Linear and Quasilinear Elliptic Equations

Linear and Quasilinear Elliptic Equations

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31. GEORGE LEITMANN(ed.) .Topics i n Optimization. 1967 MASANAO AOKI. Optimization of Stochastic Systems. 1967 32, J. KUSHNER.Stochastic Stability and Control. 1967 HAROLD 33. 34. MINORUURABE.Nonlinear Autonomous Oscillations. 1967 Variable Phase Approach to Potential Scattering. 1967 35. F. CALOCERO. Graphs, Dynamic Programming, and Finite Games. 1967 36. A. KAUFMANN. and R. CRUON.Dynamic Programming : Sequential Scientific Man37. A. KAUFMANN agement. 1967 E. N. NILSON,and J. L. WALSH.The Theory of Splines and Their 38. J. H. AHLBERG, Applications. 1967 Statistical Decision Theory in Y. SUNAHARA, and T. NAKAMIZO. 39. Y. SAWARACI, Adaptive Control Systems. 1967 BELLMAN. Introduction to the Mathematical Theory of Control Processes 40. RICHARD Volume I. 1967 (Volumes I1 and I11 in preparation) LEE. Quasilinearization and Invariant Imbedding. 1968 41. E. STANLEY 42. WILLIAMAMES.Nonlinear Ordinary Differential Equations i n Transport Processes. 1968 MILLER,JR. Lie Theory and Special Functions. 1968 43. WILLARD F. SHAMPINE, and PAUL E. WALTMAN. Nonlinear 44. PAULB. BAILEY,LAWRENCE Two Point Boundary Value Problems. 1968 Variational Methods in Optimum Control Theory. 1968 45. Yu. P. PETROV. 46. 0. A. LADYZHENSKAYA and N. N. URAL’TSEVA. Linear and Quasilinear Elliptic Equations. 1968

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Linear and Quasilinear Elliptic Equations O L G A A. LADYZHENSK.AYA and N I N A N . U R A L ’ T S E V A LENINGRAD STATEUNIVERSITY, LENINGRAD, U.S.S.R.

Translated by SCRIPTA T E C H N I C A , I N C .

Translation Editor: L E O N EHRENPREIS COURANT INSTITUTE OF MATHEMATICAL SCIENCES NEWYORKUNIVERSITY, NEWYORK,NEWYORK

ACADEMIC PRESS

New York and London

1968

COPYRIGHT 0 1968, BY ACADEMIC PRESS INC. ALL RIGHTS RESERVED. NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM, BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS.

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United Kingdom Edition published by ACADEMIC PRESS INC. (LONDON) LTD. Berkeley Square House, London W . l

LIBRARY OF

CONGRESS CATALOG CARD

NUMBER:67-23164

Originally published as : “Lineynyye : ksazilineynyye uravneniya ellipticheskogo tipa” by Nauka Press, Moscow, 1964.

PRINTED IN THE UNITED STATES OF AMERICA

Preface

During the past half century, linear second-order elliptic equations on bounded regions have been studied, if not exhaustively, at least with reasonable completeness and the fundamental questions concerning them have received rather simple solutions. In the works of Giraud and J. Schauder in the thirties, it was shown that the basic boundary-value problems a r e solvable for such equations under the assumption of sufficient smoothness of the coefficients and of the boundary of the region. Then, they were interpreted from the standpoint of functional analysis. This approach was initiated by the article [9] of K. Friedriches in 1934 on semibounded extensions of symmetric elliptic operators. This article and further studies of Friedrichs, S. G . Mikhlin, M. I. Vishik, and others during the late forties showed that the solution of the classical boundary-value problems for elliptic equations (we a r e speaking only of second-order equations) was equivalent to solving equations of the form x + A x == f , for a completely continuous linear operator A , in certain Hilbert spaces constructed from the quadratic form of the principal symmetric part of a differential operator. This point of view concerning boundary-value problems was of help in the investigation of the convergence of the approximation methods of Ritz, Galerkin, and others. True, such an approach merely proved the existence of socalled generalized solutions (poor a s regards differentiability properties) known only to have first-order derivatives. The question of the existence of higher-order derivatives of these solutions (at least the second) and with it the question a s to whether they satisfy the equation in the classical sense was answered on the basis of the above-mentioned works of Schauder, Giraud and others, so that a clear-cut and simple investigation of boundaryvalue problems was not to be obtained within the framework of functional analysis alone. Thus, it turned out that this was directly connected with the problem of the domain of definition of the closure in 4 of elliptic operators. This question was brought up by I. M. Gel’fand at his famous seminar as one of the central questions i n VI I

PREFACE

Vlll

the study of elliptic equations from the point of view of the overall theory of linear operators in a Hilbert space. The answer to this question was given in 1950 by one of the authors of the present book in the article [55] for the Laplace operator and its iterations under various boundary conditions. (Somewhat later, another solution to the question was given by Mikhlin in [80] for the Laplace operator on a sphere under the first boundary condition.) In 1951, Caccioppoli [79] solved the problem for the general second-order elliptic operator L under the first boundary condition, and 0. A Ladyzhenskaya [lo] (see also I l l ] and [12]) solved it for the same operators and their iterations under various boundary conditions.* The answer turned out to be simple, and, in the case of the first boundary-value problem, it was independent of the coefficients of the operator. Namely, the closures lead to functions possessing generalized first and second derivatives in L,(Q) In a way, this completed the study of the basic boundary-value problems for elliptic equations in a bounded region from the point of view of functional analysis. At the end of the forties, the convergence of the method of finite differences was investigated for these problems (cf. works by D. M. Eydus and 0. A. Ladyzhenskaya). These investigations produced both a simple method of proof of their solvability and a procedure for actually carrying out the calculation of the solutions. Thus, we seemed to have a rather thorough understanding of linear equations and we needed to direct our attention to a study of nonlinear problems. However, in the course of the study of these problems, it became clear how very incomplete our knowledge was regarding linear equations (in the case of an arbitrary number of independent variables), especially regardingthe dependence of the differentiability properties of the solutions on the differentiability properties of the coefficients of the equation. Out of all the previous results for n > 2 , there was only one that was impecc-ble in all respects. This was Schauder's theorem (supplementing Caccioppoli) regarding the solvability of the Dirichlet problem in the Holder spaces Cl,",where 1 >2 (for the notation, cf, Section 1, Chapter 1) for equations with coefficients and inhomogeneous term in Cl-*, In this theorem, all the hypotheses were necessary because of the actual nature of the matter at hand. It ensures classical solvability of the Dirichlet problem for equations whose coefficients and inhomogeneous term are continuous functions in the sense of Holder.

.

..

*The articles [79, 101 gave rise to a series of articles on ways of obtaining so-called strong estimates "up to the boundary" in L , and L , and in Certain other functional spaces for solutions of higher-order elliptic and parabolic equations and systems under various boundary conditions. Today, this series of works i s close to its natural completion. Hormander showed that such estimates are possible, generally speaking, only for elliptic and parabolic operators.

PREFACE

IX

However, this result alone does not enable u s to investigate more o r less general classes of nonlinear equations. It was necessary to study the solvability of boundary-value problems and the differentiability properties of solutions of linear equations with discontinuous, though bounded, coefficients. The results and methods already in existence yielded almost nothing in this direction. The studies in the forties and fifties by Morrey, Bers, Nirenberg, Vekua, and others were devoted to an analysis of these questions for the case of two independent variables. A reasonably complete solution of these questions was obtained with the aid of exceedingly fine tools, namely, the theory of quasiconformal transformations and the theory of generalized analytic functions. Unfortunately, these methods do not admit generalizations to the case of three or more dimensions. There was reason for feeling that the results themselves for many-dimensional problems were fundamentally different from the case of two independent variables. A new stage in the study of many-dimensional linear equations started with the articles by de Giorgi [7]and Nash [8]. In a certain way, this stage reached its culmination in the works by Stampacchia [17], Morrey [l], and the authors of the present book [5, 6, 13, 14, 33, 151. In the articles [14, 15, 331, examples were constructed showing that the conditions under which the various properties of generalized solutions of linear equations with discontinuous and, in general, unbounded coefficients were established in these works a r e precise. Chapter 3 is devoted to an exposition of the results of these articles and also to the results of the earlier works by Schauder, Friedrichs, Mikhlin, and Ladyzhenskaya on the solvability of boundary-value problems in the spaces G X(for 1 3 2 ) and Wt (for l > l ) . The first chapter is an introductory chapter in which the notations used in the book are explained and examples are given that show rather clearly the scope of the desired and possible theory of linear and quasilinear equations (this theory being then constructed in the present book); the basic results of the book and their intended development are expounded. Chapter 2 includes various analytic propositions that resulted largely from the study of differential equations though not formally related to them. The greater part of that chapter is devoted to a study of the so-called 23, classes of functions that obey certain integro-differential inequalities. It turned out that it is just these classes of functions which a r e closely connected with second-order elliptic equations. However, for the most part, the book is devoted to quasilinear equations. These have almost a s long a history as do linear equations with variable coefficients. Two hypotheses, the nineteenth and twentieth problems of Hilbert, have determined the basic direction of the study of such equations in the present century.

X

PREFACE

The first of these asserted that all solutions of elliptic equations a r e analytic functions of an independent variable provided the functions generating the equations a r e analytic functions of their arguments. In the transition from equations with analytic coefficients to equations with coefficients that a r e differentiable a specified number of times, this hypothesis acquired a different, more complete formulation: The differentiability properties of solutions of elliptic equations within the region of their existence a r e determined by the differentiability properties of the coefficients of these equations and they a r e independent both of the smoothness of the boundary conditions and of the particular functional space in which these solutions a r e originally found. The twentieth problem asserted that the variational problem on the determination of a function satisfying a given boundary condition and yielding the smallest value to a given semibounded convex functional of the form

J’ F ( x , u , u,)dx

(a regular functional,

a s it is called) always has a h u t i o n if we seek it in a sufficiently broad class of functions. This problem is closely interwoven with the problem of solving and studying boundary-value problems for quasilinear second-order elliptic equations. For this more general problem, the twentieth hypothesis of Hilbert is reworded a s follows: If we assume that a weaknorm (for example, the maximum absolute value) is bounded on all possible solutions of a boundary-value problem, this problem has a solution. Both hypotheses required more precise formulations. Here, it would be natural to seek agreement of the assertions of both problems in the sense that the class “of all solutions” in the nineteenth problem ,must coincide with that “sufficiently broad class of functions” which included the solution of the variational or the boundary-value problem. Tn the solution of these problems, people turned in opposite directions. In the nineteenth problem, attempts were made to weaken the a priori smoothness required of the solution. On the other hand, in the twentieth problem, efforts were made to prove the existence of solutions possessing as much differentiability problems a s possible. The first work devoted to the nineteenth problem of Hilbert was the famous paper by S. N. Bernstein [21](see also [81])in which it was shown that three times continuously differentiable solutions of nonlinear analytic equations (here, we are speaking only of second-order elliptic equations) a r e analytic functions of their arguments. Later, H. Lewy, E. Hopf,.and J. Schauder showed that this was true for .all solutions of nonlinear equations possessing Holdercontinuous second derivatives and for all solutions of variational problems possessing Holder-continuous first derivatives. (For the

PREFACE

XI

case of two independent variables, Morrey showed that the condition of continuity of first derivatives in the solutions of variational problems can be replaced by boundedness.) Investigation of the twentieth problem proceeded primarily along the lines of variational problems. So-called direct methods were developed for solving such problems. This direction was begun by the works of Hilbert, Lebesgue, and Courant. With the aid of these methods, one can show from general and comparatively simple considerations that these problems have so-called generalized solutions, that is, functions possessing generalized first derivatives that are rn-summable, for some tn > ; 1 (Under very broad assumptions, this was done by Tonelli and Morrey.) Tt was believed that this very class of generalized solutions is that “sufficiently broad class of functions” of which one spoke in the twentieth problem of Hilbert. However, we know that this class is considerably broader than the class of “all” solutions for which mathematicians have proven the validity of the nineteenth Hilbert problem. It was necessary either to restrict it and prove the solvability of the variational problems in the restricted class o r to show that the differentiability properties of the generalized solutions a r e in fact better than the differentiability properties of all the comparison functions, that they are determined only by the differentiability properties of the integrand, and that such solutions are unique at least in the small; that is, their local variation leads to increase in the value of the functional. In addition to the direct methods of investigating the solvability of boundary-value problems, mathematicians developed various topological methods. Of these methods, the most general and flexible is the topological method of Leray and Schauder [27], which was preceded by the investigations of Bernstein (see [Sl])on the solvability of the Dirichlet problem for quasilinear equations by the method of continuation along a parameter. With the aid of these methods, it was shown that the solvability of a boundaryvalue problem is determined by the possibility of obtaininga Priori inequalities of a definite type for all possible solutions of the problem in question. Thus, to prove that the Dirichlet problem for quasilinear equations has a classical solution, we need to have an a priori bound for the Holder norm of the first derivatives of all its possible classical solutions. Thus, in the early forties, the investigation of the solvability of nonlinear problems, in particular, the nineteenth and twentieth Hilbert problems in their extended formulation was reduced to finding a priori bounds f or the classical solutions and to studying the differentiability properties of generalized solutions of the variations problems described above. It was understood that their solution depends not only on the smoothness of the

PREFACE

XI I

functions that appear in the problem but also on the nature of their nonlinearity-on their behavior as their arguments are increased without bound. This is t r u e even when the functions generating the equation are analytic. Gradually, mathematicians showed what restrictions on the behavior of these functions were necessary for various purposes, and investigators tried to prove their sufficiency. Up to about 1956, all investigations of nonlinear problems “in the large” had primarily to do with the case of two independent variables, and for this case, a large number of devices and methods were developed. (Most of these were two-dimensional, inapplicable for three and more independent variables). Bernstein gave a prion. inequalities for equations not containing the unknown function explicitly. Leray, Nirenberg, and others obtained a Priori inequalities for certain broader classes of such equations. Morrey, A. G. Sigalov, Cesari, and L. Yound, and others investigated questions of continuity and the classical nature of the generalized solutions of two-dimensional variational problems under different assumptions regarding the integrand. However, even for two-dimensional problems, despite the great amount of first-rate writing, the meeting point in the study of the ninteenth and twentieth H i l b e r t problems. (This was made under supplimentary, clearly superfluous, assumptions on the integrand.) We have not yet obtained a priori inequalities regarding the classical solutions of quasilinear equations with no assumptions on the given functions except those that were considered natural. Very little was done in the case of many-dimensional problems. The articles [7, 8, 61, 67, 88, 891, which appeared in the y e a r s 1956-1958, were the first ones in the directionof the study of manydimensional nonlinear problems “in the large.” (There have been a number of articles considering various particular classes of quasilinear equations, close to linear in some sense o r other.) Of these, the only definitive result regarding the classical solvability of a nonlinear variational problem was the result of d e Giorgi [7] and Nash [8], which has to do with the functional under the assumption that

F(u,)dx e

...,

where v and p are positive constants independent of p = ( p , , tn). (We note that, for two independent varip,,) and E = ( E l , ables, this functional was studied by Berstein almost sixty y e a r s ago.) The solution of the many-dimensional variational problems

...,

f o r functionals of the general form tions regarding F ( x .

U.

e

F ( x , u. u,)dx, under a s s u m p

p ) that seem “natural” was given by the

PREFACE

Xlll

authors of the present book in 1959 (see [ 5 , 61). The article [5] provides the desired conclusion to study of the nineteenth and twentieth Hilbert problems referred to above. It turned out that for regular variational problems it was necessary in the case of both problems to take a s “all solutions” and “a sufficiently broad class of solutionsy’ the same class of functions, namely, the generalized solutions given by direct methods. Their differentiability properties depend only on F ( x . u. p ) and they a r e unique solutions “in the small;” that is, their local variations lead to the increase in the value of the functional. Together with the variational problem, the authors studied the class of more general quasilinear equations than Euler’s equation. Among all quasilinear equations, the class of equations whose principal part is a divergence occupied a special place. For this class, the question of generalized solutions of the question of classical solvability in the large” of the Dirichlet problem was investigated with the same desired completeness. Here, we present a second method of solving both these problems. This method makes it possible to relax a s much a s possible the requirements on the smoothness of the unknown functions (it is expounded in [ZO]). After the solution of these problems, an investigation was made of the solvability “in the large” of the Dirichlet problem for general quasilinear second-order equations and certain classes of linear and quasilinear systems of such equations [67, 38, 391 and the solvability in the large of various boundary-value problems for quasilinear equations whose principal part is a divergence [62]. With the aid of specially constructed examples, it was shown [14, 15, 331 that the conditions under which this w a s done a r e brought about by the nature of the matter. The greater part of these last results a r e immediately proven for solutions of parabolic equations. These results and proofs a r e given in [39]. A l l these works [ 5 , 6, , .] by the authors of the present monograph are alike in method. The basic idea of the method consists, first of all, in replacing the equationby an integral identity and then shifting to integro-differential inequalities containing sufficiently many numerical parameters. (The second step in each case is performed by a special choice of arbitrary function appearing in the integral identity.) Second, the method consists in proof of the boundedness or satisfaction of a Holder condition on the part of functions satisfying these inequalities. It turned out that these inequalities are satisfied not only by the solutions of elliptic equations themselves but also by their derivatives, and the information contained in them is quite sufficient for obtaining all necessary a priori inequalities. Study of functions satisfyingthese inequalities produced injection theorems of a new kind and, together with

.

XIV

PREFACE

the first step (their derivation), led to a comparatively simple and uniform solution of the problems mentioned. This same method made it possible to solve analogous problems in the more difficult case of parabolic equations (see [33, 391). It permeates almost the entire book. Exceptions are: (1) Sections 1-12 of Chapter 3, which a r e devoted to linear equations and in which the results of Schauder on the solvability of boundary-value problems in C,, for 1 >/ 2 and the results of Friedrichs, Mikhlin, and Ladyzhenskaya on the solvability of boundary-value problems in W i ( 8 ) . for 1 > 1 a r e expounded; (2) the results of Rado and von Neumann on saddle-form surfaces; (3) Section 2 of Chapter 5 on the existence of generalized solutions of variational problems (Morrey’s result); and (4) Sections 5-7 of Chapter 9 , which describe Mozer’s method of bounding the Holder norm of solutions of linear equations, Nirenberg‘s bound for the Holder constant for first derivatives of solutions of linear equations in two independent variables, and Morrey’s method of bounding the Holder norm of generalized solutions of twodimensional variational problems with integrand of quadratic growth. The results of Bers and Nirenberg on linear equations with arbitrary, not necessarily continuous, coefficients in two independent variables are proven in Section 1 7 of Chapter 3 by the same general method mentioned above without using quasiconformal transformations o r the theory of generalized analytic functions. From the works of Bernstein, we use basically one (extremely fruitful) idea, namely, that of examiningtogether with solutions of an equation, certain functions of them, and the relations that these functions satisfy a s a consequence of the given equation. This book does not expound the methods and the apparatus with the aid of which Morrey, Miranda, Nash, Aleksandrov, and Stampacchia work. It has very little overlap with books devoted to differential equations, including the bookby Miranda, which summarizes the work done up to 1955 regarding the study of linear and nonlinear elliptic second-order equations. Primarily, this book reflects the investigations of the authors that have been made in the last few years. The only work that had a direct influence on the authors is the article by de Giorgi, from which they used the idea of treating solutions u ( x ) on the sets ( x ; u (x) > k) n K , , where k and K, denote an arbitrary number and an arbitrary sphere (instead of the set u ( x ) > k ] a s was done previously) and one simple though very useful inequality (namely, inequality (3.4) of Chapter 2). Basically, the authors relied on their own understanding and their own devices and methods of investigating differential equations. The reader should not worry about a certain sameness in the methods with the aid of which we solve these problems, but should look at it a s an attempt to have a uniform style, the attainment of which was not easy. I I

Contents

PREFACE

vi i

1. INTRODUCTION

1

1. T h e B a s i c Notation a n d Terminology 2. A d m i s s i b l e E x t e n s i o n s of t h e C o n c e p t of Solution of Linear and Quasilinear Equations

3. T h e B a s i c R e s u l t s a n d T h e i r P o s s i b l e Development

2. AUXILJARY PROPOSITIONS 1. 2. 3. 4. 5.

1

9 27 39

Some Simple I n e q u a l i t i e s F u n c t i o n s in t h e Class Wf,, ( Q ) F u n c t i o n s i n t h e C l a s s e s W),, ( 9 ) and W:,, (Q) Other Auxiliary P r o p o s i t i o n s B o u n d s for III~IXI u ( x ) 1

39 41 50 56 70

6. The F u n c t i o n Class % , ( Q , M . y. b, 1) 4 7. T h e C l a s s e s % , , , ( P U S , , .. . ) and %,(QuS,. ...) 8. T h e C l a s s e s 3 : ~

81

3. LINEAR EQUATIONS

106

1. Solvability of t h e Dirichlet Problem in t h e S p a c e s

Cl, (D).I ;>

2

2. Schauder’s A Priori E s t i m a t e 3. T h e S o l v a b i l i t y in C2,m (Q) of Other Boundary-Value Problems

4. Generalized S o l u t i o n s i n Inequality

90 95

w/; (9).T h e

F i r s t Fundamental

5. Solvability of t h e F i r s t Boundary-Value Problem in Wi (Q) 6. T h e Second and T h i r d Boundary-Value P r o b l e m s 7. Interior E s t i m a t e s in L, of t h e Second D e r i v a t i v e s of a n Arbitrary F u n c t i o n in T e r m s of t h e V a l u e s of a n E l l i p t i c Operator Applied to I t

xv

107 116 134 138 149 160 162

xv I

CONTENTS

8. T h e Second F u n d a m e n t a l I n e q u a l i t y for E l l i p t i c O p e r a t o r s 9. On t h e Solvability of t h e F i r s t Boundary-Value Problem

,,

in t h e S p a c e W . 3 , (")

183

B e l o n g to W : ( Q )

185 190

10. Conditions Under Which G e n e r a l i z e d S o l u t i o n s in W: (9)

11. O t h e r Ways of P r o v i n g t h e Second F u n d a m e n t a l Inequality 12. C o n d i t i o n s Under Which G e n e r a l i z e d S o l u t i o n s in W l B e l o n g t o Ci, for f >, 2 13. The B o u n d e d n e s s of G e n e r a l i z e d S o l u t i o n s i n W : (9) 14. C o n d i t i o n s Under Which G e n e r a l i z e d S o l u t i o n s in W : (Y) B e l o n g t o Co,

I

15. C o n d i t i o n s for B o u n d e d n e s s of max I vu I and uXl la for ~

G e n e r a l i z e d Solutions in W : (9) 16. Diffraction P r o b l e m s 17. The Case of T w o Independent V a r i a b l e s 18. Two-Dimensional Saddle-Shaped S u r f a c e s

4. QUASILINEAR EQUATIONS WITH PRINCIPAL P A R T IN

DIVERGENCE FORM

1. Bounded G e n e r a l i z e d Solutions. Continuity in t h e S e n s e of Holder

2. U n i q u e n e s s in t h e Small 3. A Bound for maxl vul Y'

4. 5. 6. 7.

169

A Bound for max 1 V u I Over the E n t i r e Region Q

G e n e r a l i z e d Second-Order D e r i v a t i v e s A Bound for t h e Norms I u II. Where 12 1 B o u n d s for t h e Maximum A b s o l u t e V a l u e s of G e n e r a l i z e d Solutions 8. E x i s t e n c e T h e o r e m s

193 195 20 1 203 205 223 238 244 245 254 258 266 270 277 285 29 1 318

5. VARIATIONAL PROBLEMS 1. Statement of t h e P r o b l e m s 2. F i n d i n g F u n c t i o n s T h a t Minimize t h e F u n c t i o n a l I (u) 3. F i n d i n g a Bound f o r the Maximum Absolute Value of

318 321

S o l u t i o n s of Variational P r o b l e m s

327 331

Solutions

334

of G e n e r a l i z e d Solutions

335

4. Proof of H o l d e r n e s s of G e n e r a l i z e d S o l u t i o n s 5. T h e Theorem on U n i q u e n e s s in t h e Small of G e n e r a l i z e d 6. Further Investigation of t h e Differentiability P r o p e r t i e s

6. QUASILINEAR EQUATIONS OF THE GENERAL FORM 1. A Bound for t h e Norm I ul,, in T e r m s of t h e Norm JUI,.O 2. A Bound for max I vu I ~

338 339 349

CONTENTS

370 379

3. E x i s t e n c e T h e o r e m s 4. Two-Dimensional P r o b l e m s

7. LINEAR SYSTEMS OF ELLIPTIC EQUATIONS

p8

a n d IIu

386 386 389

1. G e n e r a l i z e d S o l u t i o n s in w ~ ( u ) 2. A Bound for n i a x I u 1 3. A Bound for I u I, Q 4. A Priori Bounds f o r I u I,, =,

XVI I

,Iw;:)

5. Solvability of the Problem ( l o l ) ,(1.3) i n t h e Classes C I ,,. (Q 6. Differentiability P r o p e r t i e s of G e n e r a l i z e d S o l u t i o n s

8. QUASILINEAR SYSTEMS 1. A Priori Bounds for t h e Norms I u I l , a , u , Where l > l , in T e r m s of max (u.vu 1 J

2. A Bound for I u ,.I a 3. T h e E n e r g y l n e q u a l i t y a n d a Bound for max I V u I on the

395 398 402 404 406 407 409

4. A Bound for maxl vu 1

413 417

5. E x i s t e n c e T h e o r e m s

420

Boundary

9. OTHER DEVICES FOR OBTAINING BOUNDS FOR THE

HOLDER CONSTANTS FOR SOLUTIONS AND THEIR DERIVATIVES

1. T h e Case of t h e Simplest E q u a t i o n 2. Bounds on Holder C o n s t a n t s for S o l u t i o n s of E q u a t i o n s 3. 4. 5. 6. 7.

( L i n e a r a n d Q u a s i l i n e a r ) with P r i n c i p a l P a r t i n Divergence Form Bounds on t h e O s c i l l a t i o n s of t h e D e r i v a t i v e s of Solutions of E q u a t i o n s with P r i n c i p a l P a r t i n Divergence Form E q u a t i o n s Not in Divergence Form Moser’s Method of Bounding I u. .1 Q, for S o l u t i o n s of L i n e a r Equations Nirenberg’s E s t i m a t e Morrey’s Method for F i n d i n g a Bound for t h e Holder C o n s t a n t for S o l u t i o n s of Two-Dimensional Variational Problems

10. OTHER BOUNDARY-VALUE PROBLEMS 1. Formulation of t h e Problems a n d t h e G e n e r a l Procedure for Solving T h e m

423 425 429 442 443 446 452 456 459 459

CONTENTS

XVlll

2. Quasilinear Equations with Principal Part i n Divergence Form 3 . Quasilinear S y s t e m s

465 48 1

BIBLIOGRAPHY

484

INDEX

493

1 Introduction

This book is devoted to second-order elliptic equations. We shall study the boundary-value problems associated with such equations in various functional spaces and we shall investigate qualitatively and quantitiatively the relevant differentiability properties of arbitrary solutions of these equations. We shall begin our exposition with a discussion of examples that make it possible to draw fairly accurate outlines of the possible theory regarding these questions (Section 2) and with a listing of the basic results of the book together with the development desired for them (Section 3). Let us first introduce a number of concepts, notations, and definitions that we shall be using throughout the entire book. 1. THE BASIC NOTATION AND TERMINOLOGY E, denotes an n-dimensional Euclidean space; x = ( x l , ..., x,) denotes an arbitrary point in it. Throughout, n is assumed to be 2 or greater. Q denotes a bounded region in En;that is, an arbitrary open connected set contained in some sphere of sufficiently great radius. S denotes the boundary of Q . H denotes the closure of 8, so that PUS. Q' denotes an arbitrary strictly interior subregion of 9, SO that the distance between Q' and S is always positive. K, denotes an arbitrary sphere of radius p in En; x, = mes K,.

a=

Q, = K,

n 8,

p =(pl. ,pl

. . .,

x=(xl.

prJ.

=(i 1=1

1x1

p:>,* 1

.. ., X,j.

=(iq. 1=1

2

INTRODUCTION

A l l the functions and quantities considered in this book will be real unless the contrary is explicitly stated. a (x,

If,

. .., .r/l. It. . . . , x,,, u .

p ) = a (xl.

u ( x . u , u,\.)= a (xl,

I.'] ux

..

-. P , J ,

. . . .. q.

If u ( x ) is some function of x , then

osc ( u ( x ) ; Q } is the oscillation of u ( x ) on x , that is, the difference between essential max u ( x ) and essential min u ( x ) . Q 9 Sometimes, we shall write max u ( x ) instead of essential max 11 ( x ) , Q Q and we shall always write u2, ~ instead of ( u , ~ ) ~ . Furthermore, 1

u(k)(x)=-[u(xI,

Xk+hxk,

Axk

1

u - (x)=-[u(xI. Axk

(k )

v, p.

E,

.-.,Xn)-

- u (XIS

..., x k . .

* * s

-u (XI.

*

x,r) -

..

1 .

xk

.

*

9

xk.

* *

9

xn)].

AX^, . . .. x,,)].

6, a,, 0 and 1 denote positive constants. a positive nonincreasing continuous function defined

v ( t )denotes

for t >O. p ( t ) denotes an arbitrary nondecreasing continuous function defined for t >/ 0. denotes the Kronecker delta: Si = 0 for I # 1, 6f = 1. In the equations that we shall study, we shall encounter exd [ a ( x , u ( x ) , u x ( x ) ) ] . This means that, in pressions of the form -dx1

d

we need to take into account the evaluating the derivative presence of x i not only in the first set of arguments x , but in the other two a s well, that is, in the functions if ( x ) and u, ( X I , so that

Here and in all that follows, the appearance of the same index used twice indicates summation from 1to n. In particular,

T H E B A S I C NOTATION AND TERMINOLOGY

3

Sometimes, when there is no danger of confusion, we shall replace d the symbol for a total derivative dx, with the more widespread symbol

-.

d dxi d

Thus, in a linear equation, usually we shall write the d

terms -( u f ,( x ) u x , ( x ) ) in the form -(ul, ( x ) ux,(x))though even here, dxf dXi in the differentiation, we need to consider x f in both arguments, a l l ( x ) and u,,(x). In what follows, we shall encounter various constants defined by quantities that a r e known to us from the conditions. We shall denote these constants by a lower-case letter c with various subscripts. Where there is no danger of confusionand where the value of the constant in question is of no significance, we shall drop the subscripts on the c, so that even in a single proof the letter c with the same subscript o r even with no subscript at all may be used to denote different constants. In other cases, when we need to emphasize the dependence of a constant on some quantity o r other, this dependence will be shown explicitly. A function u ( x ) is said to be of compact support in Q if it vanishes in a neighborhood of the boundary of Q. Its support (carrier) is the closure of the subregion 8' in which u ( x ) is nonzero. This closure is contained in the interior of Q. The classes 23, (Q, . . .), 23" (Q.. . .), etc., with different parameters a r e defined and studied in Sections 6-8 of Chapter 2. If .L is an operator in a Hilbert space, we shall denote its closure by 1. We shall denote the domain of definition of L by D (L)and its range of values by R (L). n is an outwardly directed unit vector (relative to Q) normal to S d denotes differentiation in the direction n. at any point on S. We shall use the following notations for the functional spaces that we shall encounter. L,(O) denotes the Banach space of all functions on Q that a r e measurable and that a r e m-summable with respect to Q with rn > 1. The norm in this space is defined by

Measurability and summability a r e always understood in the sense of Lebesgue. The elements L,(Q) a r e the classes of equivalent functions on Q.

4

INTRODUCTION

Generalized derivatives a r e understood in the way that is now customary in the majority of works on differential equations. Different but equivalent definitions of them and their fundamental properties can be found, for example, in [l] and [2]. Wf,(Q) denotes the Banach space of all elements of L,(Q) that have generalized derivatives of all kinds of the first 1 o r d e r s that a r e m-summable over Q. The norm in W',(P)is defined by

where the symbol Dck'denotes an arbitrary kth derivative of U ( X ) with respect t o x and 2 denotes summation over all possible kth Iki

derivatives of u. For regions with "not too bad" a boundary, Wk(Q) coincides with the closure in the norm (1.1)of the set of all infinitely differentiable functions in 9. This will be the case, for example, with regions having a piecewise-smooth boundary (the definition of this will be given below). Sometimes, we shall write Wf, instead of Wf, (Q), especially when the region 9 needs subsequent precision. Wf,(Q) denotes the set of elements Wf,(Q) that a r e of compact s u p p p t in M. Wf, (Q) denotes the closure in the space Wf, (52) of the set of all infinitely differentiable functions of compact support on 9 , in particular,Wi/:,( Q ) c Wf, (Q). Wi,,,(S2) denotes the subspace of the space W:(M)which is the closure of the set of all twice continuously differentiable functions on 9 that vanish on the boundary. We shall say that the function n ( x ) satisfies a Halder condition with exponent a , where a E (0. 11, and with Halder constant M in the region SL if 1 S U P p -' osc { U , 9,)= M. (1.2) where the supremum is over all components P ,< Po.

Qiof all Q, such that

C,..(P) is the Banach space the elements in which a r e all funcu ( x ) that a r e continuous in P having finite 1 u I(*),P. The norm in Cu, is defined by the equation

tions

(a)

luIa,,=yx

l u l + IuI(*),P-

(1.3)

The domain of definition of the elements Co, (8)can be thought of a s extended to S in such a way a s to be continuous from within. Here, the points on S a r e regarded a s classes of paths that a r e equivalent in Q, and u may turn out to be multiple-valued on S.

THE BASIC NOTATION AND T E R M I N O L O G Y

5

A function u ( x ) belongs t o C , , ( Q ) if it belongs toC,, “(G‘) fo r all

Q ’ c Q.

C , , ” ( 8 )is a Banach space the elements of which are functions that a r e continuous in Q and have continuous derivatives in SZ of the first 1 orde rs and for which the value of

is finite.

Equation (1.4) defines a norm in C,, ,@). The elements C,,(G)may be assumed continuous and 1 times continuously differentiable in s. For this, we need t o extend the definition of u ( x ) and its derivatives onto S as continuous functions. C,,.(Q) is the set of functions belonging to C,, ,@’) for all Y ’ c Q . The Banach space C,,o(SZ) and-the linear set C,,,(Q) a r e defined analogously. The norm in C,,,(M) o r , what amounts to the same thing, in C,(M) is defined by the equation 1

-2

IU1f,9=I~II,0,e-k=0

2 max I W ) U1, (*)

(1.5)

,(c)

In brief, C,, consists of all functions that a r e 1 times continuously differentiable in s. For brevity, we shall denote C0,,(q simply by C(@. This is the Banach space of all functions that are continuous in s , where the norm is max I u ( x ) I.

-

Let us’compare C,, @! and C,,o(% It is easy to see that C,,,(Q: c C , (Q.The converse does not hold. We know (cf., for example, [l]) that C,, (S)c W’,(Q) for arbitrary finite m , so that the elements C,,,(Q) have generalized first derivatives in 52 and, for these derivatives, essential max I Du I = I u I, e

3.

Let S , denote a portion of the boundary S of the region 51. Let us denote by Cl, a (&US,) the set of functions belonging to the spaces Ct,=(al), where 8,is an arbitrary subregion of Q lying at a positive distance from S \ S,. Let us agree to assume that the symbol C,,.is used only for positive values of a less than unity. C, Wk (Q), and the other functional spaces referred to above ar e assumed complete. Cl (Q) is the set of all functions in C, (s)that are finite in Q. Later, we shall also encounter the classes of functions 0, (g)for

(a),

1=1.

2.

6

INTRODUCTION

,-

The - set 0, (q is larger than the set Cl, (52) though - smaller than C,, (Q), In contrast with an arbitrary element of C , (Q), its elements For these, the norm have a first differential at every point in is finite. The functions in C,,,(Q) the first derivatives of 14,o Q which are functions poEessing a first differential at every point of P a r e elements of O,(Q). We introduce a number of definitions and notations dealing with regions, their boundaries, and functions defined on these boundaries. Throughout the entire book, we confine our attention to regions possessing “piecewise-smooth boundaries with nonzero interior angles” or, more briefly, “piecewise-smooth boundaries.” By this term, we mean a region Q the closure of which can be repreQ, = 0 and where each sented in the form P = s l U . . . US,, where

,

a.

,

n 1

of the akcan be mapped homeomorphically onto the unit sphere (or cube) with the aid of functions 2‘; ( x ) , for 1 = 1, . . , , n and k = 1. . . . , N, that belong to the class I Z , , ~ ( and ~ ) have the property that the Jacobians of the transformations d ( Z k ) a r e bounded from below by

I

~

d(x)

I

a positive constant. We shall say that the boundary S of a region Q (or a portion S,of it) satisfies condition (A) if there exist two positive numbers a, and e, such that, for an arbitrary sphere K, with center on S (resp. on S,) of radius p ,< a, and for an arbitrary component 6,of the intersection 8, of the sphere K, with 9, the inequality mes 6,4 ( 1

- e), mes K,

holds. Let xo=(xy, . . . , x:) be an arbitrary point of the boundary S of the region Q. We shall call (yl. . . ., y,) a local Cartesian coordinate system with origin at the point xo if Y and x a r e connected by the equations y, = aZIk ( x k - x i ) , for I = 1, ,n , where a l pis an orthogonal numerical matrix and if the y,-axis is directed along the outer (with respect to M) normal to S at the point x,,. We shall say that the surface S belongs to the class C,, for 1 > 1. a E 10, 11 if there exists a number p > 0 such that the intersection of S with the sphere K, of radius p with center at an arbitrary point x o S is a connected surface the equation of which in the local Cartesian coordinate system (yl. . . ., y,) with origin at the point xo is of the form y, = w (y,, . . . , y,,), where w ( y , , . . . y,-,) is a function of the class C!,a in the region D constituting the projection of K , n S onto the plane y, = 0. Suppose that a function p (s) is defined on a surface S of class Cl,II for 1 > 1. We shall say that p(s) is a function of the class C,, .(S) if a s a function of ( y , , *. y,-,) it is an element of C l , a ( D ) .We shall

...

(1

.

.,

T H E B A S I C NOTATION A N D TERMINOLOGY

7

of all take for the norm 1 'p 1 , a. the greatest of the norms I 'p (y) points xoof the surface S. lf 'p is defined over all s a n d 'p (x) E Cl,a (525 for 1 > 1, then, on the boundary S of a region Q belonging to the class C k a , it defines a Ix=sc s belonging to the class C,,=(S) (this holds function 'p (s) ='p (4 also for a portion or the boundary S). The converse is also true: If 'p (s)E C,, a (S) and S E C l ,a for I > 1, then 'p (s) can be extended to the entire region 5! in such a way that the extended function 'p(x)belongs to C l ,I (Q). Furthermore, this extension can be made for all functions 'p ( x ) in C,, o. (S) with the aid of the same construction in such a way that the norms I 'p (s)11, o, and I 'p ( x ) ti , a, will be equivalent. In view of this, it is immaterial whether we speak of defining the function 'p on S or on all H and whether we speak of its norm I 'p II, R , S or 1 Q II &. However, for I = 0 , these statements do not apply to functions defined on S E Cl, Out of two possible though different definitions, we shall find it more convenient to take the one under which and I 'p ( x ). .1 is conserved. Thereequivalence of the norms I 'p (sj,I. fore, we define 1 'p (s) . .1 a s follows:

I'p l a , =max 'p (s) + sup p-a S€S

osc ('p (s);

s;):

(1.6)

where the supremum is over all components S: of all K, n S, where P < P"' Sometimes, instead of stating that "the boundary Sof the region P is a class C,,a surface," we shall say that ' ' Q is a class C l , m region." It is easy to show that this definition of belonging to the class C I , a on the part of 8 isequivalent to the following: The region Q is a class C , , a region if 9 can be represented a s the union of aN(not necessarily disjoint) for each finitely many regions G , of which the intersection of its E-neighborhood Gi with Q can be mapped homeomorphically onto the unit sphere with the aid of the functions z: (x), for I = 1 , n and k = 1 , * . N , which belong to the space C,, (@k n 2) and which have the property that the Jacobians d ( z k ) exceed some positive number 6. e .

.

.,

...,

.,

lml This second form of the characteristic of the smoothness

Sts

more convenient for defining regions belonging to the class W,. Such regions are defined just a s regions belonging to the class C l , according to the second definition except that the condition z: (xj E CI, pkn 9 is replaced with the condition ( x ) E W:,(ai n G). To characterize them in terms of the functions y,, = ~ ( y , ., . ., Y,,-~) giving the equations of portions of the surface S, we need to introduce new concepts of W',spaces with fractional valuesfor 1. This should also be done for describing the boundary properties of functions of the classes W:,(Q). Not wishing to load the book down with an exposition

a

INTRODUCTION

of these questions, we shall use only the second definition for c l a s s Wf regions and, when investigating the solvability of various boundary-value problems in WL spaces, we shall give the boundary values ~ ( xon ) S with the aid ofthe function ‘p defined from the s t a r t for all g. The results established in connection with boundary properties of functions of WL(Qj spaces and the possibility of extending ‘p (s) from S onto s1 in the form ‘p ( x )in WL (‘2) make it possible to use either of the two forms without loss of accuracy (cf. [82, 83, 841 interalia). We shall often say “let u s straighten out a portion of the boundary S in a neighborhood of some point x0 of it, introducing new COordinates.” This means that, in a neighborhood of the point X‘’, we introduce new regular coordinates (in general, not Cartesian) z, = z , ( x ) for I = 1 , , n in such a way that the equation for the portion of the boundary S in these coordinates will bez,=O and the portion of the region Q adjacent to it will be located in the halfRegularity of the new coordinates zl, z,means space z, >O. that there is a continuous one-to-one correspondence between the new and old coordinates, that the functions z l (x) for i = 1, n are

. .,

differentiable, and that the Jacobian

..., ...,

%).is nonzero in the x-

...,

...,

region in question. Furthermore, the functionsz, (x) for I = 1 , n (and hence the inverse functions x l (2) for 1 = 1 , n) possess the smoothness indicated. The coordinates with such properties for surfaces of the class Cl, or I >, 1 in a neighborhood of xo S can be introduced a s follows: From x , we shift to local Cartesian coordinates )’ and from y to the new variables z according to the formulas ‘1

The cylinder

will, for small p and 8 , belong to G-. Close to the point x“, these coordinates will be regular and the functions z1( x ) will belong to

,.

Throughout the entire book, we shall encounter assertions that such and such a constant i s defined by such and such quantities. Tn the great majority of cases, we shall not point out the exact relationships, which will either be sufficiently obvious or of no particular interest to us. For example, we shall not point out their

A D M I S S I B L E EXTENSIONS

OF T H E CONCEPT

9

dependence on the number of dimensions n although in many places such a dependence will exist. We shall, for the most part, point out the dependence on a region only when we need to consider regions of sufficiently small dimensions and where the dependence on the measure of the region is significant.

2. ADMISSIBLE EXTENSIONS OF THE CONCEPT OF SOLUTION OF LINEAR AND QUASILINEAR EQUATIONS The basic objective of the present book is the study of secondorder linear and quasilinear elliptic equations. It i s convenient to group them into four classes: equations of the types (2.1), (2.2), (2.3), and (2.4). ‘In connection with these equations, we a r e interested primarily in the principles governing them and the relationships connecting known functions of the equations and the solutions of these equations-relations that are general for all equations belonging to the particular type in question and that can be characterized quantitatively with the aid of numerical characteristics of the known functions, namely, their norms in the spaces L,(Q), W: (Q), o r Cl, Therefore, these characteristics (both of the solutions and of the known functions that appear in the problems) must not depend on the particular equation and must be expressed in t e r m s of belonging to one of the spaces just mentioned. We shall always* assume that the equations are uniformly elliptic. What this means will be defined below with the aid of inequalities (2.5)-(2.7). A s the last few decades have shown, it is not expedient to confine oneself to consideration of classical solutions alone. We shall begin our investigations with a clarification a s to what extensions in the concept of solution of linear and quasilinear equations a r e admissible under the different assumptions regarding the functions generating the equations. We shall consider generalized solutions admissible for a given class of equations if these generalized solutions preserve those theorems on the uniqueness of the boundary-value problems that hold for classical solutions. For second-order elliptic equations, we shall take the Dirichlet problem a s our criterion. We know that, for such equations, the theorem on the uniqueness of the solution of this problem in the class of classical solutions is valid “in the small;” that is, the Dirichlet problem has only one classical solution if the domain of definition of these solutions is sufficiently ~

*Theorems 8.9 of Chapter 1V and 3.6 of Chapter VI are exception.

10

INTRODUCTION

small. (In the nonlinear case, we sometimes impose restrictions on the magnitude of the deviation of the trial functions and their derivatives.) Clearly, this property, which is inherent in the classical solutions, should be preserved for the generalized solutions introduced. Thus, we shall call a class of generalized solutions of secondorder elliptic equations “admissible” if the uniqueness theorem for the Dirichlet problem is valid for it “in the small.” We shall begin our search for admissible extensions of the concept of a solution with examination of linear equations of the form (2.1), the coefficients a,,. a,. b, and a being discontinuous functions. a (a,,ux,)=O, which constitute a special For equations of the form ax1 case of this class, this class is known; namely, they a r e the generalized solutions in W;(Q). For general equations of the form (2.1), we shall find the restrictions on a i , b l , and a that a r e necessary for admissibility of such generalized solutions. We shall analyze further the case of Eqs. (2.2) with nondifferentiable coefficients a , , ( x ) ; then, with the same point of view, we shall study quasilinear equations of the forms (2.3) and (2.4). It turns out that, for nonlinear equations, the admissibility or nonadmissibility of the different extensions of the concept of a solution depends not only on the smoothness of the functions a , ( x . u , p ) , a ( x , u , p ) and a , , ( x , u . p ) generating the equations (as is the case with linear equations) but also on their behavior a s 1 u]-+cc and I p I +m. Such a preliminary analysis was necessary in nonlinear equations even for the study of the solvability of boundary-value problems within the framework of classical formulations because it also established what a pn-ori bounds of the classical solutions a r e impossible. In the present section, we a r e concerned with the necessary restrictions on the given conditions of the problem, and for this we shall construct appropriate examples. The necessity of the restrictions indicated below should be understood in the sense that if one of these restrictions is weakened (for example, we replace membership in L, with membership in Lq,,where q’ < q), then the class of equations in question will include one with a solution not possessing the property in question (the property for which necessary conditions a r e sought). This necessity does not rule out the possibility that, for narrower classes of solutions possessing certain special properties, these restrictions can be weakened o r replaced with others. Here, we shall give another series of examples providing conditions that a r e necessary for an arbitrary admissible generalized solution of a given class of equations to belong to the spaces C,,, or C,,I or W i , etc., and to satisfybounds on the norms of these spaces.

A D M I S S I B L E E X T E N S I O N S O F T H E CONCEPT

11

The necessary restrictions that are given on these examples are also sufficient. Proof of their sufficiency occupies the central point in the present book. The examples shown below give precise outlines of the possible theory of boundary-value problems for equations of the type (2.1) with discontinuous coefficients and for quasilinear elliptic equations in the spaces W:(Q),W: (Q), Wi (Q) fl Co, (Q), Ct, (Q), etc., where 51 is an arbitrary bounded region, under the assumption of their uniform ellipticity. Let u s look at equations of the following forms: n

n

(2.3)

and n

assuming that they are uniformly elliptic. (2.2), this condition takes the form

For Eqs. (2.1) and

...,

:,(here and below) where v and p are positive constants and a r e arbitrary real numbers. For Eqs. (2.3), this condition is of the form n

n

12

I NTRODUCTlON

and, for Eqs. (2.4), it is of the form n

(2.7) where rn is a number greater than 1. Let u s begin with Eq. (2.1). Let u s compare it with the integral identity

where 7 is an arbitrary smooth function that vanishes on the boundary S of the region Q. If u ( x ) i s , for example, a twice conand all the coefficients in tinuously differentiable function on (2.1) are smooth, then it is easy to see that the relations (2.1) and (2.8) are equivalent for it; that is, (2.8)follows from (2.1), and (2.1) follows from (2.8). However, the identity (2.8) is meaningful for functions u possessing derivatives of first order only and f o r nondifferentiable functions a,). a,. f,. b,, (I,and f. The principal t e r m in it is a,jux,qxfd x . The quadratic form f aijTx d x correspond-

n

.U

P

1

)

ing to it is defined, on the basis of (2.5), on an arbitrary function 1 E W:(&). Therefore, a natural class of generalized solutions of

Eqs. (2.1) under conditions (2.5) is the class of generalized solutions in W:(Q). We shall call an arbitrary functiop u ( x ) in W:(Q)that satisfies the identity (2.8) for arbitrary q in W:(Q)a “generalized solution of Eq. (2.1) in the class Wi(Q).9y However, such an extension is inadmissible if the coefficients a t , 6, and a in (2.1) possess strong singularities. Let u s agree to express our restrictions on the character of admissible singularities in the coefficients in the form of their membership in the spaces L, for different values of s. The following example shows u s that it is necessary to impose restrictions on a,, bt, and a in such a way that there will be uniqueness in the small for the generalized solutions W:(Q) and the size of the region in which the uniqueness theorem is valid will be determined by the quantities II L s , ( Q ) . II bi II Ls2(Q). 11 a llLs,(e) but will be independent of the other characteristic of the coefficients (for example, the absolute values of their integral continuity, etc.). Let u s consider the function u ( x ) =In I In r 1, where /n

A D M I S S I B L E EXTENSIONS O F T H E C O N C E P T

13

For this function,

In the sphere K, = ( r ,< R ) , where R < 1 , it obviously belongs to W:. It can be regarded as a solution of any one of the equations

where bl = - F ( r )xi In r . or n

These equations are also satisfied by the function v(x)=InlInRI, where R is an arbitrary positve number less than 1; this function coincides with u on the boundary of the sphere KR. Thus, we do not have uniqueness "in the small." The coefficients aI, b,, and a in (2.9) are S, s, and (s/2 summable with exponents respectively, where s is an arbitrary number less than n for n > 2 and s =n for n = 2. Thus, such singularities are inadmissible. They can be removed by requiring that

However, if we wish for the size ofthe region in which the theorem on the uniqueness of the Dirichlet problem for (2.1) is valid to be determine9 only by the numbers v and p in (2.5) and (2.10 ), then q in (2.10 ) must pe assumed greater than n even when n > 3 ; that is, instead of (2.10 ), we need to assume

-

11%

biIILq(P)4P n in (2.10) is necessary for the validity of the theorem on uniqueness of the type indicated. In Section 4 of Chapter 3, it will be shown that this assumption together with (2.5) is sufficient also. There, we shall show that, for every specific operator L of tpe type (2.1) with coefficients satisfying conditions (2.5) and (2.10 ), there exists a number m,,> 0 such that, for those regions Q' c Q such that measure Q',<m,,, the Dirichlet problem for L has no more than one solution; however, in view of what was sa,id above, m,cannot be determined from v and p in (2.5) and (2.10 ). It is determined by v , p , and the "moduli of the integral continuity" of the coefficients, =E(P). A s soon a s the coefthat is, the quantities llu;, b;, u ''L; (Kp)

ficients a r e fixed [and satisfy (2.10 ')I, the quantities ~ ( p )a r e uniformly small for small values of p and they approach 0 a s p + 0. Then, the fact that ~ ( p ) approaches 0 ensures uniqueness of the Dirichlet problem in regions of small measure. It will also be shown in Chapter 3 that the assumptions (2.5) and (2.10) ensure validity of all the Fredholm theorems for the Dirichlet problem for (2.1) under quite broad assumptions regard' ing the inhomogeneous terms f and f,and boundary values 'p (s) of the solution. These assumptions o n f , l i , and 'p a r e also necessary for the existence of a solution in the class W : ( Q ) ) . This is also ensured

A D M I S S I B L E EXTENSIONS OF THE CONCEPT

15

by conditions (2.5) and (2.10') with the same qualification a s in the discussion of the uniqueness theorem. We pose the following question: To what classes L,(Q)must the functions a,, 6,. 3, f,, and f belong in order that all generalized solutions u ( x ) of Eqs. (2.1) in W : ( Q ) be bounded functions? To ascertain the necessary conditions, we again take the function u = In I In r ( and regard it in the sphere r ,< R < 1 as a solution of any one of the following equations: Au = F ( r ) ,

AU -, nf=, ,( nr )r ,u=O.

Au - -(A) a = 0, dxl

r*Inr

h---(a dx, r*lnrlnllnrl

(2.11)

where F ( r ) has the same meaning as above. It is easy to see that in these equations f E L- (KR).f,E L, (KR) and a E L ,(KR)but a , E L, (K,& 2

2

Therefore, (n/2)-, n-, (n/2)-, and n-summability o ff, f i , u, and a, respectively does not ensure boundedness. of the generalized solutions. Therefore, the requirements (2.10) on a , and a and the requirements

(2.12)

on f, and f are necessary. The first of Eqs. (2.9) shows that a necessary requirement on the 6, is their q-summability over 52 with q >n for n > 3 and q > n for n = 2. However, if we wish, as fo r the uniqueness theorem, not just boundedness but a bound max lgl in terms of some weaker n o r F of u (for example, IIulir,ce,) and constants v and 1.1 in (2.5), (2.10 ), and (2.12) characterizing the norms of the known fyc t i ons , we need t o assume q > n also for the bl in condition (2.10 ). Otherwise, as is easily seen, the operators LR provide an example of the impossibility of such a bound for q =n for the entire set of equations (2.1). Thus, we see that the requirements (2.10) and (2.12) a r e necessary for the validity of the principles stated. In Chapter 3, we shall prove their sufficiency. Furthermore, it turns out that they are also sufficient for an arbitrary generalized solution u ( x ) in W:(Q) of Eqs. (2.1) to be not only bounded but continuous in the sense of Hblder and for its norm where Q' c 8 is bounded from above by a constant depending only on (1 u I(r,(o) and on the numbers v and 1.1 in (2.5), (2.10), (2.12) (and, of course, on the distance Q/ from the boundary S if we make no assumptions regarding the regularity of 3' and u .)sI For a specific equation of the type (2.1), all this can be done under a weaker assumption concerning the 6, for n >3 , namely, n-summability of the 6, over 8.

16

INTRODUCTION

But the bounds max IuI and 1u1, will then depend on the modulus of the integral continuity of the b , ( x ) in Ln(Q)>that is, on the quantities

Thus, we see the necessity of conditions (2.10) for the construction of a sensible theory of generalized functions in W : ( Q ) for equations of the form (2.1) with discontinuous coefficients ail that satisfy inequalities (2.5). tn all that follows, we shall assume that these conditions a r e satisfied. We shall not stop for details regarding the critical case of q = n >/ 3 The methods presented below enable us to include it also. The necessary explanations for this a r e given in Sections 4 and 5 of Chapter 3. Detailed supplements could be given in other places of that chapter and in Chapter 3, but we shall not do this lest we overburden the exposition. Let u s clarify the properties that must be possessed by the coefficients of the entire set of operators L of the form (2.1) in order for them to be bounded operators in L2(Q)belonging to W:(Q). We shall express these properties just as above in terms of the membership of the coefficients in the sets .L,(Q)and Wj(8). Here, their form will be of a single type for all dimensions n 2. It is easy to verify that these properties [not including condition (2.5)] a r e of the following form:

.

>

Specifically, a s the embedding theorems show, membership of u ( x ) in > 2 , the derivatives u,,cL zn (Q); n-2 2n (Q); for n = 2 , the derivatives u,, a r e for n > 4, the function u LWi(S1) implies the following: For n

I-summable over $52 for a&!trary exponent 1 ; for n = 2 or 3 , the function u ( x ) is continuous on for n = 4 , the functionu(x) is 1summable over Q for arbitrary exponent 1. Using these facts and H'dlder's inequality, we find a bound for the norm in L 2(Q) of every t e r m in L. These bounding inequalities include the norms of the coefficients L in (2.13) and I(uJI4z(e). Since the embedding theorems that we have been using and H'dlder's inequality a r e precise, this confirms the necessity of the restrictions (2.13). It will be shown in Sections 6 and 7 of Chapter 3 that conditions (2.13) a r e also sufficient for every generalized solution of Eq.

ADMISSIBLE EXTENSIONS OF THE CONCEPT

17

(2.1) in W:(Q) with right-hand member

in L, (Q) to belong to W i ( Q ' ) , where 8' E Q. Suppose that conditions (2.5) and (2.13) for L in (2.1) are satisfied. Let us see under what restrictions on the coefficients and the inhomogeneous t e r m s arbitrary solution of Eq. (2.1) in W i ( Q ) will belong to Cl, (Q). The example (I

ArX=A(r~+h--2)r~-~

indicates the nedessity of the condition (2.14)

where the quantity q - n > 0 depends on the Holder constant a. If the t e r m s au and (dal/dxl)u in Lu are related to the inhomogeneous t e r m and if we remember that u is bounded, then the requriement (2.14) indicates the necessity of the additional condition (2.15)

It turns out that the restrictions (2.13)-(2.15) are also sufficient. Specifically, in Section 15 of Chapter 3, we shall show that when conditions (2.13)-(2.15) are satisfied, all generalized solutions u of Eqs. (2.1) in W ; ( W belong to C,,,(Q) and that their norms 111 II,atP,, where Q'c52 can be bounded from above by an expression involving the constants v , q , and M in conditions (2.5), (2.13)-(2.15), J1u!lw;(u),and the distance from 8' to 2. These same quantities also determine the size of the exponent a > 0. The following formulation of conditions (2.10), (2.12), and (2.13)(2.15), which shows their natural connection, is of interest. Suppose that u is a generalized solution of (2.1) in W i ( Q ) . The set of functions up= u and u k = u, k , where k = !, n, can be regarded a s a generalized solution in W : ( Q )of the following system:

...,

18

INTRODUCTION

which may be written more concisely

Since Eqs. (2.1) constitute a particular case of systems of the form (2.16), conditions (2.10) and (2.12), which a r e necessary for Eqs. (2.1), will also be necessary (from the standpoint of the purposes mentioned above) for the entire class of systems (2.16). For (2.16), they take the form

I l 4 U L p ) < @J*

nB:llL*

-2

(Q)

< a.

(2.17) (2.18)

If we recall that A:, and B; a r e formed from the coefficients in Eq. (2.1), it becomes clear that conditions (2.17) and (2.18) correspond to conditions (2.13)- (2.15). In Chapter 7, it will be shown that conditions (2.17) are sufficient for admissibility of generalized solutions in W:(Q)and that conditions (2.17) and (2.18) a r e sufficient for all such solutions to belong to CO,=.This also holds for the entire class of systems of the form (2.16). Let us turn to equations of the form (2.2), assuming that U , ~ ( X ) in the general case a r e nondifferentiable functions satisfying only inequalities (2.5). It is impossible to derive an identity of the type (2.8) to such equations. Therefore, for such equations we cannot consider generalized solutions u possessing only first derivatives. Their generalized solutions must have second derivatives. It turns out, however, that this is insufficient. To clarify the given question, let us look at the following example. The function u = ri defined in the sphere KR: ( r 4 R } satisfies the equation* Lu = a l / U x l x , where a l j =S{+bT

XlXj

and b = -

u,, =irA-'x1. u,

'The fact that (2.19) with a,, = 81 noted in [95].

(2.19)

+-.

For this equation,

1

,=)\rk-

1 /

= 0,

4

n-1 1--h

[()\-2) x i x j

XixJ

+' { r 2 ] .

6 (r) 7 has a solution of the form u = u ( r ) is

ADMISSIBLE EXTENSIONS O F T H E CONCEPT

19

For b > - 1, that is, for A < 1 , Eq. (2.19) is elliptic (inequalities (2.5) are satisfied). The derivatives u x , are p-summable over UR, for p

< +i.For

< *%,and

I1

the derivatives uxI are q-summable, for

values of A close to unity, p is close to n and q is

close to infinity. However, we cannot consider this function u = rA for A < 1 as an admissible generalized solution of Eq. (2.19) since Eq. (2.19) is also satisfied by the function v = RA, which coincides with r Aon the boundary of the sphere UR. It is possible to exclude this case in various ways. For arbitrary n 2 2 , let u s look at those generalized solutions u that have essential max I V u ( 2 , inequality (2.20) is proven for an operator L in (2.2) under the condition that the aIi are continuous. Here, the constant c in (2.20) depends on the modulus of continuity a,, ( c depends also on the region Q , but, for example, can be chosen the same for all spheres KR, where R ,< R,,). A l l attempts to prove this inequalityfor n > 2 for arbitrary bounded a,, or even to show that, for smooth a,,, the constant c depends only on v and ~1 in (2.5) have been unsuccessful. And, from the example (2.19), we can see that this is impossible. Specifically, for the function u = r A- RAin the sphere K,, the norm llull w z is of order 2(

R)

Therefore, in (2.20), the constant c will increase without bound a s R -+ 0. (For the case of n = 2, this example yields nothing since

20

INTRODUCTION

u = 1).- RA, where A < 1 , does not belong to W', ( K R ) and, a s we know, for n = 2 , inequality (2.20) is valid for all bounded at, satisfying inequalities (2.5).) In inequality (2.20), we can get rid ofthe term IIuIILlce.if we know that the point A =0 is not a point of the spectrum for t,more precisely, if we know that the problem .Lu=O, u J s = O has only the zero solution in the class W i ( Q ) . In this case, the operator L has a bounded inverse in L, (Q). Therefore, )Iu 1) L l ( e ) is bounded in terms of

! J L U I I ~ ,for ( ~ ) an

arbitrary function u in Wg(&)that vanishes on S. In particular, if the coefficients L in (2.19) a r e smooth, then A = O is not a point of the spectrum for L in (2.19). Therefore, for arbitrary u in Wz(Q)that is equal to 0 on S,

11 11 w;(q 4

I @j%,

(2.21)

llL* (9)'

Let us show that the constant c in (2.21) depends not only on v and p in (2.5) but also on certaincharacteristics of the smoothness of the a l l , in other words, that it cannot be general for the entire set of smooth coefficients ail that satisfy (2.5) with the same 'Y and p. Let u s average both members of Eq. (2.19) with the aid of some sort of nonnegative infinitely differentiable kernel wp ( Ix -- Y I 1 (see 11, 21). We write the result in the form where

but

The coefficients

u l j Pa r e

infinitely differentiable but

f, E L, (KR) and II fpll L2( K R ) a s p-0.

--+

0

This last follows from the inequality Jdx{

KR

IX-Y

s

Up(

I 0, inequality (2.21) holds; that is,

But it is clear from what was said above that the constant cp in this inequality approaches 00 as p - t o , so that it does not depend only on v and p . Let u s turn now to quasilinear equations. We begin with Eqs. (2.4) in their general form. We have already pointed out in the linear case the need for the requirements of smoothness of the first derivatives for generalized solutions to be admissible if we know, for example, only that the second derivatives of these solutions belong t o L,(Q). We shall show that we still need to require this of generalized solutions of Eqs. (2.4) no matter how smooth the functions aiJ(", u , p ) satisfying conditions (2.7) are. Let us take the function u = r ? where )i < I , in the sphere KR. A s we saw above, this function satisfies Eq. (2.19) and hence the equation (2.23)

since IVu12=F elliptic for

)i

U

and xl = 3 r2-4 Equation (2.23) is uniformly A

V - 2



where f ( t ) is an infinitely differentiable positive function of t 0 that is equal to t for t to> 0 and that satisfies the inequality f ( t ) t for all t 0. The function u =r* satisfies the equation

>

>

>

(2.24)

22

I NTRODUCTlON

in the sphere

because in that sphere,

Therefore, f( ( V u ( 2 ) = 1 V u 1 2 , and Eq. (2.24) coincides with (2.23). Inequalities (2.7) for ail a r e satisfied, so that (2.24) i s uniformly elliptic. On the other hand, the function u = RX,for 1

t o u-2

R=(p)

'


0. This is caused by the fact that a ( x . u. p ) increases faster than does l p i 2 a s IpI +m. Consequently, such an increase in a is inadmissible, a fact expressed by inequality (2.25). Let u s consider the question of generalized solutions of quasilinear equations (2.3) whose principal part is a divergence under the assumption that these equations a r e uniformly elliptic. First, let u s suppose that the a i ( x , u , p ) and a ( x , u , p ) do not have unbounded singularities for finite values of x , u , and p . (At the end of this section, in Section 7 of Chapter 4, and in Chapter 9, we shall consider the general case.) Equation (2.3) can be replaced with the identity

J [a, (X'

u , ux) TXf - a ( x . u . ux) li] nx = 0.

(2.27)

P

which contains only the first derivatives of u. We begin by investigating bounded generalized solutions, that is, those for which essential max 1111 < CL The basic t e r m in (2.27) is a, ( x , u , ux) qx. The q u a d r s i c form

'%(a p p , corresponding to it determining dpi i

the type of equations increases like ( p I ma s IpI -00. Therefore, it is natural t o consider those bounded generalized solutions u the derivatives uxf of which belongtoL,(Q), that is, that belong to Wf,(Q).

We denote this class of functions by 9l. The integral

24

INTRODUCTION

is defined for arbitrary u and 7 in 92. This same class is the class of admissible functions among which we seek a solution of the variational problem for the functional

when we know that F ( x , integral

u. p )

increases a s

/PI"'

a s p +w

in (2.27) to be defined also for arbitrary functions

is necessary to require that the inequality

u

. For the

and q in B, it

l a ( x . fJ. P ) I < P ~ l ~ I ) ( l + I P I ) m

(2.28)

be satisfied for arbitrary p . Tf we write (2.3) in expanded form, we see that (2.28) is a consequence of (2.25). However, (2.25) requires something additional. Specifically, for (2.3) it leads to the condition that the o r d e r s of growth of the functions

with respect to ipI must not be greater than m ; that is,

In Chapter 4, it will be shown that conditions (2.6) and (2.29), which express the uniform ellipticity of (2.3) and the necessary consistency in o r d e r s of growth with respect to p of the functions a f ( x . u . p ) , n ( x . u , p ) and their derivatives, are sufficient for u s t o obtain all the necessary a priori bounds of the norms I u I , , . of the solutions of Eqs. (2.3) and to use them to prove the classical solvability "in the large" of boundary-value problems for (2.3). In addition, generalized solutions of (2.3) in % a r e investigated. In particular, it is shown that they satisfy a H'dlder condition if inequality (2.28) and the inequalities

A D M I S S I B L E E X T E N S I O N S OF T H E C O N C E P T

25

a r e satisfied for them. These last inequalities are weaker than (2.6). The theorem on the uniqueness of the Dirichlet problem "in the small" is valid for them if, in addition to (2.6) and (2.29), the corresponding conditions on the derivatives of a ( x . u , p ) with respect to u and p are satisfied, that i s , if

These examples and our disFussions emphasize the necessity of restrictions of the type (2.29 ) or (2.29) (assuming that (2.6) is satisfied) if we wish to consider generalized solutions in 94'. However, if we wish to study a broader class of generalized solutions of equations (2.3), namely, solutions in W:,, (9) n L,(Q), where q

> z rnn - ~ ,

with, in general, unbounded essential max J u I , we need to

impose conditions other than (2.29), which are more stringent, on the behavior of the functions u , ( x . u . .p) and a ( % , u , p ) for large values of u a n d ( p ( . To clarify these requirements, let u s consider the following example. First, let u s again take the function u = rk and Eq. (2.26). If )i is negative, then E will be negative also, that is, condition (2.28) will be satisfied [for (2.26) with m = 2 ] . The solution itself belongs to

Wi ( K R ) (and, in fact, to W h (KR)withm < &)if n > 2 and A i s small. Nonetheless, the theorem o r uniqueness "in the small'' is violated. From this it i s clear that for the uniqueness of generalized solutions in W f , (8)for (2.3) when conditions (2.6) are satisfied, we need

to make a more stringent requirement than (2.28) on the function a ( x . u. p).

In the example of the function u = r - A , equations AU = u ( x , U . u ~ ) , let us find bounds for la(x.

110

a ( x , u, p)

for n

>2

(2.31)

in the form

P)I S c C p ( x ) ( l + Jul")(l '9 ( x ) E L, (Qh

where A > 0, and the

+ lp12-e).

(2.32)

that a r e necessary for generalized solutions in W: ( 8 )n L,(Q), where 2n

to be admissible for Eqs. (2.31). [ F o r (2.31), m = 2 and 2n /( n - 2 ) is the limiting exponent of embeddability of W i in L .I The constants s, a , and E are to be determined. Assuming that tke function u = r - A , for )i > 0, is q-summable, where q 2 n / ( n - 2 ) , let

4>/n:2,

>

26

INTRODUCTION

u s regard it as a solution of equations of the form (2.31) with different a ( x , u, p)=cr-Bo(uIpIIplpa. Those exponents B, thatarethenobtained must be excluded. The function u =r-l satisfies the condition (2.33)

where c1 =cA-”,

c=-A(n-A-2),

po+kpl+(A+

1)P2=A+2.

Let us assume p2 > 0 and Po, PI > 0. The greatest exponent p2 is ob*+2 1 =1 + Since tained when Po = (3) = 0. It is equal to p2=

-.*+

1

u

E L,, we have A
I +L. We eliminate this n+q

example, requiring that B2
0, setting 6,=0. The exponent

The greatest lower bound

Po

for 0 < A < n/q is equal to

To eliminate the function r - * , we require that

Let us now choose B2 > 0 and Po > O that satisfy inequalities (2.34) and (2.35) respectively. Then,

27

THE B A S I C RESULTS

Consequently, to conditions (2.34) and (2.35) we need to add the restriction on 8,:

til < 2-P*-PBo

4+1-ez=(2-Pz)(:+

q - 1 - 7 4P o

in order to exclude the solution +of Eq. (2.33) with A in (0, n / q ) . Thus, this example shows that, in inequality (2.32), we need to assume that (2.36)

that the function y ( x ) is s-summable over Q , where

E

and that the exponent

+4

(E

- 1)

for

c
2 of the first boundary-value problem. Also expounded a r e the analogous results of Miranda and Fiorenza in connection with other boundary-value problems. Primarily, they consist in the following: If the coefficients and the inhomogeneous t e r m in Eq. (2.2) are smooth functions belonging to C l ,a, where 1 >/ 0, then the boundary-value problems for them (with accuracy up to the spectrum) are solvable in the classical sense, more precisely, in the space C1+2,2.These remarkable results are final results that cannot be extended by future discoveries. Until very recently, they were unique from the point of view of the nature of the dependence of the differentiability properties of the solutions on the differentiability properties of the coefficients of the equations. However, they provide nothing for equations with coefficients or inhomogeneous t e r m s that a r e not smooth functions, that is, that do not belong to C l ,=, for 1 0. For such equations, we are lacking both the results themselves and the methods of proving them. Seeking their solution within the classical framework is futile: Such equations can fail to have twice continuously differentiable solutions. The greater part of Chapter 3, Sections 4-17, is devoted to a study of linear equations with nonsmooth and, in general, unbounded coefficients. A s characteristics of the properties of the coefficients and the inhomogeneous t e r m s in the equations, we have chosen the question as to whether they and their possible generalized derivatives belong to the spaces L , (8). Regarding the solutions themselves of such equations, we show that they belong to the spaces W: (for l = l , 2 ) , C I , o , and C l ,a (for 1 0). A s the examples in the preceding section show, the greater number of these results cannot be improved ( i n t e r m s of the spaces

>

>

THE BASIC R E S U L T S

29

that we have chosen). Of course, this does not exclude the existence of other relationships, expressed in t e r m s of other functional spaces, between the properties of the coefficients and the inhomogeneous t e r m s in the equations and their soluions. In Sections 4-6, we study the solvability of boundary;value problems for equations in divergence form in the space W 2 . The results a r e a generalization of the results obtained in 1950 by Friedrichs ([9] inter a l k ) , Mikhlin [94], Vishik ([69] inter a l k ) , and others. The determination of the solutions of boundary-value problems with the aid of integral identities (which vary in form depending on the functional space to which these solutions belong) and the procedure for finding them (expounded in Sections 4-6), were first presented by one of the present authors. The central idea of inverting the principal part of the equation with the aid of Riesz' theorem on linear functionals belongs to Friedrichs [9]. In Sections 7-11, we study the solvability of boundary-value problems in the space W,". In Sections 7-10 and 12, we expound and somewhat generalize the results of the articles [lo-12, 551 by Lady zhenskaya. Sections 13-15 contain recent results. They are devoted to a study of boundedness and continuity in the sense of Holder of generalized solutions of linear equations (2.1) and the derivatives of these solutions. The results of these sections constitute a special case of analogous facts established by the authors of the book regarding the solutions of parabolic equations [13, 141 and the solutions of quasilinear elliptic equations (cf. [4-61). They a r e expounded here independently with the appropriate simplifications. Propositions similar to the theorems in Sections 13 and 14 were also established by Stampacchia [17, 471 and Morrey [16]. The first results of this type were obtained by de Giorgi [7]and Nash 181. In Section 16,we investigate the solutions of diffraction problems. In the last two sections, namely, 17 and 18, we take up the case of two independent variables. This special case occupies a special place in the study of many-dimensional problems. A number of authors, especially, S. N. Bernstein, E. Hopf, Morrey, L. Bers, L. Nirenberg, and 1. N. Vekua, have found various bounds and representations of the solutions that made it possible during the middle fifties to study rather completely linear equations with discontinuous bounded coefficients. B e r s and Nirenberg did this with the aid of generalized analytic functions and quasiconformal transformations. We prove and slightly generalize their results with our own method, a method that made it possible to solve the problem in the general n-dimensional case. We shall point out certain directions taken in the study of linear equations that seem interestingto us and that a r e immediately

30

INTRODUCTION

connected with the questions that we are discussing. F i r s t of all, this is a study of equations of the form (2.2) with n >/ 3 with no assumptions made regarding the continuity or differentiability of the coefficients ail. The results expounded h e r e have to do with equations of the form (2.1). Tn the case of nondifferentiable ail, such equations are not equivalent to equations of the form (2.2). For such equations (2.2), we still do not know in what functional space the Dirichlet problem is solvable in the sense of Fredholm. Shall we encounter here the unfortunate circumstance that this class depends on the coefficients of the equation? At one time, it appeared that the self-adjoint extension in the sense of Friedrichs depends on the coefficients afj of the highest derivatives even when these a r e differentiable. Such misgivings, however, turned out to be wrong (cf. [ l o ] and Sections 7-11 of Chapter3). The example (2.19) of Chapter 1 and the r e m a r k s made following it regarding the impossibility of inequalities of the form (2.20) and (2.21) show that, for (2.2) with discontinuous a , j , the Dirichlet problem is not solvable in W i ( Q ) with an arbitrary inhomogeneous t e r m f(x) in L,(Q) if n > 2. However, it may be that a highly esthetic proposition of the following type is true: The Dirichlet problem for these is solvable in W i (Q) for arbitrary f i n L, (8)provided h = 0 is not a point of the spectrum. At the present time, we know for such equations only the results of Aleksandrov [3] regarding the theorems on the uniqueness of their solutions in the space W z ( Q ) and the results of Cordes [88] dealing with the case of small dispersion of the eigenvalues of the form aijE,Sj. The second direction in the study of linear elliptic equations was the obtaining of results analogous to the results of Sections 7-15 of Chapter 3 for linear elliptic equations and systems of high o r d e r s with principal part in divergence form with nonsmooth coefficients. When the coefficients are sufficiently smooth, Schauder’s “gluing” method (consisting in the reduction of all difficulties to equations with constant coefficients) made it possible to generalize the results on the solvabilit of boundary-value problems in the HGlder classes Ci, (G) and W;Y, (Q) (with p > 1 ) to the case of arbitrary mth-order elliptic systems with nz 41. But what will be the case when this “gluing” is impossible because of insufficient smoothness of the coefficients? To what functional spaces will the solutions of the various boundaryvalue problems belong? For general systems, we have results analogous to the results of Sections 4-6 of Chapter 3. Results of the type 7-15 are proven only for the systems in Chapter 7. A third direction is in the study of boundary-value problems in unbounded regions and the search for esthetic forms of posing the boundary conditions at infinity in the case in which we have a continuous spectrum. The most complete results in this direction have been obtained up to now only for equations of the form

T H E BASIC R E S U L T S

-Au + q

31

( x ) u = f in regions the complement of which with respect to the entire space is a closed bounded set. Finally, we note that the analysis that we make in the present book of second-order equations with discontinuous coefficients was directed toward a clarification of the conditions that need to be imposed on the coefficients and the inhomogeneous t e r m s in the equations (where possible, not only sufficient but also necessary conditions!) under satisfaction of which for the entire class Of such equations the boundary-value problems will be solvable in a completely determined functional space independent of the particular coefficients o r inhomogeneous t e r m s in these equations. However, such an analysis does not answer, for example, the following question: Suppose that we know only that the function q ( x ) in the operator Lu == - A u + q ( X I II is locally summable. What properties does L possess as an operator in the Hilbert space L, (E,)? Because of the ‘‘poor differentiability properties” of q ( x ) , L is not defined in a natural manner even on smooth finite functions. Clearly, both the original domain of definition of L and the domain of definition of its self-adjoint extensions will depend on q ( x ) . We have excluded such cases from our considerations since we had quite a different purpose in mind, namely, that of finding those operators L for which all the characteristics a r e independent of the particular form of the coefficients generating them. Chapter 4 is devoted to so-called quasilinear equations with principal part in divergence form. ln that chapter, we make a complete analysis of the properties of the solutions of such equations, beginning with generalized solutions and ending with classical ones. In the beginning, the investigations are made within the domain of definition of the solution without any assumptions regarding the behavior of the solution close to the boundary. Then, we study the behavior of the solution of the first boundary-value problem near the boundary. h Chapter 10, we do the same for other, more complicated, boundary-value problems. The class of quasilinear equations with principal part in divergence form possesses a distinctive feature of its own. For this class, it is possible to construct a theory of generalized solutions having only first-order derivatives-something that cannot be done for other quasilinear equations not belonging to it. This class of equations is rather broad. lt includes Euler’s equations for variational problems and various nonlinear equations encountered in mechanics. The results of Chapters 4, 9, and 10 follow a definite pattern. A s the examples of the preceding section show, all the basic assumptions a r e due to the essential nature of the matter. The results of Chapter 4 were proven by the authors of the book and published for the most part in the articles [5, 6, 18-20], The results of Chapter 1 0 are being published in detail now for the first

32

INTRODUCTION

time. A resume of them is given in [62]. These articles were preceded by a rather large number of articles devoted to quasilinear equations though, in all of them, in the case of n > 2 , it was assumed that the equations were, in some sense or other, close to linear. For the case of only two independent variables, much m o r e was accomplished (see the articles by Bernstein [Sl], Hopf [58], Schauder and Leray [27], B e r s and Nirenberg [28-301, et ul.). Even for this case, the propositions expounded here are new. The reader may acquaint himself with what was doneprior to the works which constitute the basis of the present book by consulting the extremely valuable book by Miranda, which summarizes the material written during the last fifty y e a r s on elliptic equations and the survey articles by Sigalov, V. I. Plotnikov, and the authors of the book [34, 6, 32, 151, which are devoted to the variational problem and to elliptic equations. The investigations of Chapters 4, 9, and 5 can be extended in various directions. The one that seems most interestingto us is the study of higher-order equations and systems with principal part in divergence form in the sense of Chapter 4. The investigations of Chapters 7 and 8 are carried out in just this direction. Another development of Chapters 3, 4, 9, and 10 can be carried out along the line of replacing the spaces L,and WfIlthat we have chosen with others. (The articles 198, 1011 and others a r e in this direction.) These classes a r e naturally connected with equations but they are not the only ones possible. Specifically, it is their choice that conditioned our requirements that the functions generating the equation grow like a power with respect tolVul. A third direction is the relinquishment of the condition of uniform ellipticity and agreement of the o r d e r s of growth (two such cases are considered in Section 8 of Chapter 4 and Section 3 of Chapter 6). However, if we wish to proceed in this direction, it is worthwhile to construct, first of all, a reasonably complete list of examples, one that would outline the situations that we should expect and the actual facts, not just those that a r e obtained f r o m the method chosen by the investigator. The articles [96-981 are carried out along these lines. Chapter 5 is devoted to a regular variational problem for functionals of the form J(u)=

J F (x. u . u x ) d x ,

(3.1)

0

which served as the starting point for the investigations on which the present book is based. Regularity implies ellipticity of the Euler equation

THE B A S I C RESULTS

d

33

(3.2)

corresponding to (3.1). The basic direction in the study of this problem (just as in the study of nonlinear elliptic equations in general) is related to the 19th and 20th problems of Hilbert (cf. [64]). In the 19th problem, it is asserted that all solutions of Eq. (3.2) are analytic with respect to xif F ( x . u , p ) is an analytic function of its arguments. In the 20th problem, it is asserted that the variational problem on the determination of the minimum of the functional (3.1) that is semibounded from below under the condition that always has a solution if we seek it in a sufficiently broad class of functions. The formulations of both problems have required greater precision in certain respects: In the first place, it was necessary to understand just what was meant by ccal199 solutions of Eq. (3.2) in the 19th problem and just what was meant by “a sufficiently broad c l a s s of functions” in the 20th problem. In the second place, it was necessary to clarify whether the function F ( x , u . p ) did not need to possess, in addition to a certain smoothness and convexity with respect to p [this last being equivalent to ellipticity of Eq. (3.2)], certain other properties in order for the assertions of these problems to be valid. Bernstein constructed examples (cf. [ 811) showing that supplementary restrictions on I-‘ were indeed necessary. He formulated them in the form of a series of conditions on the behavior of F ( x . u , p ) and its partial derivatives a s

Roughly speaking, they reduce to the fact that F behaves, for large values of I p I, like a polynomial in x , u, and p of degree m > 1 in p . We shall refer to these requirements together with the requirements for a certain degree of smoothness and semiboundedness below and convexity of F with respect to p as the “natural” requirements. A l l our subsequent investigations are directed toward a proof that the natural requirements are also sufficient. Along towards 1940, the efforts of a number of mathematicians resulted in establishing that the assertion of the 20th problem regarding the existence of a function that provides the minimum of the functional (3.1) is valid if we take a s the set 9t of admissible

34

I NTRODUCT ION

functions, all functions satisfying condition (3.3) that, together with their generalized first derivatives a r e 1 -summable, where 1 is such that the integral (3.1) for them is finite. (The most general results in this direction were obtained by Morrey.) We shall refer t o functions of the class W that provide the minimum (or, more generally, an extremum) for J(u) a s generalized solutions of the problem (3.1), (3.3). However, such an extension in the concept of a solution of the problem (3.1), (3.3) necessitated analysis of all s o r t s and a clarification of the question a s to whether the basic properties of an arbitrary correctly posed boundary-value problem and, especially, the theorem on the uniqueness of the solution of the problem (3.1), (3.3) “in the large” o r even “in the small” might not be lost. Also, mathematicians needed to investigate whether the differentiability properties (that is, the smoothness properties) of these solutions a r e improved a s the differentiability properties of the function F a r e improved and, in particular, whether they a r e analytic functions whenever F is analytic. In other words, it was necessary to prove (or refute) that the words “all solutions” in the 19th problem of Hilbert should be understood a s the solutions in just this very class. For the case of two space variables, Bernstein showed that all three-times-continuously differentiable solutions of Eqs. (3.2) a r e analytic for analytic F . During the forties, Hopf, Schauder, and Morrey succeeded in weakening the smoothness conditions to the extent of requiring boundedness of the first derivatives in the case n = 2 and Hdderness of these derivatives in the case n > 2. For such solutions, the theorem on uniqueness “in the small” was also established a s well a s the fact that their differentiability properties depend only on the differentiability properties of F. However, the existence theorem, of which we spoke above, yielded, for the general case, only generalized solutions, functions with considerably poorer properties. Thus, between the existence and uniqueness theorems, between the classes of functions for which mathematicians succeeded in proving the 19th and 20th problems of Hilbert, there was a wide gap, to the filling of which the greater part of the study on the variational problem (3.1), (3.3) was devoted. This work proceeded basically along the line of imposing various supplementary restrictions on F under which it was possible either to prove the existence theorem immediately in the class of sufficiently broad functions (as was done by Bernstein, Tonelli, Hopf, and others) or to improve gradually the differentiability properties of the generalized solutions of the problem (Morrey, Sigalov, Plotnikov, and others). For the general case (with n I-2 ) , Sigalov proved the existence of continuous generalized solutions. For the case of ri = ni = 2, Morrey carried out all the investigations

THE B A S I C R E S U L T S

35

of generalized functions that were desired. A l l these results are for two-dimensional problems ( n = 2). Study of the many-dimensional case ( 12 > 2 ) proceeded quite slowly. It required development of new methods. The first conclusive result regarding the classical solvability of the problem (3.1), (3.3) for n > 2 was de Giorgi’s result [7] in 1957. An analogous result could be obtained from Nash’ work [8] in the same year inconnection with linear parabolic equations. H e dealt with the simplest case, in which F ( x , u . p)is independent of x and u and has quadratic growth with respect to IpI. For this case, the problem has a unique solution “in the large.” Not long afterward, the authors of the present book gave the solution of the problem (3.1), (3.3) inthegeneral case [5, 6, 18-20]. Specifically, it was shown that, for arbitrary n >/ 2 and ni > 1 , only under natural assumptions regarding the function F is an arbitrary generalized solution u of the variational problem unique in the class 131 “in the small” and only then are its differentiability properties completely determined by the differentiability properties of F. Specifically, if F as a function of its arguments belongs to C,,m. then the solution u also belongs to C,,“. Tn particular, it is analytic when F is analytic. This result filled the gap mentioned above and led to the soltuion of the 19th and 20th Hilbert problems for all the desired class of functionals of the form (3.1). Two methods of proving these propositions were given. The first method was expounded in 1959 in the articles [5, 61, the second in the articles [18-201. Here, we shall give another, more esthetic method of proof. In [5, 6, 18-20], the investigation was primarily of generalized solutions that a r e bounded in absolute value. For just such solutions, natural restrictions on F were formulated, dealing with the behavior of F for large values of 1 p I (though not I ri I). Tf we drop the assumption of boundedness of the generalized functions, that is, if we broaden their class, it becomes necessary to strengthen the restrictions on F. This last can be done in various ways. We have chosen one of them and have constructed examples (cf. [14, 15, 331) showing what values the constants (exponents) appearing in the inequalities characterizing the behavior of F for large values of I p I and 1 u 1 must not exceed. The conditions shown by these examples t o be necessary for admissibility of the unbounded generalized functions in W l I(9)that are q-summable where q is any fixed number, are also sufficient. We did all this not only for Euler’s equations but for the entire class of equations in Chapter 4. For solutions of problems on the minimum of the functional J ( u ) , various cases were found in the article [6] and the earlier works by Sigalov [34], Morrey [36], and others in which it is possible t o give actual bounds on the maxima of the absolute

36

INTRODUCTION

values. However, it is clear that this can be done only under certain conditions. We note that, at the present time, one other method of investigating the differentiability properties of generalized solutions of the variational problems has been given (Morrey [23, 371). This method is not applicable for the entire set of values n > 2 , ni > 1 , and it is applicable only under restrictions somewhat more stringent than the natural one. An investigation of the following questions regarding the variational problems s e e m s interesting and pertinent to us: (1) The dependence of the differentiability properties of solutions of the variational problems given in parametric form on the differentiability properties of the integrand. This question has been studied only for certain particular types of integrand for 11 = 2. (2) The exact formulation and proof of Hilbert’s problems for functions F that depend on certain unknown functions and contain derivatives of order higher than the first. (3) Complete investigation of the question of existence of stationary points of functionals of the form (3.1). (4) Investigations analogous to the investigations of Chapter 5 for cases of nonpower growth of F with respect to 1 p 1. The fi rst of these seems especially interesting to us. In Chapter 6, we shall look at quasilinear equations of the gene r a l form. For such equations, a theory of classical and generalized solutions is constructed. However, it turned out, that for these equations, the class of admissible generalized solutions is considerably more restricted than the class of admissable generalized solutions for equations of the divergence type considered in Chapter 4. This restriction is due not only to the natural requirement that they have generalized second derivatives (since without this requirement it would be impossible in the general case to give a meaning to the statement that a function u satisfies the equation) but also to the additional requirements on the behavior of the first and second derivatives. The example that we have constructed in Section 2 of the present chapter shows that, if we make no requirements on the second derivatives other than their squaresummability, then, for 11 > 2, it is necessary to assume that the first derivatives are bounded (since, otherwise, the uniqueness theorem would be violated even for regions of arbitrarily small dimensions). In that chapter, we shall prove that these restrictions on the class of admissible solutions a r e also sufficient. The entire theory is reasonably constructed for such generalized solutions: Boundary-value problems have no more than one such solution if the region is sufficiently small, and the differentiability properties of these solutions a r e determined only by the differentiability proper t i e s of the functions generating the equation.

37

T H E BASIC R E S U L T S

In Chapter 6, we have confined ourselves to a consideration of the first boundary-value problem. The basic results of this chapter a r e published in [67, 38, 391. To investigate the general quasilinear equations, it was necessary to broaden still further the functional classes %, which we introduced in the study of the variational problem and of equations of divergence form since the derivatives of their solutions (to say nothing of the solutions themselves) do not, in general, satisfy the inequalities listed as the basis of the definition of the classes 23,. However, it turned out that certain functions constructed from derivatives satisfy such inequalities. From these inequalities, we can deduce that each of the derivatives individually satisfies a H’dlder condition. Chapters 7 and 8 are devoted to linear and quasilinear secondorder elliptic systems. They constitute only a portion of the entire set of elliptic systems. This class of systems was singled out according to the following principle: For an arbitrary solution q ( x ) , . . . vN ( x ) of the system obtained from an arbitrary system in this class by discarding all the t e r m s other than those containing the highest-order derivatives, the function

.

or, what amounts to the same thing, an arbitrary positive increasing function ‘p (t)of t = z ( x ) satisfies a “maximum principle;” that is, its smallest and greatest values for an arbitrary closed region occur on the boundary of that region. It is well known that this property is characteristic of a single second-order elliptic equation, and we have made it the criterion for defining a specific class of elliptic systems. Systems in this class have the special property that all the equations constituting them have the same principal parts. It turned out that, for this class of systems, all the basic propositions that we have proven in Chapters 3 and 4 for a single second-order equation remain valid. However, proofs of these propositions are based on more complicated analytical facts (the Hzlderness of functions belonging to the classes 232). A l l the restrictions under which they a r e proven a r e brought about by the nature of the problem, as the examples listed in Section 2 show. With regard to future study of systems, it is desirable to understand (by constructing appropriate examples) for what broader classes of elliptic systems any particular a priori inequalities and theorems proven in the present book remain valid. We have studied rather completely the question of the existence of generalized solutions to boundary-value problems for quasilinear

38

INTRODUCTION

equations and systems with divergence principal part that possess generalized derivatives of order equal to one-half the order of the equation (or system) or that exceeded it by 1. This was done recently in the articles by T. B. Solomyak, M. 1. Vishik, Browder, and others ([99-[ 1021 inter alia). The results obtained a r e neutral generalizations of the results expounded in Section 5 of Chapter 3 and Section 2 of Chapter 5 on the existence of generalized solutions in Wf, (52) for second-order equations. An especially simple proof of the existence of such solutions is given in a recent work by Browder [ 231. Questions regarding the “admissibilityy7 of such solutions, that is, regarding theorems on uniqueness of such solutions “in the smally’ and questions regarding the validity of the 19th hypothesis of Hilbert for them, that is, regarding the improvement of the differentiability properties with improvement in the differentiability properties of the functions appearing in the equations, a r e yet to be answered. In Chapter 9, we give another method of deriving bounds for the Hglder constants for the solutions and their derivatives of all the equations hitherto considered. Here, we combine both difficulties: The possible singularities with respect to x and the maximum possible nonlinearities with respect to u and p . (This combination is admissible within the framework of the basic method. That it was not done is due to purely methodological considerations.) The corresponding bounds obtained in Chapter 3 for linear equations and in Chapters 4-6 for quasilinear equations a r e two particular (and limiting) cases derived here. A second method which we present is somewhat simpler and “more usualyythan the basic one. (This is especially clear in the caseof parabolic equations, to which it is also applicable.) A l l analytic propositions on which it is based a r e contained in the first method. The work is made easier by repeatedly returning to the equation: In the first method, we derive from the equation certain inequalities (*) (which appear in the definition of the classes 8,n) and we prove all subsequent propositions regarding functions that obey only these inequalities. In the second method, we return several times to the equation and derive from it inequalities similar to the equations (*) both for the solutions themselves and for certain convex functions of them. In Sections 5-7 of Chapter 9, we expound certain other devices for obtaining bounds for the Hglder constants for the solutions of different classes of linear and quasilinear equations.

2 Auxiliary Propositions

In this chapter, we shall prove various propositions on arbitrary functions that satisfy certain inequalities containing integrals of these functions and their derivatives. Most of them a r e of independent value although, for the present book, they a r e used only a s part of the auxiliary analytic apparatus. Some of them were known earlier (the lemmas and theorems in Sections 1-4). Most of them were discovered and proven in recent y e a r s in connection with the investigation of the series of questions that we now answer (see the assertions in Sections 5-8). 1. S O M E SIMPLE INEQUALITIES

We shall frequently have occasion to use certain well-known algebraic and functional inequalities. Of the algebraic inequalities, we use the following: (1)Cauchy’s inequality

...

...,

which is valid for an arbitrary nonnegative quadratic form aiIEiE, with a i / = a l l and arbitrary numbers El, , En and ql, qn. (2) Cauchy’s inequality u6

which is valid for arbitrary

< - u2 + E

>0

1

62,

and arbitrary a and b .

39

40

AUXl LlARY PROPOSITIONS

(3) Young’s inequality, which is more general than the preceding:

which is valid for arbitrary > 0 and p > 1. We write it in the following form, which is more convenient for our purposes:

where E is an arbitrary positive number and a E ( 0 , 1). We have the following function inequalities:

the first of which is valid for norms of an arbitrary Banach space, and the second is valid for the scalar product and norms of an arbitrary Hilbert space. We shall also use the following inequalities: (4) Holder’s inequality

where

1 1 -+- 1. 4 4’-

and, more generally,

(5) Cauchy’s inequality

These are valid for arbitrary measurable functionsu ( x ) , v ( x ) , ut (XI, and v i ( x ) defined i n 9 and having finite norms on the right.

F U N C T I O N S IN T H E CLASS W A t f l )

41

2. FUNCTIONS IN THE CLASS Wf,(Q) We spoke above about functional spaces L, (Q) and W t (52). The space L, ( Q ) , where m >,l, is a Banach space with norm

Its elements are classes of equivalent measurable functions u ( x ) defined on an open connected set P of the Euclidean space x =( x , , ,x,J that are m-summable over that space. (Equivalent functions a r e functions that differ 'only on a set of n-dimensional measure 0.) The region P will be assumed bounded throughout the entire book. We know, for example, that sets of infinitely differentiable functions that are finite in !2 are dense in Lm(Q). The class of functions Wf,(Q), for m >l, can be defined in two ways (cf. [l, 21). Specifically, we can define it either as the set of all elements L,,, (51) possessing generalized derivatives of the first 1 orders in Q with respect to x i , for 1 = 1 , , n, that are m-summable over Q, o r we can define it as the set of all functions that can be obtained in the form of strong limits in the norm

...

...

of sequences of infinitely differentiable functions in G . Fo r arbitrary regions, the first definition probably leads to a broader class of functions than does the second. For regions Q with boundaries possessing some regularity, for example, for the regions with piecewise-smooth boundaries that we considered, both definitions yield the same class of functions. We obtain the Banach space Wf,( Q ) , if we define the norm in it by means of Eq. (2.2). Strictly speaking, its elements are not the functions u ( x ) but classes of equivalent functions on P , and when we make any assertion regarding an element in Wf, (Q), we should keep in mind that it is this particular representative that possesses the property in question. Thus, we shall use the formula for integration by parts

where n is the outer normal to the boundary S f o r functions u in W!,,(Q) and Y in W i t (Q) where l / m + l / m ' = 1. Formula (2.3) cannot

42

AUXILIARY PROPOSITIONS

be valid for all functions u and v in Wf,, (M) and W!,.(2). However, it is valid for special representatives of elements u and w in the spaces W L (Q) and W!nt(Q). These representatives of II ( x ) possess the property that they are defined as the functions in L,(EIl-J on an arbitrary smooth ( n - ])-dimensional manifold belonging to = Q u S, and they vary continuously in the norm of L, (E,f-l under a smooth deformation of E,,-,. (Analogous r e m a r k s hold for o ( x ) . ) They are obtained for u t x ) , for example, as the limits of a sequence of smooth functions u,, (x) that converge on the norm W!!,(Q) to u ( x ) since we know that the sequence u,,(w) then converges in L, on an arbitrary ( n - 1)-dimensional smooth $ , I - l c 53 uniformly with respect to the displacement En-l in s. Therefore, the assertions that a function ~ ( x )in Wf,(Q) assumes the values cp(x)E L,(S) on the boundary S and that it converges to these values in the norm L,(S) are meaningful. For more detail on this point, consult the specialized text [ l , 21 and articles [41-451. For (2.3) to be valid, we also need some s o r t of regularity of the boundary S. Clarification of the properties of S that are needed for this was the subject of special investigations. Here, we shall not give the results of these investigations and we shall use only the fact, long known, that piecewise smoothness of S is a sufficient condition for this. In addition to the spaces Wf,(Q), we shall deal with the spaces W L (a) and W;,,(Q) consisting of thoseelements of W!n(9) and Wi(Q), respectively, that vanish on S in the sense just described for the elements of wf,, (~1." The subspaces W k (Q) and Wi, (52) can also be defined i,n another way. F o r regions with an arbitrary boundary, the space Wf,(Q) can best be defined as the closure in the norm W:,(Q) of the set of all functions that are smooth and of compact support in Q. Obviously, if the function u ( x ) belongs to such a closure, then in formula (2.3) for this function and for w ( x ) in W k , (Q), the last integral, that is, the boundary integral, will be missing. The space W i , o ( Q )is defined as the closure in the norm W t ( 8 ) of all twice continuously differentiabl? functions on that vanish on S. Finally, Wk (51) can be defined as the set of all elements W!l (a) f o r which formula (2.3) is of the form

a

for arbitrary smooth v. (Then (2.4) will be valid for arbitrary v in WL,(Q).) F o r regions with piecewise-smooth boundary, all these definitions a r e equivalent. However, we note that if we wished

FUNCTIONS IN THE C U S S

43

WA(a)

to investigate only the first boundary-value probtem with homogeneous boundary condition, we could define Wj,(P) as the closure in W',(Q) of the set of smooth functions finite in 51 without any assumptions regarding the regularity of the boundary S. For the purpose of the present book, we have to know a number of facts regarding the properties of elements of the spaces W',(Q) and the relationships between convergence in the spaces W',(51) with different I , m, and n (where It is the dimension of the x-space which we assume in what follows to be 2 or greater). We shall refer to an operator that assigns to every function in the space W',(Q,) that same function a s an element of a broader space WLI(Q,,.),where P,tcQ,, or C1.,. (9,) a s the injection operator from the space W', (P,) into the space Wkt (Q,,)or Ctt, (QJ. We know sufficient conditions for such operators to be bounded or completely continuous. Here, we give one of these p-ropositions, specifically, Theorem 2.1. Let P, be a region in n-dimensional Euclidean space and let S , be a plane 1-dimensional portion of it belonging to Q,+G,,. Then the injection o p e r a t o r p m W',(Q,) into L,(S,) is and qQa, and it is completely bounded if n > m l , r > n - - 1 , continuous if q < a - m l For n < ml it 2s completely continuous fm arbitrary finite 4. For n ml ,ifcinjection operator fiom W!,,(Q,) into C,,R(Si,) is bounded f o r a