the reader. We confine ourselves here only to a few remarks concerning the proof. Theorem 3.4(a) is to be applied to all boundary integrals in (8.6) with the exception of the integrals


DzD:K!('-YJ t)#j,,(y)dYJ IyJ < mi-z.

The estimation of the Holder continuity in, say, IPI < 2 of these integrals follows in a more elementary way, since here the kernels are not very singular on t = 0. We note further that in estimating the right-hand side of (8.6) one has to distinguish the cases I < 2m, 1 2 2% in addition to the two cases for I j . In this way one estimates the Holder continuity of the right-hand side of (8.6) in 1P1 < 2. Since w(P) = 0 for /PI > 1, this estimate may then be used to estimate the polynomial V J in a fairly obvious way, leading to the desired estimate for Dlu itself.

Thus in order to complete the discussion of Theorems 8.1, 8.2 we shall verify (8.6). Suppose first that all functions concerned are of class C". Then not only are the integrals on the right of (8.6) convergent, but in fact their derivatives may be obtained by differentiation under the integral signs. and (8.8) is an immediate consequence of our representation formula (4 .7) ,

We have now to eliminate the C" hypothesis, by approximation of u by means of C" functions as in the proof of Theorem 6.1. However, this approximation is not so obvious here, and we shall carry it out. To this end we smooth out the function u by convolution with a C" kernel (the Fried- richs rnollifier). Let i ( r ) >, 0 be a C"" function of one variable with support in Iri 5 1, such that

j" j (r)dr = 1. -" Set for E , E' > 0

in t 2 0, integration being over entire n-space, and

J E , E I ~ t is C" in t 2 0, while J,u is C" in the x variables. It is easily seen that JESdu is a solution of

LJe,etu = z'DBJn,e,f~ 3

while Jeu satisfies (8.1)' with f B , q$, replaced by J 8 f b , J E + j , y .

Let Dz be any differentiation operator of order 1. According to (8.6) for C" functions we have, with v defined by (8.5), and V ( E , E ' ) a suitable polynomial of degree I,,-&- 1 with coefficients depending on E , E ' ,

D6 J , , u + V ( E , E ' ) = D1 Je,+,v

+ J" D'K,(~-Y, t)fij[J,e*zl(y, t)- J E , ~ , ~ ( Y , t ) l r = = o n ~ .

Now let E' -+ 0 keeping E > 0 fixed. Then we find, from the fact that


9. Variable Coefficients

In this section we shall treat equation (7.1) with variable coefficients in general domains, but we start by considering equations in the hemisphere 2 = ZR R 5 1, in t 2 0 and solutions u of class C:z+a, 0 < a < 1, where I,, 2 I 2 maximum order of t differentiation occurring in the Bi .

For the Operators, for f and for 4, we assume the forms L = z\ Dba,,,(P)D@,

(9.1) B, = c D:bi ,y ,@(x)m

j = l; . . , m.

F = 2 DBF,, dz, = 2 D:CDj,?, with summations over I& 5 max (0, 2m-I), lpl I d , IyJ 5 max (0, mj-Z),

Now to specify our assumptions. We shall assume (i) and (ii) of Section 7, where by the coefficients of the operators L, R j we mean now the functions as,P bi, y , p .

F, E Cu if 1 .=c 2m, F E C1-2m+a if 1 2 2m, (iii)'

@i,y E Cu if 1 < mi , ds E Cr-mi+a if I 2 mi . Furthermore we shall assume, respecting the coefficients, that the following norms are bounded by k: (aS,p(a if I < 2m, Iao,rlz-zm+a if 1 1 2ml Ibi,Y,plu if I < m,, Ibj,o,clIz--mt+u if 12 m j .

We shall assume the solution to be of class Cz+a in ZR . By solution we mean the following analogue of (8.1)': For every C" function ((x, t ) with

In addition we assume

BOUNDARY ,ESTIMATES FOR ELLIPTIC EQUATIONS 675

compact support in the interior of Z;, and for every C" function ((x) with compact support in 1x1 < R we require that

= 2 (-1)1y'S4n,,yD/((x)dx, i = 1, * * - , m.

We give now an analogue of Theorem 7.2. (The constant Ci in the statement depends only on the same parameters as the constant C2 of Theorem 7.2.)

in Z = ZR , R 4 1, and assume (i) and (ii) of Section 7 and (iii)'. Then

THEOREM 9.1. Let u be a solution of (9.2) of class

(9.3)

9rovided the right side i s finite. Here

KO = 2 IF/Jl2m-l@l,a if 1 < 2m,

KO = 1F12m,t-2m+a if 12 2m,

Kj 2 I@,,yIm,-lyl,a if 1 < m,,

Kj = l@lm,,t-m,+a if 1 2 m, . Our proof of this theorem is modeled not as much on that of Theorem 7.2

as on the proofs of Theorems 1 and 4' of [lo]. We also make use of the interior estimate of Theorem 4' (with s = 2m-Z, c = 0, in the notation of that theorem, in case 1 < Zm).

Proof of Theorem 9.1: As in previous arguments we may assume that lullfa is finite; otherwise we may first derive (9.3) for u in ZRP8 and then let E --f 0. We may also suppose that [ u ] ~ + ~ > ; for, otherwise (9.3) follows easily from (7.5) with E = #. Thus there are two points Po , Q in Z with 41Po-QI I_ d p , , d, and a particular derivative Dzzl such that

B

and for j = 1, - - , m

Y

N

N cu

(9.4)

We may take Po = (xo , to) , Q = (9, t), t 2 t, > 0. Let 1 5 $ be a positive constant which will be determined later as a

function of R, A , A , - - * , a. We distinguish several cases according to the relative positions of P o , Q.


(a) \Po-QI ZAR. Then N B 5 constant - 1-a[ulZ .

Combined with (9.4) and (7.5) this yields easily the result (9.3). (b) to 2 i d p , . Then u is a solution of (9 .2) in the sphere with center

Po and radius i d p o . Since Q lies in the concentric sphere with half the radius the result follows again with the aid of the interior estimates of [lo].

(c) IPo-QI < IR, 2AR 5 to 5 i d P o . Then as is easily seen the sphere with center Po and radius 2IR lies in ZR . Since Q is in the concentric sphere with half the radius, the desired estimate for A, and hence for [ u ] ~ + ~ , follows from the interior estimates (Theorem 4‘ in case 1 < 2 m , otherwise Theorem 4 ) of [lo].

The last case is (d) IPo-QI < 1R, to <

N

d, and t, < 21R. In this case the hemisphere 2 with center (xo , 0) and radius I = min (Q dpo , 4AR) lies in ZR ; in fact its distance from the curved boundary of ZR is at least i d p o . Furthermore the points Po , Q lie in the concentric hemisphere with radius +Y. We are thus in a position to apply Theorem 8.2 to u in 27 once we rewrite the equations as

L’(Po ; D ) u = 2 DBafl,,(P,)D,z@-? == c D @ f f l ( P ) , I S I + L I = ~

where f f l (P) = c (aa,II(Po)--fl,r(P))D”u(P)

IBl+lPl=2m - 2 a,,,(P)DP.u(JP)+F,>

IBl+lr1<2m and a t t = 0

q x , ; D ) u = z: qbj , r , l l (xo)Dau = 2 D;$&, IYl+ IPI =m,

where

These equations should of course be understood in the integral sense analogous to (9 .2) .

We claim now that by applying Theorem 8.2 we may obtain the inequality

(9.4)’

where the constant

applying Theorem 8.2,

depends only on E , tc, 1. To see this we shall only consider some typical terms which occur in

We observe first that in z” we have laA,,(Po)-aA,,(P)I f KI“ (see condition (iii)’).


Suppose 1 < 2m. Then N

rll(aA,,(Po) -aA,,(P))D~u/$’ 2 constant - F Z Y ~ I U J ~

since Y 5 Q dp and the distance of 27 from 1xI2+t2 = R2 is at least i d p ; also

~~++o~[(~,,,(P,)-~~,,(P))DPu]~’ 5 constant * K r a I ~ I Z f a .

r2m-1@1(1 aS, ,Dp z& +ralaS, D k u 1:) 5 constant - kr[zc( z+a

with similar estimates for the terms occurring in #,,y and for I2 2m.

Since r 5 ra and ZL=, [u]: Rk 5 constant - [a [&, we are led to (9.4)’.

N

Furthermore, if 181 < 2m--I, N

N

We now choose 3, as the largest number satisfying

3, 5 f , C(41R)a 5 2. (Note that with this choice of iz the factor Aka occurring in case (a) is bounded independent of R for R 5 1.) Combining (9.4), (9.4)’ and using (7.5) in the usual way we obtain (9.3), completing the proof of Theorem 9.1.

Before passing on to general domains we prove an extension of Theorem 9.1. Consider again the situation of Theorem 9.1 and let us make the following additional assumptions. Let # be an integer 2 I of Theorem 9.1. Replace (iii)’ by the stronger assumption

I;, E Cf’--l+Or if 1 < 2m, F E C*-2m+a if 1 2 2m, (iii)“

@,, E Cp-z+a if 1 < m, , @ E Cf’-m*+a if 1 2 m, . Furthermore we shall assume that the coefficients have the following norms bounded by k : laa,plp-z+a if I < 2m, \ao,p(g-2m+a if I 2 2% I b j , y , p I p - z + a

if l < mi, Ib3,0,pIz-,,,,+a if 1 L mi.

THEOREM 9.2. Under the conditions of Theorem 9.1 and under the stronger hypothesis (iii)” the solution u i s of class in .Z and

where C i depends only on p and the same parameters which enter in the de- fiendelace of C,.

The KI are defined as follows: each constant K i of Theorem 9.1 is a sum of norms of the form I l a , b . The constant K; is formed by replacing b by 9 - l f b in the terms of K , .

In proving Theorem 9.2 we shall make use of the following lemma which is proved in Appendix 6.

LEMMA 9.1. Let v be of class C1ta in the interior of ZR and be a solzGtion of

(9.5)


in Z;, for some fixed k > 0, where the fwctions v, Dxv , v, belong to C" in ZR. Then D,v E C" in ZR .

We understand (9.5), as usual, in the distributj.on sense, i.e. that, for every C" function C(%, t ) with compact support in the interior of ZR , we have

ss vDfi:cdxdt = 2 (-1)Ifll I v,DflCdxdt.

Lemma 9.1 has an L, , p > 1, analogue which can be proved in a similar way.

LEMMA 9.1'. Let v be a solution of (9.5) in ZR having first derivatives in L , in every subdomain whose closure lies in the interior of ZR . Assume that for every 6 > 0 the functions v , Dxv , v, belong to L, in Z R - 8 , Then the same is lrue for D,v.

Before proving Theorem 9.2 we mention that the last part of its proof yields the following useful result

See the end of Appendix 6.

LEMMA 9.1". Let v be a solution of an equation Lv = f

of order k in the interior of a hemisphere ZR and assume, for simplicity, that the coefficients of the equation belong to C" in the closure of ZR . Assume that the flat boundary is nowhere characteristic for the equation (which need not however be elliptic). Asszcme that the derivatives up to order j < k of the solution belong to Ca(L,, p > l), in every domain whose closure lies in the interior of ZR . Assame also that for every 6 > 0 the functions Diu, D,Diu, 0 5 i < i, belong to C"(L,) in Z R - 8 . Then the sawe i s true for Diu.

Proof of Theorem 9.2: We shall carry out the proof only for the case 9 = 1+1. To treat higher $, for instance p = l+2, we may, because of our assumptions on the coefficients, and because we have the result for p = I+ 1, write the system (9.1) (or (9.2)) in the same form, but with l replaced by I+ 1, after carrying out some first order differentiations.

To prove the theorem for the case $ = Z + 1 it suffices to show that u E Cz+I+" in ZR . The desired estimate then is to be obtained from Theorem 9.1 applied to (9. I ) , rewritten as indicated with 1 replaced by If 1. By the interior estimates, Theorem 4' of [lo], we know that u E CZ+l+" in the interior of Zx.

We start the proof by showing that derivatives of the form D x D z u belong to C" in ZR . This we do by taking difference quotients of (9.1) in x-directions. If h is a vector perpendicular to the t-axis, then for Ihl < S/2 the difference quotient


of a function g(x) defined in ,ZR is well defined for P = (x, t ) in 2B-a,z. We can see easily that uh(x) is a solution of the differenced equations

Luh = 2: DPuB,,(P)D"z~(P) = 2 DB( F ~ - u ~ , , ( P ) D ~ ' u ( P + ~ ) ) ,

Bj24h = 2 D ~ b j , r , / d ( ~ ) D ~ ~ h ( ~ J 0) = 2 D ~ ( ~ ~ y - b ~ , y , p ( ~ ) D P ~ ( ~ + h , O))] in the integral sense of course. Applying Theorem 9.1 we find that since ZXb6 is a distance 812 from the curved boundary of ZR-8i2,

luh/z4 5 constant independent of h.

Letting h --f 0 we find, since h may have any x-direction, that lD,ulZ-* is finite for any fixed 6 > 0, i.e., DzDKu E Ca in ZR .

To conclude the proof of the theorem we need only show that D:+'u E Ca. In case 1 2 2m-1 this follows from the fact that we may solve for D:+'u in terms of D:-2m+1 F and the other derivatives D,Dzu from the equation D:-2"+1(L~~--F) = 0. Thus we need only consider the case I < 2m-1.

Let us write equation (9.1) in the form

D!"-'(aD:u) + other terms = 2 D@ Fa, where n = ag," for = (0, - , 0,2m-Z), ,u = (0, * , 0, I). Because of ellipticity, a # 0. Because of our conditions on the coefficients, and because we have shown that D, Dlu E C", we may write the equation in the form

D~m-g(aD:u) = 2 D Y v , ,

where the ZI, belong to Ca in ZB . We may therefore apply Lemma 9.1 to the function v = aD:u to conclude that Dt(uDfu) E C". It now follows easily, since a # 0, that D i f l u exists and is in C" in C,, completing the proof of the theorem.

Finally we describe briefly the analogue of Theorem 7.3, i.e., Schauder estimates for a uniformly elliptic equation (9.1) in integral form in a general domain 9 which as in Section 7 may be unbounded. Here the boundary conditions are satisfied on a portion r of the boundary. We assume that in terms of local coordinates x, t near a point on the boundary r, with t = 0 on l', the boundary conditions have the form of (9.1) (or really (9.2)). It is easily seen that the maximum order of differentiation with respect to t which occurs in the B, is independent of the'particular choice of local coordinates. \Ye take I 5 I , to be fixed and not less than this number at every boundary point. Let p 2 1 be a fixed integer.

The statement of the result is very similar to Theorem 7.3. In fact, we shall assume that we have the same geometric configuration except that the boundary is assumed to be of class C k a (as defined there) where I = max (I,, p ) . Under the local mappings T , , described in the condition

I Y I c2m--1

680 S. AGMON, A . DOUGLIS, AND L. NIRENBERG

for Theorem 7.3, we shall assume that the equation and boundary conditions go over into the form (9.1) (really (9.2)) and satisfy the hypotheses (i), (ii), (iii)" of Theorem 9.2.

We shall assume that the norms J@j,rJzfl+a if 1 < m, , or I @ ) p - m j + a up if 1 2 mj and I F p l ~ ~ ~ ~ ~ if 1 < 2m, or I F1D--2m+a if 1 2 2m, are bounded by a constant K . We assume finally that u is bounded and is of class CZ+& in 9fr. Then we have

THEOREM 9.3. The solution u is of class C9+" in and (9.6) luI*D+, a 4 - constant - (K+lul?), where the constant depends only on d, K , f i and the constants occurring in conditions (i), (ii), (iii) ".

The proof of the theorem is similar to that of Theorem 7.3 using Theorem 9.2 in place of Theorem 7.2, and will not be repeated.

Again we have, in case 5D is bounded, T = $, ill = a: REMARK 1. We may refilace the term lul? by the L, norm of u.

REMARK 2. If the solution of class C p + a i s unique, then we may omit aZtogether the term 1~12 on the right of (9.6). The constant in the resuZting inequality then depends on the system bzct is independent of u.

The proofs of these remarks are the same as the proofs of the corresponding remarks after Theorem 7.3.

CHAPTER IV

APPLICATIONS O F THE SCHAUDER ESTIMATES AND COMMENTS

10. Necessity of the Complementing Condition

Up to now we have always assumed the Condition on L and the Com- plementing Condition specified in (i), (ii) of page 626 to hold. We shall now show that these conditions are necessary. Consider solutions of (7.1) in Z = ZR . The Condition on L required that at every boundary point x on t = 0 the principal part L' of L should have the property that for real t = ( E l , - * , t,J # 0 the polynomial L'(x; t, r ) have exactly m roots on either side of the real z-axis. We propose to show here that if either this condition or the Complementing Condition on the B j fails to hold at a single boundary point on t = 0, say at the origin, then it is impossible to estimate

operators have C" coefficients. In fact we will show that there is no constant [ullOfa GR-6 6 > 0, in terms of ILu[&m+a, IBjuf-,,+, and 1u1,-even if the


C such that for every C" function u(x, t ) with compact support in ZR the following holds:

(10.1) I 4 Z O + " 5 C ( l ~ 4 0 - 2 m + " + 2 I B P b o)lzo-m,+"+l~lo).

L ( P ; D ) u = L'(0; D)u+L(P; D)u , B,(x; D)u = Bi(0; D)u+B,(x; D)u.

l~tlt,+a < C(IL'(0; D)UIlo-zm+a + 2 IB;(O; Db(Z, O)lZo-m,+.

fl~~Iz,-z,+, + 2 I W x , O)I10-m,+"+/40) .

Assume first that (10.1) holds for some C independent of u. If L ' ( P ; D ) , Bj(x; D) are the leading parts of L and B, , set

Then from (10.1) we have, for N with compact support in L',

If the support of u is contained in Zr and if Y is small enough, Y 5 r, , we have

C(IL~b~z , - z rn+a+ IB~z~(z, 0 ) I~o--mj+a) 5 $IZhIl,+a+EIUIO

for some constant C. This follows, with the aid of (7.4) and the fact that the leading coefficients in Z,. differ little from their values a t the origin, if Y is sufficiently small. Thus we find that if u has support in Z, and Y 5 Y,, , ro independent of u, then

I 4 l 0 + = 5 2C(JL'(O; ~ ) ~ I z 0 - - 2 m + a (1O.l)l + 2 IB;(O; D ) u ( ~ , O)Izo-m,+~+~(C+~)IuIo

Let us now suppose that the Condition on L is satisfied but that our Condition on the Bj is violated at the origin. Then consider the function vA defined by (1.13), with I = I,, relative to operators L'(0; D), R;(O; D). The function v A satisfies the homogeneous system

L'(0; D)vA = 0, t 2 0, B;(O; D)v, = 0, t = 0 , j = l ; - . , ~ z .

Now set

(10.2) un(x, t ) = v,(z, t)T(z, 4, where origin. Since

is a C" function with support in Zro, and equal to one at the

L'(0; D)un and B;(O; D)u,(z, 0)

involve derivatives of v A only up to orders 2m-1 and mj-l, respectively, we see from the construction of vA that as A + co the norms

IL'(0; D)26A1Zo-2m+a > IB; (O; D)'A(z> O)lZo-m,+U J lUh10 J

remain bounded (in fact tend to zero). On the other hand j ~ ~ l ~ ~ + ~ blows up like I"-contradicting (10.1)'.

682 S. AGMON, A. DOUGLIS AND L. NIRBNBERG

Suppose now that our Condition on L is violated at the origin, which may happen for n = 1. Then let vA be the function defined again at the end of Section 1 relative to L'(0; D ) , B;(O; D ) with 1 = I , . If we define as before uA(x, t ) by (10.2) we find again that for 1 large uA does not satisfy (10.1)'.

We may see similarly that our conditions are necessary for the Schauder estimates for equations in integral form and also, as is shown again by the functions given in (10.2), for the L, estimates of Chapter V.

11. Differentiability for Nonlinear ]Equations

Consider a nonlinear differential equation of order 2m in (n+ 1)-space,

(11.1) F ( x , U , Du, * * - , D2"u) = 0,

which is elliptic with respect to a solution u, i.e., the first variation of the operator F with u and its derivatives inserted in the coefficients is an elliptic operator.

The latter operator is a linear operator on a function w, and is of the form

We shall prove a differentiability theorem near the boundary for a solution satisfying m boundary conditions (which inay be nonlinear)

(11.2) F,(z, U , * * , Dm3zt) = 0, j = l ; . . , m ,

for x on a portion r of the boundary. Since differentiability is a local property we shall assume that r is an open subset of an ut-dimensional hyperplane, in fact, we shall assume that the solution is defined in a hemisphere C = ,ZR and satisfies the boundary conditions on the flat part of the boundary. The first variation of the operators F, with respect to a particular function u are the operators

We first prove differentiability theorems using only the Schauder estimates, and show later how these may be improved with the aid of the L, estimates of Chapter V.

If the linear system made up of L and the boundary operators B j satisfies the conditions (i) and (ii) of Section 7, then Theorem 7.2 yields the following differentiability theorem.

THEOREM 11.1. Let I , = max (2m, ?a,) and assume that u E CZo+Or in ZR . Let 1 2 I, be a fixed integer and assume that F and F j have Holder continuous


(exfionent u ) derivatives ufi to orders I - 2m and 1 --m, , respectively, with respect to all of their argctments.

The proof follows familiar lines. With the aid of the interior Schauder estimates it has been proved (Theorem 5 of [lo]) that, in the interior of the domain, u E C2+". This was done by differencing the equation for two points, x, x+Sx and obtaining a linear elliptic differential equation for the difference quotient. The interior estimates then yielded uniform estimates for the difference quotient which were independent of 62, and letting dx --j. 0 the desired differentiability was obtained. This standard method works here, as in the proofs of Theorems 7.2 and 9.2. On taking differences parallel to the flat boundary face and applying Theorem 7.2 we obtain for fixed positive 6 a uniform estimate for the norm I of the tangential difference quotients of M from-which it follows that derivatives of u of the form D,Dzou belong to Ca in .ZR . Since we may solve for the remaining derivative Dp+lzl

of order Z,+1 from the equation F ( x , - -, Dmu) = 0- differentiated with respect to t - it follows that u E C l ~ + l + ~ in ZB . Continuing in this way we obtain the result.

Consider now a nonlinear equation in integral form such as would arise from a problem in the calculus of variations, and a solution satisfying nonlinear boundary conditions. Again we shall imagine the solution to be given in ZR and to satisfy the boundary conditions on the flat face:

Then u belongs to CQa in Z;, .

F ( x , t , u, * * * , D2"u) = 2 Dflap(x, t , zc, * - * , D z u ) B

(11.3) = 0 in y T R ,

2 DZb,,,(x, u, - - - , D z u ) = 0 on t = 0, i = 1, . . * , m.

Here summation is for 1/31 5 max (0, 2m-4, [yI 2 max (0, mj-l) , and 1 < I, is a fixed number which is not less than the maximum order of differentiation with respect to t occurring in the boundary operators. By the integral form of the problem we mean that for every C" function [(x, t ) , with support in the interior of ZR, and for every C" function [ ( x ) , with support in 1x1 < R, u(x, t ) should satisfy (analogous t o (9.2))

We assume our solution to be of class Cz+u in ZR and the first variation of the system with respect to u to be elliptic and to satisfy conditions (i), (ii) of Section 7. The first variation is described by the linear system (meant in the integral form of course)

684 S. AGMON, A. DOUGLIS AND L. N1RE:NBERG

for a function w(xJ t ) .

THEOREM 11.2. Assume that u E CE+a in ZR , Let p 2 l be a fixed integer and assume that the aB have Holder continuous (exponent a ) derivatives up to order p-l if 1 < 2m, or p-2m if 1 2 2m, while the b j , y have Holder continuous derivatives (exponent ci) up to order p-l if l < mj , or p-m, if 1 2 mi . The% u belongs to CP+“ in ZR .

Proof: From the interior differentiability theorems (see Section 8 of [ lo]) we know that u E Cp+a in the interior of Z;, . Our proof that u E Cp+a in ZR is similar to that above and to the proof of Theorem 9.2. We shall consider here only the case p = l + 1 . By taking tangential (to the flat boundary) differences of the system and applying Theorem 9.1 we show in the usual way that derivatives of the form D, Dz u belong to Ca in ZB. We have only to show that D;+lu belongs to Ca in ZR.

In case 1 2 2m-1 this follows from the fact that we may solve for in terms of the derivatives D, D z u and lower order derivatives from the

equation D:-2m+1F = 0; in this the ellipticity of the equation is used. In case 1 < 2m-1 we observe that, since the system is elliptic with respect to u, there is a term in (11.3) of the form

with a a p D : u # 0. Because of our conditions on the aa , and because we have shown that D, D z u E Ca in Z;,, we may write the equation (1 1.3) in the form (to be understood in the integral sense)

D21n-Z a(x , t, u, - * * , Dz.u)

p m - 2 ~ ( x , ~ , - - . , D ~ u ) = 2 D ” u ~ , IYI <2m-z

where the u, belong to Ca in ZR . We may therefore apply Lemma 9.1 to a to conclude that

D,a(x, t , * . * , D z u ) E Ca in Z;, . Since i3apD:u # 0 it follows easily that Di+lu E Ca in ZR. This completes the proof of the theorem.

We now indicate how the differentiability Theorems 11.1, 11.2 may be sharpened with the aid of the L, estimates of Chapter V. Using Theorems 15.3, 15.3’, we may, in most cases, replace the assumptions u E CZofa and u E CZ+Or in these respective theorems by the weaker assumptions u E C1o and U E C L .

We start first with standard interior differeintiability results [32].


THEOREM 11.3. Let u be a solution of (11.1) of class czm in a domain. Assume that F i s once continuously differerttiable with respect to its arguments and that the associated first variation, the operator L , i s elliptic. Then u E C2m+a' for every positive u' < 1. If F has Holder continuous (exponent u) derivatives up to order 1 - 2m > 0, then u E Pa.

For equations in integral form a similar result maintains:

THEOREM 11.4. Let u be a solution of the differential equation given i n (11.3) and be of class C' in a domain. Assume that the aa aye once continuously differentiable with respect to their arguments, and the associated first variation is elliptic. Then u E CQa' for every positive u' < 1. If the aa have Holder continuous (exponent u ) derivatives up to order j , then u E Cz+j+a.

The proof of the first statements in the first (respectively second) theorem is carried out in the usual way by taking difference quotients, and applying the interior L, estimates to estimate the L, norms of the derivatives of order 2m (1) of the difference quotients of u, choosing$= (n + 1) / (1- u'). I t follows from the well known results of Sobolev [37], see Lemma A5.1 part (c) in Appendix 5, that the derivatives of order 2 m - 1 ( I - 1) of the difference quotients of u satisfy a uniform Holder condition (exponent a') in any compact subdomain, and hence that the same is true for the derivatives of u of order 2m ( I ) .

The remainder of the theorems follows from the differentiability theorems in [lo].

Turning now to differentiability at the boundary, consider again a solution of ( l l . l ) , (11.2) in ZR and assume that the linear system of first variation satisfies, as before, the conditions (i) and (ii) of Section 7. Then we have the following

THEOREM 11.1'. Let I , = max (Sm, mj + l ) , and assume that u E C'1 in Zfi . Assume that F and F j have continuous derivatives zbp to orders 4 - 2% 4- 1 and I, - nzj + 1, respectively, with respect to all their arguments. Then zb belongs to C I I + ~ ' in .YR for every positive u' < 1.

Consider finally a system in integral form (11.3), with the first variation, as before, satisfying conditions (i) and (ii) of Section 7,

F ( x , t , u, - *

2 D:b,,,(z, - - ., Dz-lu) = 0

DZm u ) = 2 Dbag(x l - * *, Dzu) = 0 in ZR, on t = 0, j = 1, * * *, m;

here summation is for IBI S max (0,2 m - I ) , IyI I max (0, mj - 1 + l ) , and 1 is to be greater than the maximum order of differentiation with respect to t in the Bi.

THEOREM 11.2'. Assume that U E C ~ in .ZR, and assume that with respect to all their arguments, the aB have continuous derivatives of first order if


1 5 2nt, OY up to order 1 - 2m + 1 if 12 2m, while the b j , y have continuous derivatives of first order if 15 m,, or up to order 1 - m, + 1 if 12 mj. T h e n u belongs to Pa‘ in ZR for every positive a‘ < 1.

The proofs of the theorems follow the now familiar procedure. In each case one differences the equations in a direction parallel to the boundary, obtaining a linear system for the corresponding difference quotient of u, and applies Theorems 15.3, 15.3’ for f i = (n + 1)/(1 - a’) (see the Corollary of Theorem 15.2). In this way, in either theorem, one obtains a uniform Holder condition (exponent a’) in a compact subset of CR for the derivatives of order 1 - 1 of the tangential difference quotients of u, from which it follows that derivatives of the iorm D,Dz-lu belong to C“’ in ZR. In the case of Theorem l l . l ’ , the Holder continuity of the remaining derivative 0:. follows from the fact that by applying the implicit function theorem to the equation this derivative may be expressed as a function of the others and lower order derivatives.

In the case of Theorem 11.2’, in order to infer the Holder continuity of the derivative Diu, we set for /? = (0 , . . ., 0, 1)

a/&, t , u (x , t ) , * ’ * ) DZu((2, t ) ) = v(x , t ) . It is easily seen that it suffices to prove the Holder continuity of v ( x , t ) in ZR . From the L, estimate of the previous paragraph we see that in every ZRP6 the derivatives of u of the form D,Dzu belong to L, for p = (m + 1)/ (1 - a‘). Hence the first derivatives D,v belong to L, in ZR-8. Thus it suffices to prove only that D,v belongs also to L, in ZR-8. But one sees easily, via the differential equation, that v satisfies an equation of the form ( 9 4 , to which Lemma 9.1’ may be applied to give the desired result.

Theorems 11.1, 11.1’ together yield a general differentiability theorem at the boundary for nonlinear problems, as do Theorems 11.2, 11.2’ for many problems in integral form.

12. Existence and Properties of Solutions

12.1. We shall discuss mainly the use of the Schauder estimates of Section 7. Similar remarks apply to the estimates for equations in integral form of Section 9 and to the L, estimates of Chapter V. We consider the equation (12.1) LU = F in a (for simplicity) bounded domain Fs, with the boundary conditions

(12.2) B j U = @*, j = 1 ; - - , m ,

over the entire boundary a, and assume that these satisfy all the conditions of Theorem 7.3 with 8 = Fs, and 1 equal, say, to 1, . Our solution u, as well as


F and 0,. , are assumed to be of classes Clota, Czo-zm+a, CLo-mf+a, respectively. We say that the system (12.1), (12.2) is solvable if it admits a solution u in CZa+OL for every such F , Qi , and that uniqueness holds if the only solution in C2ofa of the homogeneous system is u = 0.

Some immediate consequences of Theorem 7.3 are THEOREM 12.1. There are at most finitely many liizearly independent

solutions of the homogeneous system (i.e., with F = Q,. = 0).

THEOREM 12.2 (COMPACTNESS). Let ui be a sequence of uniformly bounded solutions of the systems

Liui = Fi in B Bj,iu, = Qj,d 0% 9, i = l , . . . > m,

which satisfy, zmiformly for all i, the conditions of Theorem 7.3. Assume that the coefficients of the systems tend uniformly to the corresponding coefficients of L and B, , and that F , and Qj,& tend uniformly to F and di,, as i -+ 00.

Then a subsequence of the ui converges in €he norm IuJl, to a solution in Czo+a of the limit system.

The proofs of Theorems 12.1 and 12.2 are immediate. Consider the systems L, B, of (12.1), (12.2). For E , k' > 0 we define

an ( E , A')-neighborhood of the system as follows: A system 1, B,, i = 1, - - - , m, (these operators are of the same orders as L, B,) belongs to the ( E , k')-neighborhood, if the coefficients in J ( B , ) differ from the corresponding coefficients in L(B, ) by less than E , and have norms I I l o - - 2 m + a (I I l , - m , + a ) bounded by k'. Here K' 2 the constant k occurring in (iii) of Theorem 7.3.

THEOREM 12.3. Assume that uniqueness holds for the given system (12.1), (12.3). Then it is true that: (a) For any k' > k there i s a n E > 0 such that for any system 2, Bj in the ( E , k')-neighborhood of the given one we have, for 21 E CLo+a,

IUIlo+a 5 constant * (IZuIl,-zm+a + 2 I'5UIlo-rn,+a)r 5

with the constant depending only on the given system and on k'. Thus uniqueness holds in the ( E , k')-neighborhood. (b) Assume that for some k' there i s a system in every ( E , k')-neighborhood of the given one which i s solvable. Then also the given one i s solvable.

Proof: With the aid of Theorem 12.2, part (b) follows easily from part (a), and we shall only prove part (a). Its proof is similar to that of Remark 2 after Theorem 7.3. Suppose that part (a) is not true. Then there is a sequence of systems L, , B j , i , i = 1,2, e e e , satisfying the conditions of Theorem 12.2, and a sequence of functions ui in such that Iuillo+a = 1, and


oi = (lLiuilzo-2m+a + z: l B , , i ~ i l z o - m $ + a ) + 0 as i --f j

For i sufficiently large, the systems satisfy uniformly the conditions of Theorem 7.3 (with different constants) so that, by (7.8) (for (21 = D),

1 = IuilZofa 5 constant - (cri+Iuil0). By Theorem 12.2 a subsequence of the ui converges in the norm I lz0 to a solution u in Czofa of the homogeneous original system. But by the preceding inequality this limit solution u cannot be zero, contradicting the uniqueness assumption.

THEOREM 12.4. Let L be a differential operator of order 2m, and M a differential operator of lower order, such that L and L+M satisfy all the conditions of Theorem 7.3 relative to given boundary operators B j . Assume that for every set of data F E Czo-2m+a, dz, E Czo-*j+a there exists a unique solution in czo+a of

Consider the problem (12.3)

Except for a discrete set of values of 1 the system (12.3) has one and only one solution in CZofu for arbitrary F E Czo-2m+a, dz, E Czo-*L~+a. Furthermore if for some L there is at ,most one solution in Czofu of (12.3) (i.e., zmiqzleness holds), then in fact a solution exists for that A.

Proof: Let dzj be given values of the boundary expressions B, u, and let v be the solution of L v = 0, B,v = dzj on $. We may subtract v from u and thus reduce our problem to one with homogeneous boundary conditions

(12.4)

Denote by L-l F the solution w of L w = F with homogeneous boundary data. (12.5) u ~ A L - ~ M u = L-l F.

Lu = F in %, B j u = djj on @.

(L.+AM)u = F in 9, B,u = dj, on $.

(L+AM)u = F in 9, B j u = 0 on 6.

Then our equation may be written in the form

Since the order of M is at most 2m-1 we see, for u B Clo, that 1~-1Mul&, I constant I @ ,

with the constant independent of u. Since the set

]ullo+a 5 constant is compact in the Banach space Czo it follows that L-lM is a completely continuous transformation of Czo into C1o. We see furthermore that a solution u in Czo of (12.5) belongs also to so that the equations (12.4) and (12.5) are completely equivalent. The desired result follows from the Riesz theory for completely continuous operators applied to equations (12.5).

Consider now equation (12.5) for functions u in Czo.

BOUNDARY ESTIMATES FOR ELLIPTIC EQUATIOWS 689

12.2. We now describe the continuity method in a slightly more general form than the usual.

THEOREM 12.5. Let L , , B3,t be a fami ly of systems depending on a parameter t , 0 5 t 5 1, satisfying zcniformly in t the conditions of Theorem 7.3. Assume thai the coefficients of L, vary continuously with respect to t in the I I norm and those of Bj, continuously with respect to t in the 1 1 20-nz,+a

norm; here we assume that the mj are independent of t. Consider the system (12.6) i = 1, - - - , m,

for arbitrary F in Qj in C ' O - ~ J ~ ~ . Assume (a) for t = 0 and A = 0, the system (13.6) is uniquely solvable, (b) for every t there i s a complex number A, such that uniqueness holds for (12.6) with 1 = A,. Then for each t the system (12.6) has one and only one solution in for arbitrary F , G, -except possibly for a discrete set of values of A. Furthermore uniqueness for (12.6) (for any t , 1) i qb l i e s existence (for the same t , A ) .

(L,+l)u = F in B, Bj,,u = Gj on a,

QUESTION 1. Does the theorem hold if assumption (b) is dropped? QUESTION 2. For n > 1 can any two elliptic operators L o , L , of order

2m be connected by a one parameter family of elliptic operators (depending continuously on t ) L , , 0 5 t 5 1, redwing for t = 0 to the given Lo and for t = 1 to the given L,?

Concerning Question 2 we mention that for n = 1 the set of elliptic operators satisfying the Condition on L is connected. This is not difficult to show.

In general the continuity method described here raises the problem of characterizing the class of boundary value problems L, B, that can be connected to any given one by a family of such problems satisfying our conditions.

Proof of Theorem 12.5: For simplicity we treat the case that the Bj, , = B, are independent of t. As in the preceding proof it suffices to consider (12.6) with homogeneous boundary data B j u = 0 on a. Denote by

the subspace of the Banach space C1o+" consisting of those functions satisfying the homogeneous boundary conditions R, u = 0.

We may regard L, as a bounded transformation of into C10-2m+a.

Let T be the set of points on the interval 0 5 t 5 1 for which our assertion holds; by Theorem 12.4, T contains t -- 0. We shall show that T is both open and closed-and hence is the entire interval. The openness is simple. Suppose t E i' and let be such that L,+x maps &fa one-to-one onto Clo-zmfa. According to Remark 2 after Theorem 7.3, L,+x has a bounded inverse. The operator L,+,,-L, has small norm for small (btj, and it follows by a standard argument that Lt+dt+X has a bounded inverse for /dtl small. That t+dt lies in T follows now from Theorem 12.4 (with L = Ltfst+A, M = identity).


To prove that T is closed suppose that {ti> is a sequence of points in T converging to t. We will show that L,+il, is invertible; this will imply as above, via Theorem 12.4, that t E T .

Consider the operators L, = L t d . Since for each i the operator Li+A has an inverse (except for a discrete set of A ) , there is a sequence Ai --f A, such that the operators L,+& are invertible, and the result follows from Theorem 12.3(b).

12.3. We now state a general perturbation theorem for nonlinear elliptic equations. It is in fact in proving such a result that the strength of the Schauder estimates becomes apparent; we know of no other way to establish this result.

THEOREM 12.6. Consider a nonlinear differential equation and nonlinear boundary conditions (11.1)' (11.2) depending o n a $arameter z,

F,(x, u, - - - , DZmu) = 0 in 5B9 (12.7)

Fi,,(x, u, - . , Dmfu) = 0 in a, i = l , . - . , n t ,

and let uo be a solution in CZofa for t = 0, say uo 3 0. Assume that F , , F , , have continuous first and second derivatives with respect to z, and that F , , F!,, and these derivatives have Holder continuous (exponent u ) derivatives ufi to orders 10-2m and lo-mj, respectively, with resfiect to the other arguments. Asswme further that the linear system for a function w(x) ,

satisfy all the conditions of Theorem 7.3 and possess a unique solution ze, in Czo+a for every f in Czo-zm+a and 4, in CzO-mjfa. T h e n for [tl sufficiently small the system (12.7) fiossesses a unique solution in CZofa.

With the aid of the Schauder estimates of Theorem 7.3 the proof is routine. Let Wl[:f]+Wz[+l, - 0 . , +,,$I be the solution of (12.8), where W l [ f ] is the solution of Lw = f with homogeneous boundary data, and W,[+, , - - - , q37n] is the solution of Lw = 0 with the boundary data of (12.8).

Let us now write (12.7) in the form

-zLu = F,(x, u, * * , D2"u)-zLu = RCu] in 9, -zB,u = Fj,,(z, u, * * - , D2"u)-zBju = R,[u] on a, i = 1, * - a , m,

For z small the right-hand side is small, of first order in z. Using


iterations one may prove the existence of a solution in CEo+-a for t sufficiently small.

12.4. From now on we shall confine our remarks to a special linear problem, namely, the Dirichlet problem: Ri = aj-l/&zj-l, where n denotes the normal to the boundary. Here m, = j-1, E , = 2m, and the Comple- menting Condition is automatically satisfied. For convenience we shall assume that the boundary 9 is of class C". A boundary of class CPrnta can then be handled by a suitable approximation procedure.

Until recently the Dirichlet problem was treated only for strongly elliptic operators L , in the sense of Vishik [38] (see [31]), an operator L of order 2m being strongly elliptic if after multiplication by a suitable function its real part is elliptic. Thus after multiplication by the factor, we may write for a strongly elliptic operator (12.9) ( - l )rn 9 e L ( z ; E ) 2 a(z)]E12m, where a(.) is a suitable positive function.

As mentioned in the introduction, recently both Schechter [36] and Agmon [3] using L, estimates and Hilbert space theory obtained general existence theorems for a wide variety of boundary value problems. In particular, for an elliptic operator L (satisfying the Condition on L of the introduction), having sufficiently differentiable coefficients so that the formal adjoint operator L* is well defined, they proved the following: There exists a solution of the Dirichlet problem for L provided that uniqueness holds for the Dirichlet problem for L*-uniqueness being in the class of solutions having, say, square integrable derivatives up to order 2 m in the domain 9. Using this result it is easy to prove the following theorem (in the same way, using the results of Chapter V, we may prove a similar theorem for solutions having square integrable (or L,) derivatives up to order 2m in 9, assuming boundedness of the coefficients and continuity of the leading ones).

We always assume that the Condition on L is satisfied. THEOREM 12.7. If the elliptic operator L has coefficients in Ca, and if the

Dirichlet firoblem

has at most one solution in C2rn+a (i.e. uniqueness holds), then in /act it has a solt&on for every F in C" and Gj in Cam-*+lta.

Proof: It suffices to consider only homogeneous boundary conditions,

Assume first that all coefficients in L and B j are, say, C" functions. We shall make use of the following norms for functions in B, where integra-

aji = 0.


tion extends over 9, for a non-negative integer j :

(12.10)

We denote by Hi,L, (k,,&,) the closure in the norm 1 1 l[j,LB of C" functions (Jvith compact support) in 3. Denote by S the intersection of H2m,L, and Hm,L8. By the results of Schechter and Agmon the Dirichlet problem for the formal adjoint operator L*, L*v = F , with homogeneous boundary conditions, has for F in HO,La = L, a solution v belonging to S . The solution 21 can be made unique by the additional requirement that it should lie in the orthogonal (with respect to L, scalar product in B) complement of the null space of L*, i.e., in the set of solutions of the homogeneous equation (the set is finite dimensional). Denote by V the intersection of the complement with S, and denote by P the complex conjugates of the functions in V. The map L* : V +- L, has an inverse L*-l.

We propose first to find a solution u in P of Lu = F , for given F in H , ; if F is in C" in 3, the solution u belongs also to C" in $I. Our proof of the existence of such a u is based on the observation that

L-L* = &f

is an operator of order < 2m. Here E* is the operator whose coefficients are the complex conjugates of the coefficients of L*. Clearly the map L* : P + L, has a well-defined inverse L*-l. For u in P the equation (L*+M)u = F is equivalent to the equation

u --E*-lMufE*-l F . Since the operator M is of order lower than 2m the operator L*-lM is a compact operator. Hence by the Riesz theory for compact operators the last equation has a solution if and only if uniqueness holds. But if w E F' is a solution of

then it is also a solution of Lu = 0, and hence of class C" in % and so, by our uniqueness assumption, is identically zero. Thus the equation above, and hence Lu = F , admits a solution in V .

It follows then, in a standard way, that the null space of L* is in fact zero, so that V = S . Thus if uniqueness holds for L it holds also for L*.

The solution u obtained above is in C" in % provided that F is. If now we are given a function F in Ca we may approximate it by C" functions Fi , solve the corresponding equations Lu, = F, and use Theorems 12.2-12.6 to conclude that there is a solution u E C2m+u of Lu = F . Thus we have shown, for a system with C" coefficients, that uniqueness for the Dirichlet problem implies solvability.

= --E*-l&fu,

BOUNDARY ESTIMA4TES FOR ELLIPTIC EQUATIONS 693

To complete the proof, i.e., to dispose of the C" hypothesis we approximate the system by systems with C" coefficients, and the desired result follows from Theorem 12.3.

This completes the proof of the Theorem. 12.5. In view of Theorem 12.4 and Theorem 12.7 it is natural to ask

the following question (which is a special case of Question 1 after Theorem 12.5). For a given elliptic operator L of (12.1), does there exist a constant 1 such that uniqueness holds for the Dirichlet problem for the operator L-FA?

For strongly elliptic operators with sufficiently smooth coefficients the existence of such a 1 follows from GBrding's inequality (see [ll]). Recently hlorrey f29] announced that such a uniqueness result also holds for strongly elliptic L with coefficients in C". We include here an elementary proof of a slightly more general result, Theorem 12.8. At the same time, Rrowder has also found a proof of Morrey's uniqueness result.

DEFINITION. We shall say that L is weakly positive semidefinite if

(12.9) ' where L' is the leading part of L.

An immediate consequence of (12,9)', via Fourier transforms, is that for any xo in 9 the operator L(x0 ; D) with constant coefficients satisfies

( - l )m W e L'(x; 9) 2 0 for real 8,

(12.11) W e ( L ' ( Z 0 ; D)C, C ) 2 0 for every C" function 5 with compact support. Here ( , ) represents the L, scalar product in 9.

In particular, any strongly elliptic operator, i.e., one satisfying (12.9), is weakly positive semidefinite.

THEOREM 12.8. Let L satisfy condition (1) and be weakly positive semidefinite. Assume that the coefficients of L are bounded and that the leading coefficients are continuous. For 1 sufficiently large positive we have

(12.12) llull~m,L2 I constant * I I ( L + A ) U ] ~ ~ , ~ ~ for 6 S, where the constant i s independent of u; hence we have uniqueness of the Dirichlet Problem for L f l .

Here we have used the notation occurring in the proof of Theorem 12.7;

As an immediate consequence of Theorems 12.8, 12.4, 12.7 we have the following statement which contains the results of [29].

COROLLARY. Let the operator L in (12.1) have coefficients in Ca(%) and assume that it i s weakly positive semidefinite. Then for all except possibly a discrete set of values of 1, the Dirichlef problem for the operator L + 1 is uniquely solvable. Furthermore uniqueness implies solvability.

s = H2m,L, n H,,LS *

694 S . AGMON, A. DOUGLIS AND L. NIRENBERG

The corollary may also be proved without the aid of Theorem 12.7, using instead the existence theory for strongly elliptic operators, and Theorem 12.3(b), for a weakly positive semidefinite operator may be approximated by solvable, strongly elliptic operators with C" coefficients.

In proving the theorem we shall make use of the integral estimate proved in Chapter V (here we take the case f i = 2)

(12.13) I I 4 IEm, L, 5 c (I ILUI I:, La + I I 4 I:, L,) 9 24 E s,

where the constant c is independent of u, We may assume that c 2 1. Proof of Theorem 12.8: We have

(12.14) I [ (L+A)ulI:,La = IILU~I:,L,+~J We(LG, f.t)kJ2IIuII;,~,. If we can show that

1 2c

(12.15) 22 9:e(L.u, a ) L - - l l ~ l l ~ ~ , L ~ - ~ ~ ~ l l ~ l l ~ , L ~ ~ - c ' l l ~ l l : , L ~ 8

where C' depends only on L, then the inequality (12.12) follows; for, combining (12.14), (12.13) and (12.15) we have

from which (12.12) follows for L sufficiently large. The proof of (12.15) has some similarity to the proof of Garding's

inequality [ll] for the non-constant coefficient case. From (12.11) we obtain immediately: If L satisfies the conditions of Theorem 12.8, and if its leading coefficients differ from their values at a point Po by less than E, then for u in S

(12.16) g'e(L% 24) 2 - ~ I I ~ l 1 2 m , L , I I ~ 1 1 0 , L ~ - ~ I I ~ l I z m - . 1 , L p l l ~ l I 0 , L ,

where C depends only on the bound of the coefficients of L. This comes from writing L ( P ; D) = L(Po ; D)+error term, and applying (12.11) to L ( P , ; D); the contribution of the error term is clearly bounded from below by the right-hand side of (12.16).

In the following we shall use the notation Ci,k to mean an expression which may change from case to case, but which is bounded in absolute value by constant - I I U ] ~ ~ , ~ , I I U ~ ~ ~ , ~ , , the constant differing from case to case depending only on L. C,, C,, . * * will denote constants depending only on I..

Let

UOUNDARY ESTIMATES FOR ELLIPTIC EQUATIONS 695

be a partition of unity in %, the w j being real C" functions of compact support such that, in % n support of any oj , the leading coefficients differ from their values at a point by less than 1/4c. (That this is possible follows from the continuity of the leading coefficients, but it may clearly be possible under weaker hypotheses.) Then

21 ge(Lu, u) = 2iz 2 92e(wjLu, w , ~ ) = 21 2 Be(L(wjzc), ojzc) i

+izCz,-*,o . Applying (12.14) to the term We ( L ( w ~ M ) , o j u ) we find

(1 2.17)

For the last terms we have

(11.18)

+ C 3 l I 4 l L , L, By a known estimate (see Lemma 14.1) we have

1 C,I I4 I L , L, 5 l l Z 4 Li, t,+C41 I4 1:

so that, inserting in (12,18), we find

Inserting this, in turn, into (12.17) and using the fact that c 2 1 we obtain (12.15) with C' = C,.

We conclude with some remarks about equations in integral form. We mention first that the preceding results have analogues for equations in integral form which we will not bother listing. However, we shall state the


analogue of Theorem 12.7. Again assume the boundary to be C" and B, = aj-yanj-1.

THEOREM 12.9. Let L be written in integral f o rm as in Theorem 9.3 with 1 2 m-1. Irz terms of local coordinates (2, t ) with the boundary represented by t = 0, the equation thus has the fo rm of (9.2). Assume that there is at most one solution of the equation of class Clf". T h e n there exists a solution for F , i?? C" and Qj in C1-j+l+".

The proof is similar to the proof of Theorem 12.7. We shall mention some consequences of the theorem. Consider L

written in integral form with 1 = m:

(12.19) Lu = Dba,,,Dpu, IbL I t l Sm

where all coefficients belong to C1+". Assume that the operator is weakly positive in the sense that for every C" function 5 with compact support in 5D the inequality

(12.20)

holds with c a fixed positive constant. Let @jj E Cm--i+Or, j = 1, - - - , m, be given functions on the boundary. We seek a solution in Cm--l+" of

(12.21)

THEOREM 12.10. 1) T h e system (12.21) admits a unique solution u in Cm-l+a. 2) If all the a,,, belong to Ck+" and Qj E C1c+m--i+a, then u belongs to Ck+m-l- ta .

Proof: The second part of the theorem follows from Theorem 9.3. To prove the first part we see from the conditions on the coefficients that we may also write the equation (12.19) in integral form with 1 = m-1. By Theorem 12.9 we need only prove that the solution uo of class Cm-l+a of the homogeneous Dirichlet problem is zero. But by Theorem 9.3 this solution is of class Cm+a, and applying (12.20) which extends t o such functions we find that u,rO.

Suppose now we are given L in the form

(12.22)

with Qj E Cm-j+". Assume that all the coefficients belong to Ca and that for


1pI 2 m, aS E CIBI-m+l+a. Assume also that the operator L is weakly positive, i.e., for every C" function C with compact support

J" LC z ax 1 c 1 l[12dx

for some positive constant c. Then as an easy consequence of Theorem 12.10 we have

THEOREM 12.11. The system (12.22) has a unique solution u which belongs to C2m+a in % and to Crn-l ta in 3.

13. Schauder Estimates for Semilinear Equations

Consider an elliptic operator L and boundary conditions B, satisfying all the conditions of Theorem 7.3. Here we shall take 59 to be a bounded domain, r = $, 2l = %, and, for simplicity, I = 2,. Consider the system

LU = F in 9, B,u = @, on a, j = 1 , - * -, m,

where now we permit F and @, to depend on u and some of its derivatives

F = F ( x , U, * * * , D2"+l@), @, = @,(x, U, * * , Drnj-lu 1.

Nagumo [30] has shown for a second order equation that it is still possible to obtain Schauder estimates analogous to those of Theorem 7.3 provided the nonlinear terms do not grow too fast. This is done by using Lemma 5.1 in its original form rather than its watered down form (7.4). This can be carried over directly to the general situation. We wish to estimate lull,+a in terms of lul, and known data.

Suppose that for any function u ( x ) E CZota inserted into F the resulting function

satisfies P&) = F ( x , u(z) , * - * ) D2m-%(x))

l ~ u l l o - 2 m + a 5 Kl+P (A, where (a) I is a finite sum of terms each of which is a finite product IT [u]: with ai < 1 , fa and fiiai < Z0+u, (b) J is a similar finite sum of finite products with each a, < Zo+a and 2 $*ai = l,+a; K is a constant, and p(S) is a fixed function defined for positive S which is o(S) as S --f w (12.6). This property would for instance be satisfied by a function F which consists of a finite sum of terms each of which is a finite product n, (Diju)*f times a smooth function of x, with i, < 2nz and qi positive integers, and 2 i j q j < 2m.

698 S. AGMON, A. DOUGLIS AND L. N1RE:NBERG

Similarly about Gj we assume that for any function u E C1ofa inserted into cDj the resulting function

satisfies 6j,u(x) = c D j ( X , u ( x ) , * * -, Dm,-lu(x))

I&j,uIl,-m,+a 5 KI+pdJ)- Then we have

THEOREM 13.1. If a bound for lulo is known it is possible to estimate

The proof of the theorem follows rather directly from Theorem 7.3 with the aid of Lemma 5.1, and will be left to the reader.

It is also possible to replace p ( J ) in the assumption by constant times J but the result then holds-or at any rate this proof evidently works-only if the constant is sufficiently small, the degree of smallness depending on the size of Iul0. (See Nagumo [30] for a careful analysis for a second order equation.)

l~tl lo+a *

CHAPTER V

L, THEORY

14. Constant Coefficients

The letter p will denote a fixed number > 1. We first define some integral pseudonorms and norms analogous to those in Section 5. We consider functions in a domain 3 in (nf1)-space which, for simplicity, is assumed to be either a half-space x , + ~ > 0 or a bounded domain of class C2 as defined in the discussion preceding Theorem 7.3, and we define

We define by Hj,Ls (see Section 12) the completion of C" functions in 3 with respect to the norm 1 ) J [ i , L , . Clearly, Hj,LD forms a Banach space.

We shall make use of the following analogue of Lemma 5.1 (see, for instance, [31] for a proof).

LEMMA 14.1. Suppose i ==c j , then for a n y E > 0 there is a constant c de$endi.lzg only on E, i, j , + and 5D such that


For functions 4 defined on the boundary ($ of %, and for j a positive integer, we introduce related classes of functions, seminorms and norms similar to those in (3.13)'.

Hi-l/s, L, is to be the class of functions r$ which are the boundary values of functions z, belonging to Hj ,LP in 9. In this class we introduce the norms

(14.2)

where in both cases g.1.b. is taken over all functions v in H,,L, which equal $ on the boundary.

From now on .we shall usually write the abbreviated forms I C4] 1,-1/2, , llul/j etc., omitting L, .

The following special result describing the effect on the norm ~~r$~~,.-l,s,Ln by a change of independent variables will be needed.

Let y be defined on the boundary of a.domain 9 and assume that y vanishes outside a subset y of the boundary 6. Assume that there is a ( (n+ 1)-dimensional) neighborhood U of y such that n 53 can be mapped in a one-to-one way onto the closure of a hemisphere ZR in (nf1)-space, with D n mapping onto the flat part of the hemisphere. Assume that the mapping T, together with its inverse, has continuous derivatives up to order j bounded by a constant K. We consider the hemisphere ZR as lying in our half-space t 2 0.

l[411~-l/D,Lp = g.1.b. IPIlj,L,

~ ~ $ ~ ~ ~ - l / t l , ~ D = g.leb. IlvllS,L, J

The function y goes over into the function

+(x) = VJ(Wx, 0)).

LEMMA 14.2. There is a constant C depending only on K , 9, j , n , and the distance from y to the boundary of U such that

C-lIIyIIj-l/p,L, 2 l l+ l l i - l /D,Lp 5 ' l l ~ l l j - ~ / n , ~ , . Proof: We shall prove only the first inequality, the other following in

the same way, Let v ( x , t ) be a function belonging to in the upper half- space t 2 0 which equals +(.) on the boundary and satisfies llzlll, 5 2~~+/~j-l~,. Let u(z, t ) be a C" function defined in t 2 0 with support on ZR, and such that u = 1 on Ty. Under the mapping T the function v(x, t ) o ( x , t ) goes over into a function u with support in U n 93, which equals y on y. One easily verifies that

Ilull, 5 constant - llvll,, the constant depending only on K and the distance from y to the boundary of U , and the lemma follows.

In deriving L, estimates up to the boundary we shall restrict ourselves


to solutions of elliptic equations in a bounded domain 9, satisfying the boundary conditions on the entire boundary. In general, the problem being local, one can

(a) obtain estimates near the boundary, analogous to those in Sections 7 and 9, for solutions satisfying boundary conditions on merely a poytion of the boundary,

(b) consider unbounded domains. (a) and (b) give rise to technical points similar to those, treated in

detail, for the Schauder theory. I t is only in order to avoid these, and to shorten the discussion, that we make the restrictions. Furthermore, except for some special results, we shall not consider equations in integral form. These may be treated by similar arguments with the aid of Lemma 9.1'.

In this section we consider only the constant coeificient elliptic operator L, of (6.1), containing only terms of order 2m. We recall first the basic L, estimate used in deriving the interior L, estimates (see [32, 19, 71).

THEOREM 14.1'. If has comfiact support and if its derivatives of order 2% belong to L,, p :> 1, then

j' 1 ~ 2 m u l p d x 5 constant - j' 1 ~ u 1 , d x .

Furthermore, if 1 i s an integer > 2m, and if I[LZ~]I~-~~,~, i s finite, so i s I[@lll,L, P and

I [%I 11, L, 5 constant I [Lul I2-2m. L, . Here the cowtants d e f i e d only on p , n and the conshad A of (1) in the introduction.

The first statement follows from the representation

D 2 m ~ = D 2 " r * Lzd + constant * Lu, where ;i: denotes convolution (in all variables), and l' is the fundamental solution (4.2); the kernel D2mr satisfies the conditions of the Calderon- Zygmund theorem, which yields the result. The second follows by the usual difference quotient procedure.

We mention, in passing, that for the equations in integral form of Section 8 (see (S.l)'), namely

LG = 2 oaf,, 5 2m-Z, for some fixed l < 2m, if u and f a all have support in the unit

sphere, and if u has derivatives of order Z in L, , the estimate

is valid. This is proved again from


cg constants,

by applying the Calderon-Zygmund theorem to the terms with [,'?I = 2m-1, the other terms involving less singular kernels being handled by more elementary means. Equations in integral form will not be mentioned again in this section.

We consider now, in the half-space, the system (6.1), or

(14.3) Lz4 = f , B j u = $ j ,

t > 0, t = 0, i = 1, - - , v a ,

with constant coefficients. Let 1, = max (2m, m,+l), and let I be an integer 2 I , .

THEOREM 14.1. Assume that a ( P ) vanishes for /PI 2 1 and belongs to H I , L , in the half-sflace t 2 0. Then

(14.4) llullz d constant. (l l f l lL-2m + 2 ll$~lIz-m#-1/11)> i

where the constant defiends only on 1, p and the churacteristic constant E . Proof: We see easily that it suffices to prove this inequality for C"

functions, and to prove that 1 [ZCI I~ ,~ , is bounded by the right side of (14.4). Our proof makes use of the representation formula (6.3)

DZ% = D'v + 2 I,, and (6.5), (6.5)'. Here v is given by (4.5). Since the kernels D Z m r satisfy the conditions of the Calderon-Zygmund theorem we have

I[v]ll 5 constant - [vN]Iz-zm 5 constant * I [ f ] I g - z m It follows that

I [Dk-mj-l yjI 11-1/11 I I [vI 11 5 constant * 1 [/I I z-zm *

To complete the derivation of the inequality we show that for each I j

(14.5) i r ~ ~ i i ~ 5 constant - 2 wX(+j-~j)i i l- l / l l . lB1=z--ml-l

This together with the preceding inequalities yields the desired estimate for l[zt]l l = ] [z&][~,~, . ConsiderI,given,say, by (6.5). Itmaybewrittenin theform

where I j , r = DlA("+g-Z+md/2I( ,,~ * DkA('-mj--l)/2 ($j-yj)*

The estimate (14.5) then follows by applying Theorem 3.4(c) to this representation of the function I i , k . In a similar way we treat I , in the form (6.5)'. This completes the proof of Theorem 14.1.


The following is a simple

COROLLARY. Under the conditions of Theorem 14.1 the representation formula (4.7) holds with I, re9laced by 1,.

15. Lp Estimates for Equations with Variable Coefficients

We consider first the system (7.1) in a hemisphere C = Z R , and let u be a solution with support in C,, r < R. Let I, = max (2992, m,+l) and let 1 be a fixed integer 2 I,.

We shall assume conditions (i) and (ii) of Section 7 and, in place of (iii), (iii)L, : The coefficients of L belong to Cz-2” and have I Ik--8m norms

bounded by k, while the coefficients of B j belong to C’-”f and have I la-m, norms, on t = 0, bounded by K. F and the functions cPj are assumed to have finite I I I I l-,m and I [ I I z-m,-l/p norms, respectively.

In the following, constants r, C , , C, , * depend only on E , k , 1, p and the modulus of continuity of the leading coefficients of L.

THEOREM 15.1. If ~ [ u ~ ~ z l is f inite and if r 5 r1 , then Ilu]lz is also f i n i t e and

Proof: We shall assume that we already know that I IuI I is finite; for, this may be derived from (15.1) for I = I, by the usual trick of taking difference quotients, as in Theorems 6.2, 7.2.

Using the notation of Section 7 we write the system (7 .1 ) in the form

(15.1) 1 IUI l z I C,(l IF1 C I l@d t z--nz,-l/p+ IIul l o ) .

L’(0; D ) u ( P ) = F(P)+(L’(O; D)--L’(P; D))u(P)--L”(P; D)u(P) = f ( P )

= b*(x), i = 1, * * ) m,

‘15.2) H ~ ( O ; D ) U ( Z , O ) =@,(x)+(B;(O;D)-BB;(X;D))U(Z, O)-BB:,’(X;D)U(Z, 0)

and proceed to estimate l l f l l z - 2 m l ~ ~ j l l z - . m , - l ~ D . If

L‘ (P;D) = 2 aa(P)D@, 181=2m

then we easily see that

Il(L’(0; D ) - Jw; D)).(P)IIZ-2rn 5 constant - (max laa(P) -aa(0)I~IIz411z+~IIuIIz-l).

b Furthermore,

IIL”(P; D)%(P)[lz-gm S constant - KIIuII,-, . Combining these we find


I I f I1 1-2m 5 1 l Fll r-zn+C2(l I 4 11-1 (15.3) + max la/@) -aa(O)l3l lull,). B

To estimate some of the terms in 4j we have, from the definition of the norm,

[ I ( q o ; D) - B;(x; D) - B;’(x; D) )u (x , 0)ll l -m,- l /a

I - constant * I/(Bi.(O; D)-BB;(x; D)-Bi.‘(x; D))%(X, t)llz-,n+ . Since the first derivatives of the coefficients of H i are bounded by k we find easily that this expression is bounded by

Thus constant + K(rl I4 I 1+ I Iul ll-l).

I 1451 I 2--m,- -1 /D S I I@) I I l-m,-lip +GI I4 11-1 + C2 rl I t 6 I i 1 . (15.3)’

Applying Theorem 14.1 to (15.2) and using (15.3), (15.3)’, we obtain the inequality

II26IIZ 5 C,[IIFII1-2m + t: I I @ j I I z - m , - l / D ~ I I ~ ~ I z - l i

+ (r + max I ~ ~ ( ~ ) - - ~ ( O ) l ~ ) l l ~ l l z I B

Since the aB are continuous there iS a constant rl such that C,(r + max laB(P)-aB(0)12) 5 $ for r = r l ,

B so that for Y (= r1 we have

llull‘ 5 2C3(llFIl$-2rn + t: l l ~ ~ l l z - m , - ~ / P + l l ~ l l ~ - ~ ~ .

~C3Il4Iz-1 5 ~II~I IZ+C,I l~I I l l *

j

By Lemma 14.1

Inserting this into the preceding we obtain the desired inequality (15.1). The following well known analogue for interior estimates is proved in

the same way as Theorem 15.1 using Theorem 14.1’ in place of Theorem 14.1.

THEOREM 15.1’. Let L be the operator occurring in Theorem 15.1 and let H be a ficnction, in (n+l)-space, belonging to and vanishing outside a .@here of radius r. Then if r < y1 (possibly a different constant than the one occzming in Theorem 15.1), ic satisfies (15.1) (wiih a different constant C, and with no boundary terms).

From now on we use rl to denote the smaller of the two constants called Y, in Theorems 15.1, 15.1’.

We turn now to a general domain 9 which for simplicity we take to be bounded. As mentioned in Section 14, it will be clear from the discussion that our results can be extended to a wide class of unbounded domains. Let u be a solution of


(15.4) LU = F in 9, B,u = @# on 9, j = 1, * * a , m,

L and B j being differential operators of orders 2m, nt, ; set 1 - max (2m, m,+l) and let I 2 I, be fixed. We shall assume the boundary ? - 21 of 3 to be of cla.ss Cz in the following sense: The boundary $ can be covered by a finite number of ((n+ 1)-dimensional) neighborhoods Ui such that each Ui n 5lj can be mapped in a one-to-one way onto the closure of a hemisphere Z R i , Ri 5 1 in (n+l)-space, with ui n 56 mapping onto the flat face of the hemisphere, by a mapping T , which together with its inverse has continuous derivatives up to order I bounded by a constant K .

Concerning the equation and the boundary condition we shall assume first of all that L is uniformly elliptic, and that under each mapping Ti the system (15.4) goes into a system (7.1) in ZRi satisfying the hypotheses (i) and (ii) of Section 7. In addition we assume that the coefficients of L belong to PZm and have I l z - - 2 m norms bounded by k , while the coefficients of Bj belong to Cz-m~ and have I lz-m, norms (defined say as the maximum, of these norms over the flat faces of ZRI) bounded by k . F and the functions cDj are assumed to have finite I I I I and I I I I z-m,-l/p norms, respectively, (see (14.1), (14.2)).

THEOREM 15.2. If IIuIIz, i s finite, then IJu]lz i s also finite and

(15.5) I I 4 I E 5 ~(IIFl~Z-Zm + 2 I1@P31Iz-~,--l/lp+lI~lIO), 5

where c defiends only o n K , E, k, I , f i , the domain 9, and the modulus of con- t inuity of the leading coefficients of L.

Proof: Let zfv o, E 1 be a partition of unity in 3 by C" functions mu having the following property: If the support of co, is not in the interior of 9, then it is in the interior of one of the U , , Ui(,) . Furthermore we require that the support of co, , and the image of n support of o, under the mapping Ti(,, , is contained in a hemisphere of radius r, , where r, is the constant entering in Theorems 15.1, 15.1'. We intend to show that for each CT, [Io,,ullc is bounded by the right side of (15.5) plus c * I I U I I ~ - ~ . It follows that llullz 5 2, ~ / c o , ~ ~ / ~ 5 the right side of (15.5) (with a suitable constant C ) plus c * IIuIIZ-, . With the aid of Lemma 14.1 the inequality (15.5) is then obtained in the usual way.

With the aid of Theorem 15.1 we shall treat the case that the support of mu is not contained in the interior of %. The other (interior) case, which is similar, is handled in the same way using Theorem 15.1'.

Let us consider a fixed LU, . Under the mapping Ti(,, , o, goes over into a function o with support in Zrl, u goes over into a function v which we shall consider only in the support of co. Consider first the case I = I , .


Because of our assumption on the mappings T , (15.6)

Let us denote the operators L, B, , after the mapping Ti(,, , by 1 and Bj. According to Theorem 15.1

( [ w u t ~ ~ l z 5 constant - II0v11~.

(1 5.7) I lovl I/, s Cl (I IZ (w4 I I I--2m + 2 I IS, (W.1 I I r-m,-1/9+ I I 4 lo). We easily see that

I IZ (wv) I I I-% 5 constant (I I Fl lz--2m+ I I4 IZ-d? (15.8)

IlovIl,, S constant - 1(u1),, . From Lemma 14.2 it follows also that

(15.9) I I B, (mu) I I Z-mj-l/9 5 constant ' I I B, (mu .) I I z-m,--1/9 *

Combining these inequalities we find (15.10) llwUuIl2 5 constant * (IIFII~-z~+II@IIz-I + ~ I I ' j ( ~ u ~ ) I I ~ - - m j - l / p ) ' , Thus the desired estimate for I l w u ~ l l Z will follow from the estimate

(15.11) I I Bj(wv u ) I I Z-m,-l/p 5 constant * (1 l@jI I l-m,-l/pf I IuI IZ-1). Now on $

Bj(w,u) = wu@j + 2 c j ,~ , yD'w , ,D'~ , lSl+lrl5m,

IY 1'9

where the functions c , , ~ , ~ are of class Cz--"f. These coefficients may be extended into the support of o, as functions d,,B,r in C2--"f because of our conditions on the boundary $. Therefore, by definition,

IICj,J. y o P u I lZ--ntj-l/9 5 I Id,,, y D'wuDW 12-m,

5 - constant I I U I / ~ - ~ . To estimate Ilw,,@,]Iz--m,-l~p we note that since IldijllZ-mj-1/9 is finite there is a function v, in 29 which equals @, on the boundary and which satisfies II%IIz-nr, 5 2ll@Al/,-m,-l/9 * Therefore

llwu@jll/,-m,-~/p 5 l l ~ u ~ j I l ~ - m j 4 constant * llvjIl~-m, 5 constant - ~l@gllZ-m,-l/p.

Combining these inequalities and using the triangle inequality for these norms, we obtain the inequality (15.11). This completes the proof of the theorem for I -- I, .

Suppose now that I = I,+ 1. Having a bound for I J U I I ~ - ~ we can obtain bounds for the right-hand sides of (15.8) and (15.11), and hence also for (15.9). Applying Theorem 15.1 we again see that (15.7), and hence (15.10), holdsforl = l l+l . Using (15.11) weobtain thedesiredestimatefor Ilouullr.


Knowing that now IIg1II,+, is finite, and having an estimate for it, we prove in the same way that I I U ~ ~ ~ , + ~ is finite, and obtain the desired bound for it. And so on.

In analogy with Remarks 1, 2 after Theorem 7.3 we have here: T h e term llullo on the right of (15.5) m a y be replaced by the L, norm of u,

and it m a y be omilted altogether if the solution having derivatives afi to order 1, in L, is uniqae (in either case the constant e has to be changed).

We mention again that with the aid of the examples in Section 10 we see also that the Condition on L and the Complementing Condition are necessary conditions for the L, estimate to hold.

We conclude with some corollaries of Theorems 15.1, 15.2. The first follows from Theorem 15.2 with the aid of a special case of well known results of Sobolev ([37], see Lemma A5.1 part c, in Appendix 5 ) . It will be clear that a more general form of the following corollary can be derived from the results of Sobolev.

COROLLARY. Under the conditions of Theorem 15.2, if f i exceeds the dimension n + 1, so that u = 1 - (n + 1)/p > 0, then Ia]l-l+a i s mujorized by the right-hand side of (15.5).

The following results are used in the proofs of Theorems ll.l', 11.2'.

THEOREM 15.3. Consider again a solution of (7.1) in .ZR, and assume that all the conditions of Theorem 15.1 are satisfied with the excefition that u i s no longer assumed to vanish near the curved boundary of ZR . Then, for every r < R, (i) 1 IuI If. i s finite, (ii) if C(x, t ) i s a C" function defined in .ZR which i s identically one in Zr and zero outside ZtR+,.)12 we have

~ l u ~ ~ ? 5 collstant ( ~ ~ ~ F l ~ ~ - 2 ? 7 L f z llc(xJ o)@jll&7T2j-1/P + ~ ~ u ~ ~ ~ l - l ) X v

where the constant is indefiendent of u. A similar inequality holds, in fact, with 1 IuI I$, on the right replaced by

I[u1\?, but we shall not bother proving this. For I = I , the theorem is proved (we omit details) by applying Theorem

15.1 to the function [u ; the proof for the general case is similar. Because of its application in the proof of Theorem 11.2' we state the

following analogue of Theorem 15.3 for equations in integral form, omitting the proof. Consider the system (9.1) in ZR , written in a somewhat modified form.

L = 2 B @ a p , p ( P ) D p B, = z ~ Y , h , y , 8 ( W 6

F = 2 Dfl Fp , = 2 Dz@f,7 withsummationsover 1/31 S m a x (0,2m-Z), lpl 5 I , IyI 5 max (OJwzj-l+l),


IS/ I: I - 1, and i = 1, * * -, m. Here I is to be greater than the maximum order of t differentiation occurring in the B, .

In place of (iiijL9 we assume (iii)i9: The FB belong to L, if I < 2m, F E H2-2m,L9 if I 2 2m; for every

function ((x, t ) as above, [(z, 0)Ipj,, has finite 11 111-1/, norm if I I m j , while c(z, O ) @ , E H ~ - ~ , + ~ - ~ / , if I > m, . Concerning the coefficients, we assume that the are continuous for Ip1+ lpl= 2m, and that the following are bounded by k : (aB ,Jo if I < 2m, lulp,Jl-2m if I 2 2m, jbj ,y,bll if I 5 m,,

THEOREM 15.3'. If ( 1 ~ 1 1 ~ i s finite, then for any Y < R and C(z, t ) as above l'j,0,blt-m, if > mj .

we have m

IluIIf. i constant * (2 R j + IluIIf3)' i=O

with the constant inde#endent of u. Here

KO = 2 IIFlpIlo if I < 2m, k, = IIFIIl-a, if I1 2m,

and for j = 1, * * a , m

hj = 2 \IT(%, O)@j,rII1-1/, if 15 mi >

r kj = lIC(z, O)@j ,y l I~ -n ,+ l -~ /~ if 1 > mj

For completeness, though no use is made of it, we add the following interior analogue of Theorem 15.1' for an equation in integral form

Lu = 2 DBaAsD'u = 2 Dpfb,

2m - I, lpl 5 I, for some fixed I < 2m. We assume, for simplicity that the f s and Dpu, Jpl 5 I , belong to L, in 3, that L satisfies (1) and has bounded ccefficients, and that the leading coefficients alp,,, i.e. those with IBI + 1p1) = 2m, are continuous.

THEOREM 15.1". For any compact subdomain 5% of %) we have

Appendix 1

Proof of Lemma 2.1

Consider the kernels K , , given by (2.6)', (2.6)" and choose for y the contour composed of the circular arc It1 = ZC, Y m t 2 1/2C and the chord joining its extremities (see (1.2)). For s 2 m,+q+l the homogeneity of

708 S. AGMON, A. DOUGLIS, A N D L. NIRENBERG

DsK,,, asserted in the lemma then follows obviously from (Z.S)', (2.6)". We see also from these formulas that it suffices to establish (2.13) for points P = (5, t ) on the unit hemisphere 1x12+t2 = 1, t > 0. Since for 151 = 1, z on y we have INk(5, z)/M+(E, z)l 5 C l ( E ) (see (1.11)) the inequality (2.13) for s < m,+q+l follows immediately.

The letters C,(E), C,(E) , * - - will denote constants depending only on E of (2.12), while Cl(s, E ) , C,(s, E ) , ' - - depend also on s.

If s 2 m,+q+l we find, setting s-m,-q = 0,

(Al . l )

where F ( t , z) is an analytic function of t, z on 161 = 1, z E y such that F and all the derivatives of order 5 I are baiinded by a constant depending only on E and 1. In order to prove (2.13),

(A1.2) IDsqqI 5 C(S, E l , for the case s 2 m,+q+l it will suffice to establish for (x, t ) on the hemisphere and for all s 2 mj+q+l the estimate

(A1.3)

the boundedness of the derivatives on the unit hemisphere following by integrating the derivatives twice from some fixed point on the sphere.

Because of ( A l . l ) the estimate (A1.2) certainly holds on the part of the hemisphere with t 1 +, so it suffices to consider the ring R: lsI2+t2 = 1, 0 < t < 8. Let C ( v ) be a C" function on the interval f-1, 11 such that Jll 5 1, C -= 1 for JrJ < &, 5 = 0 for 2 5 1 1 ) S 1. We may then write (A1. l ) in the form

= Il+I,, where the two integrals define I1 and I , .

We first estimate I, . Since the integrand vanishes except for Ix we see easily that there we have, for z E y , (x. [+t.rl-' 5 C,(E).

1121 s c: Jl*,-l .F, IF([, z)lldtldwl: = C,(S, E ) .

El > 8 Hence

Consider finally I,. Let q- = T,E be a rotation in E, which takes x = (xl , * * * , xn) to (1x1, 0, - * -, 0). Making the change of variable q- = TOE we have


Denote by q' the generic point (qz , - * , qn) in En-l. 7 Then

Integrating by parts IS-1 times with respect to q1 we find

. j (;)w-l [F(T,1~,t)S(T11~I)(1-~~)("-3)'21 d r l ,

Y lxIr1 +t t Since now 1x1 > Q on the ring R , and since the numerator of the last integrand is bounded by C,(s, E ) in absolute value, we find

Combining this estimate with the previous ones we obtain (A1.3) - proving the lemma.

Appendix 2

Proof of Theorem 3.2

We restate the theorem for K = K, , i.e., K, = 0. Let

(A2.1) u(xJ I ) = K ( z - - y , ' ) f ( y ) d y = * f with

satisfying (3.2), (3.3). Then if f i s in L, , #J > 1, so i s u(x, t), for each t, and

where c depends only on n and p , i.e., i s independent of t.

functions f (x) with compact support.

(A2.2) IUlL,.* d C K l f l L , 9 t > 0,

As one easily sees it suffices to establish the estimate (A2.2) for C"

'The following has to be modified somewhat when n = 1 or 2.


Consider first the case n = 1. For our given kernel K we have D(x) = K‘ (sign x), where K’ 5 K is a constant. To treat it we shall rely upon the classical result due to M. Riesz: for KO($, t ) = z / ( x a + t 2 ) , the inequality

K O * f l L , , t = Ipo (x -Y? tlf(Y)dyjL,*t 5 W l f l L , 1 < f J < o O ,

holds, with c ($ ) depending only on fJ. Now consider the function Y(x, t ) =K(x, t ) - K ’ K o ( X , t ) .

It has finite L, norm in the variable x, and this norm is independent of t. But then we have

1K * f lL , , t 5 IK ’Ko * flL,, t + IY * flL,, t

5 K C ( P ) l f l L , + IYIL,,tIflL, 5 constant - ~ l f l ~ , .

Thus Theorem 3.2 is proved for n = 1. This fact will be used in treating the higher dimensional case.

Turning now to arbitrary n > 1 we treat first the case of an odd kernel Q ( x ) = -D(-x) (for this case we require only that D be measurable and bounded by K). The formula (A2. l ) may be written in the form

u(x, t ) = jK(-% t ) f (x+y)@

in polar coordinates. Because D is odd we may write this in the form

Let I ( $ ; q) denote the inner integral. The kernel

satisfies the conditions of Theorem 3.2 for n = 1. Applying the theorem we find that the integral of 11 (2; q ) 19 along the line parallel to q through some point xo is bounded by a constant times the integral of lf(x)Ip along the same line, the constant depending only on n and 9.

Now I ~ I ~ , , ~ 5 constant * K [ jdo , , j 11(x; q ) 1 ~ d 3 ~ ] ” ~ .

Integrating first with respect to lines, in the x-space, parallel to q and then with respect to axes’perpendicular to these we find, by the above, that


l 4 L , , t r constant * K l f l L , *

This completes the case of an odd kernel. We may again assert as before that Theorem 3.2 is proved in general for kernels that are odd in 5.

Since any kernel K can be written as the sum of an odd and an even kernel it suffices now to consider an even kernel Q(x) = Q(-x). We may assume that f is of class C" and has compact support. Consider the kernels (not involving t )

In [9] Calderon and Zygmund proved that, if

f m ( 4 = Km * f = j K , ( ~ - Y ) f ( Y ) ~ % the integral being the limit, as E + 0, of the integral over /z--yl >. E, then

Furthermore,

where A , depends only on f l and n.

f = z: K , * f, .

IfmlL, 5 AplilL, I 1 < $ < m ,

We may therefore write u = K * f = 2 K * K , * f , = SI K(s-y , t)K,(y-z)f,(z)dzdy.

The functions Tm(z, t ) = K * K , are clearly homogeneous of degree --n and odd in 2. It may be shown, as in [9], that the kernels r, satisfy the conditions of Theorem 3.2 and that

u = 2 r, * f, .

IUIL,. t 5 constant * K ZI IfmlL,

Having established Theorem 3.2 for odd kernels, we find

I - constant * K * AplflL,, completing the proof.

Appendix 3

Proof of Theorem 3.3

We first prove a related, but more elementary, result which, however, would be sufficient with the aid of Lemma 9.1' for obtaining all our L, estimates for solutions of elliptic boundary value problems.

THEOREM A3.1. Suflfiose all the co-nditions in Theorem 3.3 to be satisfied with the excefition of condition (3.3). Then each x derivative D,u belongs to L, in t > 0, and

7 12 S. AGMON, A. DOUGLIS AND L. NIRENBERG

IDr4O,L, 5 CKlf l l - l / , ,L , I

where C depends only on p and n.

exists a function v(x , t ) in t 2 0 with Proof: From the definition of the seminorms for f it follows that there

(A3.1) v(x , 0) = f(.) and I v l l , L p 5 W 1 - l / , , L p *

Since f E L, it also follows easily that v(z, t ) E L, on every hyperplane t = constant, and that

(A3.2) (j Iv(x, t ) lPdz ) l iP 5 constant * tl-l/p as t + 00.

From the identity &(v(y, s )K(x -y , t + s ) )

= D,v -K(s -~J , t+s)+vD,K we obtain, after integration with respect to s, y over the strip

(A3.3)

-u(x, t ) = - J ~ ( y , O)K(x-y , t )dy

From (A3.2) and the inequality ID,K(y, t)I f constant - (ly12+t2)-(n+1)/z

we find readily that the last, n-dimensional, integral is O(T-("+l)/p) as T-t co. Hence letting T j. co we obtain the formula

-Diu = ~ J s , o D s v ( y , S ) * D,K(x-y, t+-s)dyds

+ j ~ s > o D , v ( Y , s) - D,K(x-!/, t+s)d!/ds.

Since the kernels D,K, D,K satisfy the conditions of Lemma 3.2 with, in fact, J2 = constant, we obtain, on applying the lemma,

IDiulo,L, 5 constant - lv[l ,Lp.


In virtue of (A3.1) the theorem follows. Proof of Theorem 3.3: In virtue of Theorem A3.1 it is only necessary

to establish the estimate

(A3.4) I4UIO,L, 5 constant * lf l l-1h.Lp - We proceed as above. Let v again be the function satisfying (A3.1). We obtain from (A3.3) the representation

Since D,K satisfies the conditions of Lemma 3.2, we have as before

f I , lO,L, 5 constant * ID,VI,,L, 5 constant - l f l l - l /p*L, *

Hence to prove (A3.4), and thus the theorem, it suffices to prove that in t > O

(A3.6)

where (A3.7)

n

11210,L, 5 constant * z\ IDivlo,L,t i=l

1 2 ( G 4 = /j->o V(Y, s)DZ;K(--y, t+s)dy ds.

(A3.8) S D t K ( Y J t )dy = O, t > 0.

We observe that, in virtue of Lemma 3.1, condition (3.3) implies the condition

The inequality (A3.6) then follows from the following lemma, applied to the kernel J ( z , 1 ) = D;K(x, t ) . (Also Theorem A3.1 follows directly from this lemma.)

LEMMA A3.1. Let J ( P ) be a continuous kernel, homogeneous of degree - (n+2) , in t > 0, which i s bounded by K on I PI = 1 and satisfies the condition: for every t > 0

(A3.9) J- J(2 , t )dx = 0.

I(x, 0 = ~ J s > o v ( Y , s )J (x-y , t+s)dyds

Let v ( x , t ) have finite I ll,L, norm. Yhen the function

satisfies the inequality

(A3.10)

where C depends only on p and n. Proof: We may assume that K = 1, so that

(A3.11) IJ(P)I 5 i q - n - 2 .


The word “constant” will be used to denote various constants depending only on n.

We distinguish two cases. Case A ; n = 1. With x now a single variable introduce the function

t > 0.

Clearly L is continuous on t > 0, homogeneous of degree -2 and satisfies D,L = J . We claim, furthermore, that

(A3.12)

By the homogeneity it suffices to verify this for t = 1, 1x1 5 1 and for 1x1 = 1, 0 < t < 1 ; the former case following trivially from (A3.11), we consider only the second case. By (A3.11) we have

Next it follows from (A3.9) that

As in the case for x = - 1 we find that this is bounded in absolute value by a constant. Thus (A3.12) is established.

Returning to I we have, on integrating by parts,

By (A3.12) the kernel L satisfies the conditions of Lemma 3.2 with Q = constant, which then yields the desired estimate (A3.10).

Case B ; n 2 2 . We claim first that J may be assumed to satisfy the additional conditions

(A3.13) xi J (x , t ) d ~ = 0, t > 0 , i = l;-- ,n.

Indeed if (A3.13) does not hold we may replace J by n

J o (x, t ) = J (x, t ) - z\ ci Mi (x, t ) , t=l

where


and the constants ct are chosen so that the kernel J o satisfies conditions (A3.13), i.e.,

By the homogeneity of J the value of ct so obtained is indeed a constant. We see also from (A3.11) that JciJ 5 constant.

Since the kernels Mi trivially satisfy (A3.9) we see that the kernel J o satisfies all the conditions of Lemma A3.1 and, in addition, (A3.13). Further- more, inequality (A3.10) holds in case the kernel M i is substituted for J . To see this last fact we observe that, setting Mi in place of J and integrating by parts, we have

I = ~ ( y , s)Mi(z-y, f + s ) d y d s

and the desired inequality (A3.10) follows from Lemma 3.2 applied to the kernel M .

Thus to prove the lemma it suffices to prove (A3.10) for Jo , i.e., for a kernel J satisfying the additional conditions (A3.13).

To this end, let F(z) be the fundamental solution with singularity at the origin, of the Laplace equation A,+ = 0:

F ( x ) = constant * 1x12-n F ( x ) = constant * log 1x1

for n 2 3, for n = 2 .

The following representation formula holds almost everywhere for a function g(x) which is continuous and such that g(x)(l+\xl)l-n is absolutely integrable in the whole space:

g(.) = i i =1 D, jg(Y)DJtx-Y)dy.

Applying this to the function J ( z , t ) for a fixed t > 0 we have

where we have set J(., f) = z: D,L,(x, t) .

(A3.14) La(x, t ) = jJ(Y, f ) ' DiF(X-! / )dy*

Hence I can be written in the form

= I: j js ,oDiv(Y, s) * L,(z-y, t+s)&ds,

and Lemma A3.1 will follow from Lemma 3.2 provided the kernels L,(x, t ) satisfy the conditions of that lemma.


The continuity and homogeneity of the L, are easily established, as is a bound for lLi(z, t)I for 1x1 5 1, t > $, and we shall merely establish here the following estimate for L i t enabling us (see the remarks after Lemma 3.2) to apply the lemma:

(A3.15) ILi(x, t ) ) 5 constant * log- for 1x1 = 1, t < $. I :I For 1x1 = 1 define

n

5=1 (A3.16) .Fi(y; X) = DiF(~-y)--Di F ( x ) + 2 g9D1Di F ( x ) .

Then by Taylor’s formula we see that F,(y; x) satisfies for 1x1 = 1

JF,(y; x)l 5 constant - Iyl2 for Iyl 5 i, (A3.17) IF,(y; x)l 5 constant * (Iz-yI’-”+Iy[) for IyI 2 +.

Using (A3.9), (A3.13), (A3.14) and (A3.16) we see that, for i fixed,

L i h t ) = S Jk / , t)DiF(x-g)dy = JJ(Y, t )F i ( y ; x)&

= Jl(% t ) + J Z ( ” , t ) .

By (A3.17) and (113.11) we have

w yn-l = constant - __- dr s 0 (r2+t2)n/2


Combining these inequalities for J 1 and J a we obtain the inequality (A3.15) for L , . This completes the proof of Lemma A3.1, and hence of Theorem 3.3.

Appendix 4

Proof of Theorem 3.1A

Our proof makes use of the following LEMMA. Sllppose u(x, t ) i s once continuously differentiable in the half-

space t > 0, and satisfies fm s m e positive a < 1 and every first derivative Du,

Then lDUl 5 At"-*.

Proof: Consider two points (z, t ) , (y, z) with t 5 z; denote their distance apart by d. Then we have

(here we have used our assumption on Du, and the theorem of the mean) 2A a

5 - d-"da+A

= A (1 +&). Proof of Theorem 3.1A: Set 2 [DZfh--"fla = F. In virtue of the lemma

it suffices to show that (A4.1) ]Dz+lu(x, t)l 5 constant F - ta-l. Since f vanishes for 1x1 2 R, f and its derivatives up to order l f h - n are bounded in absolute value by constant * F. Thus we see immediately from (3.16) and the expression for u that

[DZ+lu(z, t)l 5 constant - F * (lxl2$ t2)-(zf1+A)/2 f or Iz12+S.t2 2 4R2,

from which the following estimate follows easily: (A4.2) lDZ+lu(z, t ) / 5 constant - F - ta-l for IxI2+t2 2 4R2.


Thus we need only consider the region 1x12+t2 < 4Ra. For x fixed, 1x1 5 2R, denote by P ( y ) the osculating polynomial of degree Zfh-rz of f ( y ) at the point x so that f(y)-P(y) = ~ ( l x - y l ~ + ~ - " ) as y + x . With the aid of Taylor's formula it is easily seen that, in fact, (A4.3) lf(y)-P(y)J 5 constant - F - I Y - Z I ~ + ~ - ~ + ~ . The coefficients of the polynomial, being constants times derivatives off at x, are also bounded by constant - F.

Let c(y) be a non-negative C" function which is identically one for IyI 5 3R and vanishes for IyI 2 3R+1. Write D2+lu(z, t ) as a sum, for 1x1 2 2R,

D'+,.u(x, t ) = I DC++lK(x-y, t)(f(Y)--P(y))c(x--y)dy

+ 1 Dz+lK(x-y> ~)P(y)C(x,-y)dy = I,+I, .

From (A4.3) and (3.16) we have l x - y l Z+h-n+a J (l,-y12+t2)(z+1+r4ii dy {I,] 5 constant -

Z+h-n+a dv = constant - t"-l s (1.1: 1) (Z+l+h)/Z

= constant - ta--l,

where we have made the change of variable y = x+vt.

of terms of the form In order to estimate I, for 1x1 5 2R, t 5 2R, we note that I , is a sum

coefficient - 1 JJ ( - y i - x i ) ~ ' ~ z + l ~ ( x - y , t ) t (x--y)dy

= coefficient * J J J ~ ? D Z + ~ K ( - ~ , t)C(--y)dy

with k , 5 Z+h--n. We now invoke hypothesis (i) of the theorem to ensure that these integrals are bounded fort 2 2r (this is the only place that (i) is used), so that

lIzl 5 constant - F, 1x1, t 5 2R.

This estimate, combined with the estimate above for I,, and (A4.2) yields the desired estimate (A4.1).


Appendix 5

An Improvement of the Interior Schauder Estimates

Consider a uniformly elliptic equation (A5.1) LU = F of order 2m in a domain '3 and suppose that F , and the coefficients of L, belong to C" in s;D for some fixed positive a < 1. In [lo] we proved the following interior Schauder estimate (see Theorems 1 and 4 there): if u E CBm+" in 9, then for every subdomain 8 C a C 5D

(A5.2) I U I ~ + ~ 5 constant - (IFI?+IuIP), the constant being independent of u.

We shall extend this to solutions which are not assumed to be of class Cznr+a. We shall, in fact, consider generalized solutions having derivatives in, say, L,.

THEOREM A5.1. If u has s p a r e integrable derivatives up to order 2m (in the sense of Friedrichs, Sobolev) and satisfies (A5.1) almost everywhere, then in fact u belongs to C2m+a in s;D, and hence satisfies (A5.2).

For second order elliptic equation E. Hopf [ l a ] proved that solutions which are in C2 belong to we shall follow his method of proof. In the course of the proof we shall make use of the following well known

LEbfMA A5.1 (Sobolev [37]). Let f (x) be a function with support in the .unit sphere and belonging to L, , 1 < p < 00, in (n+l)-space, and consider the function

I < n+l.

Then, denoting the L, norm in the unit sphere S, of a function have, setting a = n+l-A-(n+l)/p,

1 a (a) for a < 0, l / g / I z 5 constant llfll;: where - = - - 4 n + l '

(b) for a = 0, llgll2 5 constant . l / f l I ~ for every finite q > 0, (c) for a > 0, a # Integer, lglz1 I constant / I / I I ~ , (d) for a > 0, a = integer, ]g@ 5 constant - I l f l l $ , O S b b a , where the constants depend only on n, A, p , q and b.

We shall use the lemma in showing by successive steps that u has derivatives in L, locally, for some p > 2, and then using this information to show that its derivatives are in L, for some still higher p . Repeating this

by 11$112 we


process we find eventually that % belongs to Czm+e for some E > 0. Then we use a result of E. Hopf [14] to conclude that zd E C2m+a, It will become clear from the proof that we could have assumed u and its derivatives up to order 2m to be in L, for any p > 1.

Proof of Theorem A5.1: Since the argument is local we may suppose that D lies in the unit sphere. Let K denote a bound on the coefficients of L and on their norms I I? . Let L'(P, D) be the terms of highest order in L(P , D ) . For any fixed point Po let r ( P o ; P ) be the fundamental solution (4.2) of F. John [17] of the.operator L ' ( P o , D) with coefficients equal to their values at Po . I t is important to observe, from the explicit formula [17] for T, that for IP; = 1, D2,"T(Po ; P ) is of class C" in Po and, in fact, its I I? norm with respect to Po is bounded by a constant independent of P for IP( = 1.

For any subdomain 8 C !8 C 59 let C(P) be a non-negative function with compact support in 5D which is identically one on 8. For any C" function u(P) in 5D we have

C ( P b ( P ) = r (Po ; P-Q)L'(Po 1 Dcj)C(Q)u(Q)dQ]

so that for P E 8, j 5 2m,

D$%(P) = 1 ( D $ r ( P o ; P-Q)L((%)

+ D X P , ; P-Q)(L'(P,, D ~ ) - - L ( Q , ~ ) ) r . ) d e + c ~ L ( c ~ , where cj = cj(P,,) is a constant depending on D3, cp = 0 for j < 2m, and where for j = 2m the integrals are taken in the sense of the Cauchy principal value. If now we set Po = P E 8 we find for 0 5 j 5 2m

(A5.3) + j o j r ( P , P-Q)(L'(P, D)

where DjI'(P, P-Q) involves only differentiation with respect to the second variable. Here cj =: c,(P) is Holder continuous in P.

We claim now that (A5.3) holds also for functions u having only square integrable derivatives up to order 2m in the sense that the difference of the two sides of (A5.3) has vanishing L, norm in %. To see this one approximates such a function by C" functions and checks the convergence of both sides of (A5.3) on going to the limit in the approximation. For j < 2m the desired convergence follows easily with the aid of Lemma A5.1 and the bounds (4.3), (4.3)' for the derivatives of r ( P o ; P ) . For K = 2m it follows from certain known properties of the operator

D$u(P) = [ D5T(P, P-Q)L(Cu)dQ + cjL(&),

- L(QJ D))S(Q)u(Q)dQ,

BOUNDARY ESTIMATES FOR ELLIPTIC EQUATIONS

f -+ Tf = 1 D a m r ( P ; P-Q)f (Q)dQ. (a) The operator T takes L, boundedly into L,', 1 < $ < co (this

result is contained in Theorem 2 of Calderon, Zygmund [9]); in particular this holds for p = 2.

(b) T takes C" boundedly into Ca (this result is well known, see for instance Mihlin [23]).

Before proceeding we write (A5.3) in the form

721

(A5.4)

(A5.3)' D'u(P) = 11,j+12,j+13,j+14,j t 0 5 j 5 2nt,

where

Ii.$(P) = S D ' r ( P ; P--Q)C(Q)F(QVQ + cj F ( P ) ,

14,5(P) = S D j r ( P ; P-Q) 2 (as(P)-ab(Q))DB(Tzl)dQ; IBl=zm

here the functions C , , , G~ are in C", the ab(P) being the leading coefficients of L, and F = Lu.

E Ca in V, by the above remark. We note furthermore that

We note that, since f E Ca, also

If we now apply Lemma A5.1, using (4.3) and (4.3)', we find that for j 5 2m-1 , I j , $ E L , , i = 2, 3, 4, where l/q, = * - l / ( ~ + l ) if n > 1, or q1 is an arbitrary positive number if n = 1. Since 3 was an arbitrary compact subdomain of 3 it follows that Dju, j 5 2m-1, belongs to LU1 in any compact subset of 9. We shall now assume that this is known and consider again the terms I , , , - - - , for some fixed % and C. (We shall not look at Il,i since it is always in C".)

If we apply the L, preservation property (a) of T (see (A5.4)) to the terms 12,2m, 13,2m, we find them to be in L Q 1 . Applying the lemma to (A5.6) we find that 14,2m is in L,, for

1 a - +--. $1 n f l

Dju z L u x , 1 = ' < 2m, in any compact subdomain of 9. Again we may use this fact to improve our knowledge of ZG.

Since in any case we may take q1 > PI it follows that


Operating in this way, applying again Lemma A5.1 and the L, preservation property of I', we find in general that if D 5 u EL , , + > 1, j 5 2m, in any compact subdomain of 9, then in fact, for j S 2m,

1 U (a) D j u E Lp,p-a,~, ,+l l~-~ in any compact subdomain if - > -- , + nzf l

1 U (b) D j u E L, for any q if - = __

P % + I ' 1 U

9 n+l (c) D j u E c8 for some positive E if - < -__.

Continuing thus for a finite number of steps we find that for i 5 2m, D j u E P for some positive E. But then for j < 2m, D j u E C1.

Since D j u E C1 for j < 2m it follows from the C" preservation property (b) of T that 12,2m and 13,2m belong to C". Thus we need only consider the term The fact that this term belongs to C" is contained in the corollary to Lemma 3 of Section 2 of E. Hopf's paper [14].

Consider finally the terms 12,2m , 13,2m , J4,2m .

This completes the proof of Theorem A5.1. The method of proof also applies to uniformly elliptic equations in

integral form, (9.1) (or really (9.2)) (see also Section 8 of [ l o ] )

(A5.7) Lu = 2 Dfia8,rDfiu = 2 D f l F a , where the summation is over 181 _I 2m--I, IpI 5 I, for fixed l < 2m. Assum- ing that F,, ag,/l E C a ( C o ) for ],6] = 2m--I (]PI < 2m-Z) we have

THEOREM A5.2. If u has s p a r e integrable derivatives zlp to order l and satisfies (A5.7) in the integral form, see (9.2), then zl belongs to C2+".8

The proof of Theorem A5.2 is similar to that of Theorem A5.1, the fundamental identity (A5.3) being replaced by: for j 5 I

D't t (P) = (-1)'fll J DjD; [r(J'; P-Q)C(Q)]Ffi(Q)dQ d + 2 (-1)181 Ju5ugr(~; P - Q ) a a , s , , ~ r c ~ M Q

lfllz;Zm--1 Ircl+lYlSZ

Irl<t

*Thus in Theorems 1 and 4 of [lo] the conditions that u, belong to C V a can be relaxed to: w, has square integrable derivatives up to order t , . Similarly in Theorem 4' of [ 101 the conditions that uj belong to C'J+tj+a may be relaxed to: u, has square integrable derivatives up to order o + t j .


It is interesting that by a smoothing process one may prove directly, with the aid of (A5.2) that a solution of (A5.1) belonging to Urn belongs also to CZmta. We feel that it is worthwhile to include this proof which follows the procedure of Friedrichs in his first application of his “mollifier”.

The proof involves constructing functions which approximate u uniformly in any given compact subdomain ‘3 C C D and which have uniformly bounded norm 1 lfm+a . To this end we convolve u with a C” function of compact support (the Friedrichs mollifier) which we first describe. Let i ( r ) be a non-negative C” function of one variable with support in Irl 5 1, and such that

Jrn j ( r )dr = 1. -” Let 5-8 be a domain such that Q C ‘23 C % C ‘B. Let E > 0 be less than - - $(n+l)-% times the distance from a(5-8) to the boundary x = (q , * - , x,+~) denotes the independent variables in 3, the C” function

integration being over the entire (n+ 1)-dimensional space.

of B(%). If consider in 23

= j ( x ) . Our aim is to establish a bound for lual~m+a independent of E ; since ua and

its derivatives up to order 2m converge uniformly to u and its corresponding derivatives in ‘3 the theorem will follow. To obtain the desired estimate it suffices by the interior estimate (A5.1) to establish a bound for ILu,lF independent of E . (Clearly Iu$ 5 IuIF). To this end it suffices to establish such a bound for (Lu),~? since, as is easily seen, I (LM),~? 5 /Lz@ . Thus it suffices, in particular, to establish such a bound for a typical term in L, a (x )Dku(x ) , K 5 2m, i.e., to prove

(A5.8) Ig(x, &)I? S constant independent of E ,

where g(x; E ) = aDkue-- (aDku),

The function i ( x ) and the coefficient a(.) satisfy Holder conditions

(A5.9)

Let M denote a bound for lDnul in the set of points that are closer to $3 than to $.


Consider arbitrary points x, x1 in 8. We have

(A5.10)

We note that the first integrand on the right vanishes unless Iyr-xrl < E ,

i = 1, * * - , n, or Iyi-x:l < E , i = 1, - , n, so that the first term on the right is bounded by I I + I 2 , where

With the aid of (A5.9) we see that

Il d constant - MK /2:x1iaks' - = constant - MKlx-x'la and

I2 2Mklx -x'la + constant - MK -- kEa lx:xT 5 constant - M k ( 1 f K ) Ix-x'Ia.

Similarly the second term on the right of (A5.10) is bounded by Mklx-x' 1 a.

Here the constants depend only on n. Combining these estimates we obtain the desired Holder inequality (A5.8).

Appendix 6

Proof of Lemma 9.1

In proving the lemma we shall show that for any 6 > 0 we have

(A6.1)

where the constant depends only on n, k , R, a and 6. It suffices to prove this for C" functions v y , v in .&-)/p, since we may

approximate v in Z R 4 p by the C" functions JB,LIv (defined on page 673) for


E sufficiently small, and we see that Js8v satisfies

0: J I , a ~ = I: D" J s , a ~ v a

If then (A6.1) holds for Ja,sv and Js , svv , it follows for v and v, by letting & -+ 0.

We prove (A6.1), for C" functions in ZRFa12 by extending v across the planar boundary t = 0 with the aid of the formula (4.4), slightly modified. First we write (9.5) in a form distinguishing the 2- and t-derivatives on the right (A6.2)

where Y = (p, i), vP,{ = v, . We now extend v in a manner similar to that of (4.4) by defining for t < 0 and It1 small

N large. Here the 1, are to be such that the extended function v is of class CN-1 in the full sphere SR-a/2 : 1xI2+t2 < (R-S)2, i.e., so as to satisfy

(A6.3) N 1 5 9-1 zap(-$) = 1,

At the same time we also extend the V ~ , ~ ( ~ C , t ) for t < 0 by the formulas

VP,JX, t ) = 2A9 ( - 3̂' - v w( ' x, - ;) - and verify that (A6.2) holds for t < 0. By (A6.3) the extended functions

are of class CN-k++i-l in SR-a~2. Furthermore each norm occurring on the right of (A6.l) , taken over SRFa12 instead of ZR-812 , is less than a constant times the corresponding norm over L'R-a/2, the constant depending only on k and N .

If now, say, N = 4k we see that the function v satisfies in SR-a/2 the elliptic equation with constant coefficients

(A6.4)

in which the right member is regarded as known. Since ZRea is a compact subset of SR-a/2, we find from the simple interior estimates for equations with constant coefficients (see for instance Theorem 2' of [lo]) that (A6.1) holds with each norm on the right replaced by the same norm over SR-a/2 instead of L ' R - a / 2 . By the remark above the inequality therefore holds with the norm over ZRPal2 as in (A6.1).

Lemma 9.1' is proved by applying the Calderon-Zygmund theorem to (A6.4) in a fairly standard way; see Section 14.

(DY+ 2 D?)v = 2 D$Dt+k~v,, i + ZD?-'(Div), i =1 i =1


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