Nonlinear Differential Equations [1st Edition] 9781483278377

Table of contents :
Content:
Studies in Applied MechanicsPage 2
Front MatterPage 3
Copyright pagePage 4
PrefacePages 7-10S. FUČÍK, A. KUFNER
List of SymbolsPages 11-14
Chapter I - Some Examples to Begin withPages 15-32
Chapter II - IntroductionPages 33-72
Chapter III - The Weak Solution of a Boundary Value ProblemPages 73-168
Chapter IV - The Variational MethodPages 169-238
Chapter V - The Topological MethodPages 239-273
Chapter VI - Noncoercive ProblemsPages 274-311
Chapter VII - Variational InequalitiesPages 312-352
ReferencesPages 353-356
IndexPages 357-359

Citation preview

STUDIES I N APPLIED M E C H A N I C S

1.

Mechanics and Strength of Materials (Skalmierski)

2.

Nonlinear Differential Equations (Fucik and Kufner)

3.

Mathematical Theory of Elastic and Elastico-Plastic Bodies. An Introduction (Necas and Hlavacek)

STUDIES IN APPLIED MECHANICS 2

NONLINEAR DIFFERENTIAL EQUATIONS SVATOPLUK FUClK Department of Mathematics, Faculty of Mathematics and Physics, Charles University, Prague # ALOIS KUFNER Mathematical Institute of the Czechoslovak Academy of Sciences, Prague

ELSEVIER SCIENTIFIC PUBLISHING COMPANY Amsterdam - Oxford - New York 1980

Published in co-edition with SNTL Publishers of Technical Literature, Prague Distribution of this book is being handled by the following publishers: for the USA and Canada Elsevier/North-Holland, Inc. 52 Vanderbilt Avenue New York, 10017 for the East European Countries, China, Northern Korea, Cuba, Vietnam and Mongolia SNTL Publishers of Technical Literature, Prague for all remaining areas ELSEVIER SCIENTIFIC PUBLISHING COMPANY 335 Jan van Galenstraat P. O. Box 211, 1000 AE Amsterdam, The Netherlands

Library of Congress Cataloging in Publication Data

Fu£ik, Svatopluk. Nonlinear differential equations. (Studies in applied mechanics ; 2) Bibliography: p. Includes index. 1. Differential equations, Nonlinear. I. Kufner, Alois, joint author. II. Title. III. Series. QA371.F86 515f.36 79-19877 ISBN O-W^-99771-7 Series ISBN 0-444-41758-3

Translation © 1980 by Dr. Michal Basch Copyright © 1980 RNDr. Svatopluk Fudik, CSc and Doc. RNDr. Alois Kufner, CSc All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, mechanical, photocopying, recording, or otherwise, without the prior written permision of the publisher Printed in Czechoslovakia

PREFACE This book owes its origins to our friend and respected colleague Prof. Dr. Karel Rektorys, DrSc. His book "Variational Methods in Mathematics, Science and Engineering" represents a highly successful attempt of presenting to a broad range of readers a textbook devoted to modern methods of solution of boundary value problems in linear partial differential equations; a textbook based upon deep results of functional analysis and yet intended not only for mathematicians but also (indeed principally) for engineers. We had the opportunity to acquaint ourselves with the book of Prof. Rektorys — which covers an immense amount of material, both theory and applications — in manuscript form; it inspired us to try to write a sort of loose "nonlinear continuation" of his "Variational Methods". Our book starts where "Variational Methods" ends. Concluding his book, Prof. K. Rektorys writes (p. 553): "Nonlinear problems have remained entirely aside in our book. Their investigation is based, in essentials, on the theory of the so-called monotone operators and is in the center of interest of outstanding mathematicians at present." Let us add that the investigation of boundary value problems in nonlinear differential equations is also of central interest to physicists, engineers and practi­ tioners in general; in Chapter I, we mention several nonlinear problems which have their origins in practical applications. The increased interest in the field of nonlinear problems among the "technical public" was partly due to the fact that mathematics had developed, in (nonlinear) functional analysis, a powerful tool for the solution of such problems. The vigorous expansion of the study (more precisely: mathematical study) of nonlinear problems dates from the sixties and is principally associated with the following names: G. Minty and F. E. Browder (U.S.A.), M. M. Vajnberg and M. A. Krasnoselskii (U.S.S.R.), J. Leray and J.-L. Lions (France). In connection with the development of the methods of solution of nonlinear differential equations, one must mention Prof. Dr. Jindfich Necas, DrSc, who contributed significantly to this development, in particular to its growth in Czechoslovakia, and trained and educated a number of collaborators among whom the present authors number themselves. The number of papers on nonlinear problems — theoretical papers as well as papers biased towards applications — published annually in journals is immense. The number of published books is relatively much smaller — one might mention the books by M. M. Vajnberg [89], [90], J.-L. Lions [58], [59], H. Gajewski, K. Groger, K. Zacharias [38], V. Barbu [4]. However, all these publications are of

8

a slightly different character and are aimed at a readership different from that envisaged for the present book. Bearing in mind the example set by Prof. Rektorys, our aim was to write a text which would appeal to a broad range of readers, to mathematicians as well as to engineers. This aim influenced the style of exposition employed. Modern methods of solving nonlinear differential equations require modern mathematical tools and devices. However, we did not consider it useful to collect all the applied, and thus in view of the principal aim of the book merely auxiliary, mathematical apparatus together into a separate chapter. We therefore try to explain all the necessary material from other mathematical disciplines (as far as possible) always at the point of its actual application. We believe that this will not leave readers with the impression that functional analysis plays only a subservient role in the theory of differential equations. On the contrary, we hope that the readers will become aware of the mutual interconnections of the various mathematical disciplines. Originally, we planned to relegate all the auxiliary apparatus to footnotes. It soon became apparent, however, that we had underestimated the importance of the "auxiliary" tools: at some points of the text, the footnotes would have outweighed the text proper. Therefore we adopted this approach mainly in the opening parts of the book when recalling the elements of functional analysis. In fact, we assume that the reader has a working knowledge of analysis up to the elements of the theory of the Lebesgue integral, whilst everything else should be found in the book, albeit without proofs but with adequate references to the literature. Proofs are not always given in full. The reason is that the declared aim of this book was the exposition of the principal ideas and methods, not an exhaustive survey and presentation of assertions in the greatest possible generality. Because of this, illus­ trations are given instead of proofs at some points, elsewhere the assertion is proved under simplifying conditions, and elsewhere again reference to the available literature is made. Uppermost in our thoughts is the promotion of a discipline which we con­ sider useful, not a monograph presenting the most general assertions of the theory of nonlinear equations. (Nevertheless, we indicate several possible generalizations in the remarks, with references to the literature.) As mentioned already, the book is meant for mathematicians as well as for engi­ neers. Since "one cannot serve two masters", both groups will have their objections to the book. The mathematician will miss detailed proofs — in this we were limited by the size of the book as well as by its accessibility to other groups of readers. On the other hand, the engineer will miss the discussion of specific equations of mathematical physics; here the authors lack the experience and close contact with applications which is so pleasingly apparent in the book of Prof. Rektorys. Nevertheless, we hope that this deficiency will be rectified by the publication of the book being prepared by J. Necas and I. Hlavacek [74] which is, in a sense, an "application" of the methods presented here. Similarly, we did not feel competent to embark on a detailed exposi-

9

tion of numerical methods. Indeed, these methods could serve as material for a com­ pletely new book (see, for example, J. Cea [8]). At least we have included the infor­ mative Section 28 for which the material was kindly prepared by RNDr. J. Haslinger, CSc, to whom we express our gratitude. A few words now concerning the contents of the book. Following Chapter I, which serves as "bait" in a sense, we deal — in Chapters II and III — with the question of what we understand, in essence, by the solution of a differential equation and by the boundary value problem in a differential equation. Two types of solutions are introduced: the classical solution and the w ak solution. From the point of view of this book, the concept of the classical solution is merely auxiliary. The weak solution is illustrated by means of a considerable number of examples. In this context, we direct the reader's attention to difficulties which arise with the definition of the weak solution. Since knowledge of the concept of weak solution is necessary for the understanding of the subsequent chapters, the exposition here is at a fairly easy pace. The main results of the book are presented in Chapters IV and V. There, two basic methods are given which make it possible to single out a class of differential equations for which there exists a weak solution of the boundary value problem. Socalled coercive elliptic problems form the class in question and it can be said that the questions of the existence of the weak solutions of these problems are settled, es­ sentially, even from the point of view of numerical methods. The material of Chapter VI differs substantially from the problems investigated in Chapters IV and V. So-called non-coercive problems are studied in more detail there and new, mostly as yet unpublished, results are presented. There is a multitude of open problems in this field which we submit to the reader here; the numerical processing of these problems is still in its infancy. Chapter VII is devoted to a brief introduction to the very interesting and important field of problems concerning the so-called variational inequalities whose main ap. plication is in mechanics. From this brief outline, it is clear that by far not all types of nonlinear differential equations are discussed. For instance, evolution equations (i.e., equations representing phenomena dependent on time) are not included in the book at all. At this point, we refer to the extensive monograph by O. Vejvoda et al. [92], to the books by J.-L. Lions which include an immense amount of material, as well as to the book by H. Gajewski, K. Groger and K. Zacharias mentioned above. On the other hand, we wish to stress that using the concept of weak solution, the difference between an ordinary and a partial differential equation, which is so strik­ ing in the classical approach, is suppressed. We have exploited this especially in the illustrative examples where it is possible to use clearer and simpler apparatus for ordinary equations. Now, brief instructions on how to read the book: The text is divided into sections numbered consecutively. The sections are then arranged into paragraphs labelled by pairs of numbers so that, e.g., 13.12 stands for the twelfth paragraph of the thirteenth

10

section. When referring to paragraphs we use this notation frequently. Formulas are numbered in each section separately. In the current section, we refer to a formula simply by its number. When referring to a formula appearing in some other section, we denote it by a pair of numbers placed in parentheses so that, e.g., (13.12) stands for the twelfth formula of the thirteenth section. Figures have the same number as the paragraph to which they belong. The exposition is completed by a number of illustrative examples. Not all of these are elaborated in detail; a considerable number are, rather, exercises calling for the reader's active participation. In writing the text, we made use of the experience gained while lecturing and working in seminars at the Faculty of Mathematics and Physics of the Charles University and at the Mathematical Institute of the Czechoslovak Academy of Scien­ ces, as well as from the activities of the seminar at the Technical University of Plzeii. We therefore wish to express our thanks to all those who took part in the work of these seminars as well as to those who have in some other way contributed to the elimination of mistakes and to the general improvement' of this text. Among all these colleagues, we thank particularly Dr. Milan Kucera, CSc, and the referees of this book, Prof. Dr. Karel Rektorys, DrSc, and Dr. Oldfich John, CSc. Prague, April 1979 S. FUCIK A. KUFNER

LIST OF SYMBOLS 1. Note that throughout this book we work in the real field only, i.e., with real numbers and with real functions of one or more real variables. 2. The numbers in the first column stand for the number of the paragraph in which the symbol considered can be found. {v e V; 0>(v)\

set of all elements v from V which possess property

UN R = R1 N N0

iV-dimensional Euclidean space set of all real numbers set of all positive integers set of nonnegative integers domain in UN (bounded, in most cases) boundary of domain Q point in UN point in IR2 closed interval: {ten; a ^ t ^ b] open interval: {teU; a < t < b} half-open interval: {teU; a ^ t < b} half-open interval: {teU; a < t % b] closure of set M restriction of function v to set S

Q

dQ X = \Xi,

(*.y) [fl,fc]

M) [a9b)

M] M 1.4

v\s

26.27}► u + ,

34.2 J

u~

14.1

< , >

20.7 15.7 15.7 24.5 12.8 26.24

< , >
Q

< , >v -*

G GG

X 2 , . . . , X N)

positive and negative parts of function u symbol of duality ( stands for the value of the functional F at the point v) inner product in Hilbert space symbol of duality symbol of duality symbol denoting weak convergence symbol denoting continuous imbedding symbol denoting compact imbedding

12

6.1 7.3 14.1

coefficients of differential operators A

formal differential operator

14.1) 14.2J 14.1 7.2

A

operator determined by differential operator A

a(u9 v) a = (a l5 a 2 ,..., a^)

form determined by differential operator A multi-index

7.2

lal = =Z ai

length of multi-index a

( ^ K G)

boundary value problem

15.7

i

1

30.1 )>B(r) 33.14J 9.2 17.8 #°° j C*(&) 8.2 8.2 C*(a, 6) 8.2 C°(fi) 8.2 C*(G) 8.2 Cj(fl) C°°(G) 8.2 C°°(G) 8.2 C01(0) 8.3 C*>A(D) 17.8 C4R 12.2 12.13 CAR(p) 16.15 CAR*(p) 16.20 CAR(G) 18.5 dF(u) 18.5 dF(u, v) 21.2 d2F(u9 v, w) 30.1 d[F(x); B(r), 0] 30.14 d[Tu; B(r), O] 6.2 (> div v 16.28] 10.4 S(A) 7.2 D"w 13.2 Daw 7.2 Sku 7.2 Sku A 1.2 1.2 A2

sphere with radius r and center at the origin classes of domains in RN space of smooth functions space of smooth functions space of smooth functions space of smooth functions space of smooth functions space of smooth functions space of smooth functions space of smooth functions space of smooth functions class of functions class of functions class of functions class of functions differential of functional F derivative of functional F second derivative of functional F Brouwer degree of mapping F Leray-Schauder degree of operator T divergence of vector function v domain of operator A derivative of function u generalized derivative of function u

Laplace operator biharmonic operator

13

16.20 15.7 18.8 15.7 2.1 20.1 12.5 7.2

EG(Q) II • IIG f F'(u) g grad u H Jf x = x(AT, Jfc)

Orlicz space norm in EG functional on space Q differential of functional F functional on space V gradient of function u Hilbert space Nemyckii operator

12.7 12.7

Lp(i2) Lp(a, b)

Lebesgue space Lebesgue space norm in Lp

12.8

HI,

LJQ)

II-IU

16.20

LG(Q)

12.8 1.5 1.5 1.1

meas M Mu Nu v

norm in L^ Orlicz space norm in LG measure of set M operator on dQ operator on dQ

II-IU

1) 15.7 2.3 8.2 31.1 15.7 15.7 13.3 13.4

Q sgn a supp u T'(«) -T F= V W'"(f2) W*-P(a, b)

!3-3

IHkp'lll-II*k,p

13.5 13.6

^'"(O) W%*\a, b)

13-7

|HkP,o

16.18 WUp"{Q) 16.18 W01:™(G) 16.22 WkEG(Q) 16.22 W"LC( 1 for p = 1 special Banach space sign of number a support of function u Frechet derivative of mapping T special linear set of functions special Banach space Sobolev space Sobolev space norms in Wk,p Sobolev space Sobolev space norm in WQ,P anisotropic Sobolev space anisotropic Sobolev space Sobolev-Orlicz space Sobolev-Orlicz space Sobolev-Orlicz space norm in WkLG

14

16.24 WkE^(Q) X

il-IU x*

anisotropic Sobolev-Orlicz space Banach space norm in X dual space to space X

CHAPTER I SOME EXAMPLES TO BEGIN WITH

SECTION 1. VARIOUS NOTATIONS. LINEAR EQUATIONS 1.1. In what follows, Q is a bounded domain in N-dimensional Euclidean space RN, N ^ 1. Points in UN are denoted by x = { vy) •

Besides the derivative in the direction of the outward normal du du du — = — v* + — vyy dv dx dy

(2) w

we introduce the derivative in the direction of the tangent: du _ ST

du

~ " Vx

du Vy +

dy

( Vx

'

. ^

16

1.2. In this chapter, we present a number of boundary value problems in differen­ tial equations, most of which describe some specific physical situation. In doing this, we formulate no precise assumptions. It is to be understood that all concepts we shall work here with have some reasonably defined meaning. Thus, e.g., for the function u = u(x) defined on Q, we introduce the Laplace operator A by the formula Au

d2u d2u = 7-2 + T-2 + -

d2u + Tl->

/A, (4)

here, it is (tacitly) assumed that the function u actually possesses all second order partial derivatives which appear in formula (4). Furthermore, we introduce the biharmonic operator A2 by means of the formula A2w =

A'AK)

= Y , , ij=i dx] dx]

In particular, for N = 2 we have A2w = —- + 2 — + —- . dx4 dx2 dy2 dy4

(5)

VJ

Here, we assume once again that the function u has (continuous) partial derivatives of the fourth order. Precise formulations of the conditions and assumptions are presented in sub­ sequent chapters. 1.3. The equation of a bar. Consider a bar of length 1 which we assume is placed along the interval [0, 1], The deflection of this bar (in general, of variable crosssection or, rather, of variable modulus of elasticity, on an elastic subsoil and vertically loaded) is expressed by the function u = u(x), defined for x e [0, 1], which satisfies the following fourth-order ordinary differential equation: £

[*(*) /(*) ^ ]

+ Q(x) u = / ( , ) ,

x e (0, 1) .

(6)

Here, E(x) is the modulus of elasticity in tension (Young's modulus), l(x) is the crosssectional moment of inertia about the bending axis, Q(x) the yielding coefficient of the subsoil, and f{x) the function representing the vertical load. If the bar is clamped at both ends, then the situation is described by the conditions M(0)

= 0,

u'(0) = 0 ;

ti(l) = 0 , n'(l) = 0 ;

(7)

if it is simply supported, then the situation is described by the conditions ii(0) = 0 , u"(0) = 0 ;

u(l) = 0 , u"(l) = 0 .

(8)

17

Consequently, to solve the problem of a clamped or a simply supported bar means to find a function u = w(x) satisfying equation (6) and conditions (7) or (8), respec­ tively. 1.4. The Poisson equation. Let Q be a plane domain and l e t / = / ( x , y) be a given function on Q. We seek the function u = u(x, y) defined on Q = Q u 30 and satis­ fying the equation — Aw = / on Q . This — the so-called Poisson — equation describes a number of physical phenomena: E.g., u might stand for the deflection of a membrane (of a homogeneous isotropic material) having the shape Q, loaded by vertical forces; or, u could be the distribution of temperature under stationary heat conduction in a plate of shape Q with constant heat conductivity and with internal sources of heat independent of time (here, the function / represents either the vertical load or the heat sources). Various conditions may join the Poisson equation on dQ. For instance, the condition

4« = °t) means that the membrane is clamped along its edge, or that the edge of the consid­ ered plate is kept at zero temperature; the condition — = g dn

on

dQ ,

where g = g(x, y) is a given function defined on dQ, means that the emission of heat along the edge of the plate is prescribed; the condition du h cu = a1 on oQ „, dn where c = c(x, y), d = d(x, y) are given functions defined on dQ, means that heat exchange occurs (in a prescribed manner) between the plate and its environment. 1.5. The equation of a plate. Let Q again be a plane domain, / = / ( x , y) a given function on Q, and let us look for the function u = w(x, y) which satisfies the equation A2M

= /

on

Q.

(9)

t) The symbol v\s denotes the restriction of the function v = v(x) (defined on some set M c IRN) to the set S a M, i.e., v\s(x) = v(x),

xeS.

However, we will not always be consistent; frequently, we shall write v on S instead of v\s.

18

The function u describes the deflection of a thin plate (of constant thickness, homo­ geneous, isotropic), vertically loaded by forces characterized by the function / . Further, by d2u Mu = G Aw + (1 - a) (10) dv2 and and 5 /. \ /, \ d Vd2u d2u . 2 2\ d2u 1 /^x Nu= - _ ( A u ) + (l -a)— v x v , - — - ( v 2 - v 2 ) - — vxvA (11) dv dr \jox ex cy cy J we denote operations prescribed on dQ [see(2), (3) for the notation]; o is the so-called Poisson constant. Equation (9) together with the conditions dv

- 0 dQ

describes the deflection of a plate clamped along its edge. In connection with the conditions u\dQ = 9 ,

Mu\eQ

= h,

where g = g(x, y)9 h = h(x, y) are given functions defined on dQ, equation (9) describes the deflection of a plate which has prescribed settling of supports along its edge and a prescribed moment (in the case when g = h = 0, a simply supported plate is in question). Finally, the conditions Mu\ea = 0,

Nu\dQ = 0

describe a plate with a free edge. 1.6. The differential equations discussed in the above paragraphs were linear. However, it is nonlinear equations that we want to treat here and the linear problems were discussed as an easy introduction to the problems of the field. The differential equations mentioned up till now represent certain physical phe­ nomena. However, the mathematical description of physical phenomena necessarily entails some simplification: were all the factors to be included in the representations, mathematically unsolvable problems would frequently arise. Consequently, the mathematical description is actually no more than an approximation to physical reality. The description involving linear equations is a sort of first approximation whose advantage lies in that it leads to mathematical problems solvable by given mathematical techniques at the given moment. A more exact representation of the physical phenomenon would lead to nonlinear equations; thus, the nonlinear repre­ sentation is a further approximation which makes it possible to take additional factors into consideration. We try to illustrate this on the following example: The

19

Poisson equation from Paragraph 1.4 is a special case of the equation

"£( / c ( x ' > ' ) l)"^( f c ( x , > ' ) S) = / ( x , > ' )

on Q

'

(12)

where the given function k = /c(x, y) characterizes, e.g., the heat conductivity of the material at the point (x, y) while the function u = u(x, y) represents the distribution of temperature. If the material is such that the conductivity is constant, we obtain the Poisson equation — kAu=f on Q (k = const.). However, we know that the conductivity of the material may vary not only with location, but also with the temperature to which the material is exposed, i.e., we know that the function k may also depend on the function u and, eventually, on its derivatives as well: du du^

k=

k xru

{ ' ' '8xdy.

The problem of temperature distribution is then described by an equation of the type of (12), i.e., by the equation d {. ( du du\ du\ [k x, y ; u, — , — — 1 dx \ \ dx dyj dxj = f(x,y)

d (. ( lk[x9y;u, dy \ \ on Q.

du du\ du — ,— — dx dyj dy (13)

However, this equation is already nonlinear. We discuss nonlinear equations in more detail below. SECTION 2. NONLINEAR EQUATIONS 2.1. Let u = u(xu x 2 , ..., xN) be a function defined on a domain Q a UN. The gradient of u is an N-dimensional vector function denoted by the symbol grad u and defined as follows: / du du du' gradu = — , — , ..., — \dxi dx2 cxNj In what follows, we deal with functions defined on plane domains in the first place, i.e., for JV = 2. Thus, for the function u = w(x, y) we have \cx and

cyj

20

2.2. The nonlinear equation (1.13) mentioned at the end of Paragraph 1.6 was a special case of the equation ( ai(x> yi u> g r a d u) — ) dx \ dx)

( ai(x> y\ w> grad u) — j + dy \ dy)

+ a0(x, y; u, grad u) = f(x, y) : f)

(3)

equation (1.13) is obtained from equation (3) by putting ai(*, y\ £o, Zu €2) = il9 ^ (x> y)eQi> r iM** j ; £o> £i> £2) f (x, j ) e D 2 .

(i\

29

In this case, we formulate a rather different problem instead of the problem (5), (6): Find a function wl5 defined on Qx = Qx u dQl, which solves the equation - — (fci(*> ^ " l . g r a d t t i ) - ^ ) - — (fci(x,j>;i — ^SJCAVC1

;

;

]=/

+ |gradu| 2 ) grad u) = /(x) , | )

(2)

i = l OX;

i.e., we look for a function u = u(x) defined on Q which satisfies equation (2) for all x e Q. Equation (2) is called a differential equation of the second order (in divergent form); the functions a{ are called the coefficients of equation (2), and the function/ is called the right-hand side of equation (2). The term "coefficient of equation" does not correspond to the term which is current for linear differential equations [see Paragraph 6.3 (i)]. Nevertheless, we hope that this will not give rise to any misunderstanding. 6.2. Remarks, (i) For the time being, we understand the expression on the left-hand side of equation (2) to be a formal symbol, since nothing has yet been assumed about the functions at. This will be made up for in Section 8. t) The expression

— at(x; M, grad u) ox i is to be understood as the partial derivative (with respect to xt) of the composite function v = y(x) defined as follows:

v(x) = a,(x; M(X), grad w(x)) .

34

(ii) We have called equation (2) an equation in divergent form. This has its origin in the fact that the left-hand side of the mentioned equation is (but for its sign and the function a0), actually, the divergence of the vector a = (au a2, ..., aN) the components of which are the coefficients of equation (2)|); equation (2) can then be written as - d i v a + a0 = / , where, naturally, the vector a as well as the function / also depend on the desired function u and its derivatives. 6.3. Examples, (i) If we choose the at in (1) as follows: **(*; f o> f I> • • •> is) = f i

for

i = 1,..., iV ,

then equation (2) will take the form i=i dx(

\dxj

or, in other words, -AM = / ( * ) , where A is the Laplace operator (see Paragraph 1.2). Thus, equation (2) is the socalled Poisson equation already encountered (for N = 2) in Paragraph 1.4. This equation is a special case of a second order linear differential equation. A general second order linear differential equation is obtained if all the functions at in (l) are linear functions of the variables £0, f x , . . . , £N, i.e., if they have the form N

*i(*; £o> €i, •. •> £N) = S aiAx) Zj > i = o, I, ..., N , J=0

with given functions atj{x) defined on Q. An equation with such coefficients will then be of the form

»=lj =1 0 X j \

i.e., for x = (*lf *2» • • •» *JV) e &> w e define the divergence of the vector t? as the function dvt dv2 dvN div v = — - + — + . . . + — - . 3x x 3x 2 dxN

35

(ii) If we choose the functions a(in (1) as follows: afc

t09 tu...,

£N) = j ^ "

1

sgn {, ,

," = 1,..., N ,

^O*, y; £o> £i> $2) = m(Zl + ^2) £2, a0(x, y; Zo, Zu Zi) = °> where m = m(f) is a function of one real variable defined for t ^ 0. Then equation (2) assumes the form

We have already encountered this equation in Paragraph 2.2 (for the notation, see Paragraph 2.1). (iv) We recommend the reader to go through the specific equations discussed in Chapter I (first of all, the equations from Paragraphs 2.2, 4.1, 4.2, 4.3, 4.4) and derive the concrete form of the coefficients at for the individual equations. 6.4. General form of an equation of the second order. For the moment, consider a plane domain Q9 i.e., N = 2. Equation (2) then takes the form

a

/

du du\

d

(

du

du\

where the coefficients a,{x, y; £0) £ 1; %2) (' = 0» 1> 2) are defined for (x, y) e Q, (£o> £i> £2) e K3- Ifthe derivatives indicated in (3) are worked out (see the footnote

36

on p. 33), then this equation will assume (after some simple modifications) the form dax d2u

da2

d^t dx2

3£i dx dy

dax du d£0 dx

d2u

d2u

dax

d£2 dy dx

da2 du — + a0 d£0 dy

da, dx

da2 d2u d£2 dy2 da2 r - = /, dy

/AS

(4)

where, naturally, the functions a0, da^d^o, da{\d^l9 dajd^ (i = 1, 2) also depend on the desired function u and its derivatives dujdx, dujdy [we tacitly assume that the derivatives of the functions au a2,u which appear in formula (4) do actually exist], The Monge-Ampere equation (see Paragraph 2.4) has the form

b

2± ^ __^

dx2 dy2

82u

dx dy dy dx

the reader will easily discover that this equation cannot be written in the divergent form (3), or (4), i.e., that there do not exist any functions a09 au a2 such that equation (4) reduces to the Monge-Ampere equation. Consequently, this means that an equation of the form (3) is by far not the most general second order differential equation (in two variables); in the most general form, such an equation can be written as follows F

/ du du d2u d2u d2u d2u\ [x> y> u> — > — > —, > 7 - ^ r > T - T - ' T I \ ox oy ox ox oy oy ox oy J

=

°>

where F = F(x, y; £ 0 , £i9 £ 2 , £3, £4, £s> ^ ) *s a given function of nine variables defined for (x,y)eQ, fe^i,..,^)^7. [We obtain the Monge-Ampere equation by choosing F(x9 y; Z0> ••> Q = £ 3 ^ - UZs •] Equation (3) is thus a special case of a second order equation in two variables and, similarly, equation (2) is a special case of a second order equation in N variables (N ^ 1). However, as already seen when discussing the examples of Chapter I, equations of type (2) or (3) appear frequently in applications; therefore, it is reasonable to investigate them in detail. Although we deal principally in this book with equations which have the form (2) (i.e., with equations in the divergent form), it is sometimes possible to apply some methods used in the sequel, with certain modifications, to general equations as well.

37

6.5. Euler equations. The so-called Euler equations encountered in variational calculus belong to the class of equations of the form (2): Let the functional

/(,)=fF(*;"H * ''

38

i.e., the function F of the preceding paragraph has the form

K*> r* So, e» t2) = W{& + el) - /(*, y) So, where the function M = M(s) is determined by means of the function m = m{t) as follows: M(s) = I m{t) ds . SECTION 7. HIGHER ORDER EQUATIONS 7.1. In the preceding section, the notion of the equation of the second order was introduced. However, equations of higher order — namely the fourth order — were encountered in Chapter I (see Paragraph 4.4). Such equations will therefore be dis­ cussed in this section and the notion of equation of order 2k will be introduced. First of all, we introduce some new symbols. 7.2. Notation. Let N be a positive integer. The vector a = (a l5 a 2 ,..., ocN) whose components are nonnegative integers a,- is called a multi-index (more precisely: an N-dimensional multi-index); the number N

M = Z«/ is called the length of the multi-index a. If a is a multi-index and u = u(x) a function defined on the domain Q a RN, we denote by the symbol the partial derivative dxl1dx?...dx*NN' The number of all iV-dimensional multi-indexes of length at most k is denoted by x; we have

x.

{ (

du d2u dxN dx{

du dxx

dku) dxN)

...

Thus, the vector 5ku has x components (the order of differentiation is immaterial). The components of 5ku are ordered lexicographically, i.e., in the following way: The first component is the function u (i.e., the derivative of order zero), followed by derivatives of first order, then by derivatives of second order, etc., and, finally, by derivatives of order k. Derivatives of the same order s (s ^ k) will be arranged as follows: If a = (a x ,..., a^), p = (pl9..., pN) are two multi-indexes of length 5, then the derivative Daw precedes the derivative Dfiu if for some n from the set {1, 2, ...,iV} we have Pn,

*i = Pi for

i = l,..., n - 1 .

For k = 1, we have, e.g., x = N + 1 and Sxu = {u,gradw} = J I

w

" ox1

" , . . . " I. ox2 dxN)

For k = 2 and N = 2, we have x = 6 and u =

S7u =

{ Su du d^u d2u &u) 2 \ ' dx ' dy ' 3x ' dx dy' dy2)

t) Let us recall a classical result from differential calculus of functions of several variables which says: If a function u = u(x) defined on the domain Q c M>N has the derivatives

d2u

d fdu\

dxx dx2

dxx \dx2J

J

and

d2u

d

dx2 dxt

dx2

\dxj

and if these derivatives are continuous on Q, then the order of differentiation is immaterial, i.e., d2u

,

N

v)

d2u

f

x

(x)

dxl dx2 dx2 dxt holds for every x e Q. Naturally, a similar assertion holds for derivatives of higher order as well. In what follows, it will be seen that we are dealing with functions for which the order of differentiation is immaterial.

40

Denoting, further, by the symbol Sku the vector of all fe-th derivatives of the function u, i.e.,

4u = {DV} M = k ,

(2)

it is possible to split the vector 5ku into two parts:

The vector Sku is sometimes called the principal part of the vector Sku; in what follows, we will see that this is a reasonable definition. If we wish to emphasize that the value of the vector function 8ku at the point x e Q is concerned, we write dk u(x) . 7.3. Equations of order 2k. Let Q be a domain in RN, N ^ 1, k e N, and let aa = aa(x; £) (a are multi-indexes, |a| :g k)

(3)

be functions of N + x variables defined for xeQ,

{ e R*

[the number x = x(N, k) was defined in Paragraph 7.2]; the components of the vector £ are denoted by £fi9 where the subscript /? runs over the set of all multiindexes of length at most k, and the components ^ are ordered lexicographically as above (see Paragraph 7.2). Thus, we have

e = tf„ \fi\ ^ k]. Furthermore, let / = / ( x ) be a function defined on Q. Below, we investigate differential equations of the form

X(-i) |a| Dx(^;^"W)=/W,

(4)

i.e., we look for a function u — u(x) defined on Q which satisfies equation (4) for all x e Q. Equation (4) is called a differential equation of order Ik (in divergent form); the functions ak will be called the coefficients of equation (4) and the function / the right-hand side of equation (4). 7.4. Remarks, (i) In formula (4), summation in £

is over all the multi-indexes a

of length at most k. The expression on the left-hand side of equation (4) represents a purely formal symbol here, since nothing has as yet been assumed about the functions ax.

41

(ii) The reader will certainly have observed that the notation used for writing the equation in Paragraph 7.3 differs from the notation in Paragraph 6.1: We now deal with multi-indexes while in formula (6.2) "ordinary" summation subscripts were used. Naturally, equation (6.2) could be written in the form of (4) as well — it is a special case of equation (4) for k = 1; however, this will not be done in most cases, since the application of multi-indexes might confuse rather than clarify the matter for second order equations.f) Thus, we make the convention that for second order equations the notation of Paragraph 6.1 will be used, as a rule; the notation using "ordinary" summation sub­ scripts will also be used for equations of higher orders in cases where the number of variables remains small, i.e., for N = 1 or N = 2. Multi-indexes will be used in all cases when general considerations concerning equations of order 2/c with large or unspecified k are in question. 7.5. Examples, (i) If all functions aa = aa(x; £) from (3) are linear functions of the variables

(thus, in particular, aa = 0 for multi-indexes a such that |a| = 0 and |a| = l). Equa­ tion (4) which corresponds to this choice of the coefficients aa has the form

lD°(ED' M )=/(x)

aeM

(5)

peM

[for a e M we have |a| = 2 and, consequently, (— l) |ot| = 1]. Equation (4) is thus written in terms of multi-indexes. However, it can also be written using "ordinary" summation subscripts:

_

im^-

t) The notation which makes use of multi-indexes can be "rewritten*' using ordinary summation subscripts and vice-versa; this is rather complicated, however, and not very helpful. As far as technical problems arising here are concerned, the reader may learn more in Chapter 31 of the book [80]. See also the next paragraph.

42

i.e., we have the equation A2w = / where A2 = A(A) is the so-called biharmonic operator already encountered (for N = 2) in Paragraph 1.5. Formulas (5) and (6) represent the same equation. Comparing them shows that the notation using multi-indexes is, in specific cases, not always the most advantageous: The notation of (6) is clearer and does not call for the introduction of the multiindex set M. (ii) Let k = 2, and let M be the set of multi-indexes from Example (i). Let p > 2, and let us define the functions aa for |a| ^ 2 as follows: « « M ) = |peM I ^ r

2

( peM I ^ )

aa(x; {) = 0

for

aeM,

for

a £ M, |a| g 2 .

Equation (4) which corresponds to this choice of the coefficients aa assumes the form

SD«(|ZD"M|"-2SD/'»)=/(x);

aeAf

peM

peM

it is the equation A(|A |^2Au)=/(x), W

where we find the so-called nonlinear biharmonic operator (see Paragraph 2.5) on the left-hand side. (iii) Let k = 2 and N = 2 (and, thus, % = 6). Then we have to define altogether 6 functions, aJ[x9y;S)9

(x, y) e Q ,

{ = ({0,{!

{jjeR6

(recall that the components £ 0 , £u f2, £3* %4> is of the vector £ correspond, respec­ tively, to the functions u, dujdx, dujdy, d2ujdx2, d2ujdx dy, d2ujdy2). | ) Let us define the functions aa as follows: r> Z) = 0(f) (fa + Ks)> a ( o,2)(*, >>;£) = 0(£)(f5 + K 3 ) ,

aj(x9 y; £) = 0 for

|a| = 0, |a| = 1 ,

where #(£) = G(£l + €l + €l + £3?5), G = G(f) is a given function of the variable t > 0. t) We are guilty of a certain inconsistency here: The components of the vector £ are not numbered by means of multi-indexes although there are a multi-indexes with the coefficients aa. To be consistent, we would have to write the components of the vector { as follows: £(0,0)> £(1,0)> MO, 1)5 S(2,0)> S ( l , l ) > £ ( 0 , 2 ) •

43

Equation (4), corresponding to this choice of the coefficients aa and written using "ordinary" summation subscripts, has the following form: dx2 I

w

\dx2

2 dy2)}

\\dx2)

\dy2)

dxdyl

w

dx dy]

where V ;

\dx dy)

dx2

dy2)

it is thus equation (2.10) describing the elasto-plastic deformation of a plate — see Paragraph 2.5, where g(H(u)) was used in place of ^(u). SECTION 8. SPACES OF CONTINUOUS FUNCTIONS. SOLUTION OF A DIFFERENTIAL EQUATION 8.1. Hitherto, we have not stated what it is that we understand by the solution of equation (7.4) or (6.2). In this book, the concept of solution will be encountered in various senses; in this section, we shall define the so-called classical solution. However, certain appropriate sets of functions will first be introduced. 8.2. The spaces Ck(Q). (i) Let O b e a domain in UN, N ^ 1, and let k be a nonnegative integer. By the symbol Ck(Q)

we denote the set of all functions u = u(x) defined on Q all of whose derivatives Daw of order |a| ^ k are continuous in Q. (ii) By the symbol C°(Q) we denote the set of all functions u = u(x) defined on Q = Q u dQ which are con­ tinuous on Q. Obviously, we have C°(Q) c= c°(fl); moreover, the functions from C°(Q) are continuous on the boundary dQ as well. (iii) For a positive integer k, we denote by the symbol C\Q) the set of all those functions u e Ck(Q) whose derivatives Dau have the following

44

property: For every derivative Dau (|a| ^ k) there exists a function va e C°(Q) for which D*u(x) = va(x)

holds for

x GQ .

(1)

[In other words: The function u belongs to C\Q) if all its derivatives of at most the fe-th order are continuous inside Q9 i.e., in Q, and if it is possible to extend each of these derivatives continuously to Q, i.e., to the boundary dQ.~\ If u G Ck(Q), we shall say for short that Wu e C°(Q) for |a| g k although - strictly speaking — we can define the derivative of the function u at the point x for x e Q but not for x e dQ. By the value of Daw(x) for xedQ we mean the value of va(x), where va is the function from (l). Let us note that for N = 1, when Q is an interval (a, b), the boundary dQ consists of the points a and b and the derivative at these points is then the left-hand derivative (at the point b) and the right-hand derivative (at the point a). (iv) Let u = u(x) be a function defined on Q c UN, and let M be the set of those points xeQ for which u(x) 4= 0. The closure of the set M (in the space IR^) is denoted by supp u and is called the support of the function u. By the symbol Ck0(Q)

we denote the set of all functions u e Ck(Q) for which supp u c Q . The function u G Ck0(Q) is then identically equal to zero in some neighbourhood of the boundary dQ. (v) We put CZ(Q) = C\Ck0(Q), k= 0

c°°(S) = n C\Q) . fc = 0

8.3. The space C0tl(Q). Let u = u(x) be a function defined on Q = Q u dQ. We say that this function satisfies the Lipschitz condition on Q if a constant c > 0 exists such that for all x, y e Q the inequality \u(x) - w(^)| ^ c|x - ^| is satisfied.

45

By the symbol C0>\Q) we denote the set of all functions u e C°(Q) which satisfy the Lipschitz condition on Q. (i) The Lipschitz condition implies that the function u is continuous on Q. Moreover, it implies that the difference quotient of the function u is bounded on Q: For x, y e Q, x + y, we have !»(*) - JVr-i) ~ e < yN < a(yl9 ..., J>jv-i)} is a part of the domain Q; (d) the set «(.Vi, ••-, yN-i) lies outside the set Q.

< yN < a(yl9...9

yN-x)

+ e}

48

The set of all the domains of the above type is denoted by the symbol

Roughly speaking, it is the set of those domains whose boundaries dQ are (locally) describable by means of a function which satisfies the Lipschitz condition, while the domain Q lies "on one side of the boundary dQ" and the outside of the domain Q "lies on the other side"; see also Fig. 9.2a.

Fig. 9.2a

Now, let us illustrate the various possibilities using some more figures (all of them for a plane domain, i.e., for N = 2).

^Xf Fig. 9.2b

1) If Q is the rectangle ABCD of Fig. 9.2b, then Q belongs to ^ ° ' 1 . Indeed, if the point x e dQ is some point lying within anyone of the sides, then the corresponding surface T will always be described by a constant function if we, moreover, choose the coordinate system in a suitable way: For points along the segment CD we choose

49

as the coordinate system (yl9 y2) the original system (xl9 x 2 ), for points along segment AB we choose the system {yu y2) in such a way that yt = — xl9 y2 = — x2, for points along the segment BC we choose the system so that y± = — x 2 , y2 = xi9 etc. Naturally, none of the above systems is applicable to any of the vertices A, B, C, D; the choice of the systems for the points C and D is shown in Fig. 9.2b and the corresponding "surface" r will be described there by a piecewise linear function.

Fig. 9.2c

2) The domains shown in Fig. 9.2c do not belong to the class # 0 , 1 : The boundary of the domain QY can indeed be represented by a function in the neighbourhood of the point x 0 , but this function will not satisfy the Lipschitz condition. As far as the domain Q2 is concerned {Q2 is a circle with the segment S removed), it is precisely the points of the segment S which are the source of trouble: In the neighbourhood of the point xl9 the boundary dQ cannot be described by a function at all; the points on S can no doubt be described, but the corresponding sets M x and M 2 will both be parts of the domain Q2. Domains of the type of ^ ° ' 1 are introduced, among other reasons, because of the fact that they enable the introduction of the surface integral and the vector of the outward normal to dQ. In fact, a function a e C0,1(A) which describes a part r of the boundary dQ has — in accordance with Paragraph 8.3 (ii) — derivatives p-,

dyj

j = 1,2,...,JV-1,

almost everywhere in A. Therefore, the outward normal to dQ exists almost every­ where on T: Indeed, if y = (.Fi, •••>

yN-l,a(y1,...;yN_1))

is a point on T at which the function a has first order partial derivatives, then the vector of the outward normal v = (v l5 v2, ..., vN) to dQ has the following components at the point y e T: v

i =

! + I — V i«i \dyjj

J

— > * = 1,2,..., A T - 1 , fyi

50

The vector of the outward normal is not defined for all points y e T but only for those at which partial derivatives of the function a exist. However, since these derivatives exist almost everywhere on A, the vector of the outward normal exists almost every­ where on r . However, we may assume that the vector v exists everywhere on T: It is enough to give its definition on the set of measure zero upon which it is not given by formulas (3). If u is a function defined on dQ9 it is possible to define the surface integral: We define u(x)dS=

u(yl9...9yN-1,a(yl9...,yN-1)).

iV-l

and the integral

f "(*)IdS

Jen

can then be introduced by suitably "combining" the integrals over the individual "areas" P which cover the boundary dQ. We will not go into the details of this problem here; readers interested in these details are referred to the book [80], Chap. 28. 9.3. Boundary conditions. In what follows, we shall assume throughout that the domain Q is of the # 0 , 1 type. Now, it is possible to introduce the concept of boundary condition, whose physical as well as mathematical meaning was mentioned in Paragraph 9.1. The boundary condition is again nothing other than a requirement that the func­ tion u satisfy a certain "differential equation", this time not on Q but in fact on dQ. Here, the equation may even be a "differential equation of order zero" as illustrated, e.g., by conditions of the type of u\ea = 9 »

encountered in Chapter I. In Paragraph 6.4, we exhibited the most general form of a second order differential equation {N = 2); analogously, the boundary condition can be expressed most generally as follows: Let 5 be a nonnegative integer and a the number of multi-indexes of length s (see Paragraph 7.2); let the function (4)

*(*; i) > defined for xedQ condition

and Y\ e W9 be given, and let the function u e CS(Q) satisfy the does indeed depend upon at least one of those components of the vector rj for which we substitute derivatives of the s-th order in (5), i.e., we assume that at least one s-th derivative of the function u occurs in (5). Condition (5) is then called a boundary condition of the s-th order. 9.4. Boundary value problem. We speak of a boundary value problem if we are given 1) a differential equation [of the form (7.4)]; 2) a certain number of boundary conditions (independent in a certain sense). (What we understand by the solution of a boundary value problem will be discussed in Paragraph 10.2.) The number of boundary conditions, their order and the actual form of the cor­ responding functions # need generally bear no relation to the differential equation, i.e., to its order 2k and to the actual form of the functions aa, |a| ^ fc. However, there is a close relation between the differential equation and the boundary conditions in particular physical problems (examples of which were given in Chapter I). Because of this, we are not going to prescribe boundary conditions entirely arbitrarily here; we shall observe several principles which reflect physical reality and which are also advantageous for the mathematical investigation of the problem. The principles are the following: (a) For a differential equation of order 2k we shall consider k boundary conditions, i.e., we shall give k functions *l9 n e N . 1 — c

10.7. The linearization method. Consider the case of N = 1, Q = (0, 1), and look for the classical solution of the boundary value problem -^L + g(u(x))=f{x) for x e ( 0 , l ) , dxz ii(0) = u(l) = 0 ,

(9) (10)

i.e., for the solution of the (homogeneous) Dirichlet problem for equation (9), where / = f(x) e C°([0, 1]) and g = g(t) is a function defined for t e U which satisfies the Lipschitz condition: A constant C ^ 0 exists such that for all t, s e U we have |*(0 - 9(s)\ * C\t - ,| .

(11)

The operator equation (6) is then in question, where the operator sf is defined as follows: We have &{s*) = {UE C 2 ([0, 1]); ti(0) = ii(l) = 0} and ( ^ « ) ( x ) = -u"(x)

+ g(u(x)).

(The nonlinearity of this boundary value problem is concealed in the function g.) Now, let v be a fixed chosen (but arbitrary) function from the space C°([0, 1]) and let us look for the function u e C 2 ([0, 1]) which solves the linear boundary value problem

- S =/(*)" *(«(a)] ,

which is actually the formula for integration by parts.

70

11.2. Green's formula for higher order derivatives, (i) Let v, w be two functions from C 2 (0), where fle^0,1. Let us apply formula (1) with the function w replaced by the function dwjdxj which belongs to C 1 ^ ) . Then we have

f fSB^d,- - f^WMi)a»+ f ^ * ) , « « .

J Q d*i Sxj

J n dxj

8xt

(2)

J 8n dxj

Now, formula (1) is applied once more, this time for the subscript j and for the pair w, dvjdXi (i.e., the function v is replaced by the function dvjdx^). One obtains f 8w(x) dv(x) J f , . 3Vx) J f , . v,(x)d5. — ^ — ^ dx = w(x) ^ dx + w(x) ^CX;

Jfl 3xy

3xf

Jfl

SJCJSXJ

(3)

JaD

Substituting now from formula (3) for the first integral on the right-hand side of formula (2) — still remembering that for v e C2(Q) the derivatives d2vldxt dxj and d2vjdXj dXi are equal — the following formula is obtained:

f *?& v(x) dx -

Jn = f w(x) P & dx JQ dXidxj

+

dxtexj

f [ ^ ,(x) vflx) - w(x) ^ dx J ^ L 5*y i

v / x ) l dS . J

(4)

Formula (4) could be called Green's formula for derivatives of the second order. (ii) Now it is possible to continue: If v, w are two functions from C3(Q)9 then (l) (with the function w replaced by the function d2wjdxj dxk) implies L z

- — v(x) dx = J Q dxt dxj dxk J fl

5XJ

^ ——-■ dx + ^ v(x) vlx) dS . 5x* dx; J dD dxj dxk

The first integral on the right-hand side is then modified employing formula (4) where, naturally, the subscript i is replaced by the subscript j , the subscript j by the subscript fc, the function v by the function dvjdxi9 and, finally, Green's formula for derivatives of the third order is obtained:

JA

+

^ — v(x) dx = w(x) ^-J— dx + Sxt dxj dxk J fl dxt dxj dxk

r \p& supp v0. This fact will be exploited in the next paragraph. 11.5. An integral identity. Let the function u e C2k(Q) be the classical solution of the differential equation of order Ik X ( - 1 ) H D* aj[x; dk u(x)) = f(x) (see Paragraph 8.4).

on

Q

(8)

72

Further, let v e Ck0(Q). Equation (8) holds for all xeQ; let us then multiply equation (8) by the number v(x) and integrate the resulting equality over the domain Q: £ ( - 1 ) ' " f D*aa(x; dk u(x)) v(x) dx = f f(x) v(x) dx .

(9)

We now use formula (7) where we choose w(x) = aa(x, 5k u(x)) . We obtain I Daaa(x; Sk u(x)) v(x) dx = ( - l ) | a | | aa(x; Sk u(x)) Dai;(x) dx . Applying the above relation, equality (9) can be written as follows: £

f aa(x; dk u(x)) D*v(x) dx = [ f(x) v(x) dx ;

(10)

this relation is valid for every function v e Ck0(Q). We have thus proved the following assertion: If u is the classical solution of equation (8), then integral identity (10) holds for every function v e CQ(Q). Naturally, this assertion is not reversible: identity (10) can be satisfied even for functions u which have derivatives only up to and including order fc but do not have derivatives of higher orders. Consequently, such a function cannot be the classical solution of equation (8). Thus, identity (10) can hold for a broader class of functions u than equation (8). This fact also serves as the basis of our considerations in the sequel: The function u is called the weak (generalized) solution of equation (8) if identity (10) is satisfied for all functions v e Ck0(Q). However, more details on this topic will be found in the next chapter where identity (10) will be analyzed thoroughly.

CHAPTER III THE WEAK SOLUTION OF A BOUNDARY VALUE PROBLEM

SECTION 12. THE CARATHEODORY PROPERTY AND THE NeMYCKlT OPERATORS 12.1. In Chapter II, we mentioned that the concept of the weak solution of the differential equation

Z(-l)MjyaJLx;8ku(x))=f{x)

on fl

will be based, primarily, on the integral identity £

f aa(x; dk u(x)) D*v(x) dx = f f(x) v{x) dx

(1)

valid for every v e C£(0). When deriving this identity in Paragraph 11.5, we started from the following assumptions: usC2k(Q), veCk0(Q), feC°(Q), aa{x; Z) e C^(Q x Rx) .

(2)

Under these assumptions, all the expressions in (1) are meaningful. However, if we ignore the fact that the identity (l) was obtained from some differential equation and study it quite independently, we immediately see that assumptions (2) are un­ necessarily strong: Nowhere in (1) do we need derivatives of u of order higher than fe; we do not even need the differentiability of the functions aa; moreover, not even the continuity of the functions appearing in (1) is necessary. Let us therefore investigate identity (l) in more detail and try to make clear under exactly how "weak" assumptionsconcerning the functions aa(x; f), u(x), v(x) do the expressions appearing in (1) make sense. We first analyse the left-hand side of the identity. For a chosen fixed function v, this side can be written in the form h(x;ul(x),...,um(x))dx9

(3)

74

where Ui(x), ..., um(x) are replaced by all the derivatives of the function u up to and including order k. When then is the integral (3) meaningful? Recall that throughout this book Lebesgue integrals are used. Therefore, it is necessary that for admissible functions w^x), ..., ujx) the composite function g(x) =

h(x;u1(x),...,um(x))

be measurable on Q and, moreover, that it be Lebesgue integrable.t) In the sequel, we assume that the functions w^x), ..., um(x) are measurable on Q. The class of all measurable functions is very broad: "Almost every" function defined on an open set Q is measurable. In spite of this, it is not true that every function composed of two measurable functions need itself be measurable — and the function g = g(x) mentioned above is a composite function. For this reason, we introduce one very useful concept which makes it possible to formulate better the results concerning the measurability of the composite function g. 12.2. Definition. Let Q be a domain in UN and let h = h(x; 0 there exists an element v e M such that

t t t ) A normed linear space A'is said to be separable if there is a dense subset M in this space which is at most countable, i.e., M — {vly v2, ..., vn, ...} where vt e X.

78

is finite [recall that the symbol meas M denotes the Lebesgue measure of the set M; the infimum in (7) is then taken over all subsets M a Q which are of measure zero]. Thus, u e L00(D) if there exists a set M of measure zero such that u is bounded on Q — M. Formula (7) defines a norm on 1^(0), and L00(D) with this norm is a Banach space. Some assertions of Paragraph 12.7 can be transferred, but the analogue of asser­ tion 12.7 (ii) does not hold: The space L^{Q) is not separable. (i) The following analogue of the Holder inequality is valid: If u e LX(Q) and

veL^Q),

then

r

JJiiW^idxsHiH..

(8)

(ii) / / the domain Q is bounded and if 1 < r < s < oo, then we have LX(Q) Q 14Q) Q Lr(Q) Q L^Q) . f) 12.9. Remark. The Holder inequality (6), or (8), implies immediately that the integral aa(x; dk u(x)) D*v(x) dx Jfl is finite if, e.g., D*veLp(G),

ax(x;Sku(x))eLq(Q),

(9)

where the numbers p > 1, q > 1 are linked by relation (4), or where either p = 1, q = coovp=oo, q = l holds. At present, we do not in fact know which as­ sumptions on the functions aa and u guarantee that the second of conditions (9) be satisfied. Therefore, we present the following theorem. 12.10. Theorem. Let pl9 p2, ..., pm and r be real numbers, pt ^ 1 (i = 1,..., m), r ^ 1. Let h = h(x; £) be a function defined for xeQ and £ e Um, and let h e CAR. Denote by J^{uu ..., um) the Nemyckii operator determined by the function ft. t) If X is a normed linear space with the norm || • || x and Y a normed linear space with the norm || • || y , then the notation XQY means that (a) Xcz Y, (b) there exists a constant c > 0 such that for all u e X holds.

NlySMMIx

We say that the space X is continuously imbedded into the space Y.

79

(i) Then, for an arbitrary m-tuple of functions W;eLp.((2) (i = 1, ..., m),

jr(uu...9um)ei4a) holds if and only if the following condition is satisfied: (a) A function g e Lr(Q) and a number c ^ 0 exist such that for almost all x e Q and for all £ e Um m

+ cZmp,lr.

\h(x;Z1,...,Q\Zg(x)

(ii) / / condition (a) is satisfied, then the Nemyckii operator from the Cartesian product

(10)

operator 34? is a continuous

L„(fl) x Ln(Q) x ... x LPm(Q) into the space L r (Q).t) 12.11. Comments on Theorem 12.10. (i) In the sequel, only the following assertion of Theorem 12.10 will be used: / / condition (a) is satisfied, then the operator Jf is a continuous operator from LPI(Q) x ... x LPm(Q) into LT(Q). (ii) If condition (a) is satisfied, then it can be proved easily that the operator Jf maps the Cartesian product LPl(Q) x ... x LPrn{Q) into Lr(Q): As a consequence of f) (i) Recall that the operator A which maps the Banach space X (with the norm || • || x) into the Banach space Y (with the norm || • || Y) is called a continuous operator if the following is true: If {un}^= i is a sequence of elements from X, un e @(A), u0 e @(A), and

lim \\un - u0\\x = 0 , n-*ao

then we have

lim \\Aun - Au0\\Y = 0 .

n-*co

(ii) Let Xlf X2, ..., Xk be Banach spaces. The set X of all ^-tuples

(uuu2,...,

uk),

where

uteXi9

i = 1, 2 , . . . , fc,

is called the Cartesian product of the spaces JTlf JT2» •••> -^fc anc * is denoted by JL

==

.A.j

X .A 7 ^

•••

*^ '**■ k *

The space A" is again a Banach space if the norm on X is defined, e.g., by

\\(uuu2,...,uk)\\x

=

(i\\ui\\°xy>°; i= 1

the symbol || • \\x. stands for the norm in the space Xi and s is an arbitrary but fixed number in the interval [1, oo).

80

the familiar inequality

(dt ^ 0, r ^ 1), we have, in view of (10), \h(x; Ul(x),...,
1, choose all the p( in Theorem 12.10 the same and equal to the number p, and as r choose the number q = p\(p — 1). Then, we have ptjr = pjq = = p — 1 and inequality (10) takes the form \aJix;Z)\ ^ gJix) + ca £ \t,\'~*, (12) 1*1** where ga is a given function from Lq(Q) and ca is a nonnegative constant. According to Theorem 12.10, the function

81

will then be an element of the space Lq(Q) if ax e CAR, if inequality (12) is satisfied for almost all x e Q and for all Q e Ux, and if D?u 6 Lp(Q)

for

\p\£k.

Inequality (11) then says that

\aJix;5ku(x))l£cal The integral

will now be finite if

I

+ c.2 £ i D ^ l r 1 •

aa(x; Sk u(x)) D*v(x) dx Dxv e Lp(Q):

The numbers p and q are connected by the relation ljp + ljq = I; thus, we can use the Holder inequality (5), or (6), by which aa(x; 3k u(x)) D*v(x) dx < laJLx;8ku{x))l\\D*vl,£

^(c al + ca2 x iiD^iir^iD^i

(13)

holds (with nonnegative constants c a l , c a2 which do not depend on the functions u, v). (ii) Choose p = 1 and q = oo. Inequality (12) then takes the form \aa(x;i)\

^ga(x)

+ x.ca,

(14)

where g is a given function from L^(Q) and ca is a nonnegative number. Thus, it is immediately seen that the function w(x) = aa(x; Sk u(x)) will be an element of the space L^{Q) if aa e CAR and Dpu e Lt(Q) for |/J| ^ /c (it would even suffice if all the functions Dfiu were simply measurable). Inequality (13) now obviously holds again if we put p = 1 in it: In fact, it suffices to use the Holder inequality (8) according to which ax(x; Sk w(x)) D*v(x) dx ^

laJixid^x^pyvl,

12.13. Coefficients with polynomial growth. Condition (12) will assume a very important role in our further considerations. In fact it tells us how must the function ajx; £) behave with respect to the variable £, by setting certain limits on its growth: The function ax cannot grow faster, in the variable £, than the polynomial cx +

82

For this reason, we call the functions aa which satisfy the condition (12) functions with polynomial growth. Since we will always consider only functions 1, or q = oo for p = l.f) For the sake of completeness, we note that condition (12) can be expressed in a number of other equivalent forms; e.g., it is possible to replace (12) by the condition \aJLx; «)| ^ Six) + U £

|{,|*)'/t

with ca ^ 0 and # a e L^(D). 12.14. Examples. Condition (15) will play an important role in our further con­ siderations; we, therefore, illustrate it using the following examples. (i) Let p > 1, and put ax{x; £) = m»-2

§. = \Z.\'-* sgn {. ,

|a| £ & .

Then, obviously, aa e CAR(p) : First of all, the reader verifies easily that aa e CAR; thus, the problem is to verify that the growth condition (12) is satisfied. In fact, condition (12) is actually satisfied, since it suffices to choose ga = 0 and ca = 1 and we have

In particular, for aJix; {) -

£

we then have a.eCAA(3). t) If not otherwise stated, the following convention will be adhered to in the sequel: The pair of numbers p, q, where p ^ 1, is determined thus:

for

p > 1 we have

q = p\{p — l) ,

for

p == 1 we have