MATHEMATICAL BLEMS IN ELASTICITY HOMOGENIZATlOt'
/
O.A. Oleinik A.S. Shamaev G.A. Yosifian
MATHEMATICAL PROBLEMS IN ELASTICITY AND HOMOGENIZATION
STUDIES IN MATHEMATICS AND ITS APPLICATIONS
VOLUME 26 Editors: J.L. LIONS, Paris G . PAPANICOLAOU, New York H. FUJITA, Tokyo H.B. KELLER, Pasadena
NORTH-HOLLAND AMSTERDAM LONDON NEW YORK TOKYO
MATHEMATICAL PROBLEMS IN ELASTICITY AND HOMOGENIZATION
O.A. OLEINIK Moscow University, Korpus 'K' Moscow, Russia and A.S. SHAMAEV G.A. YOSIFIAN Institute for Problems and Mechanics Moscow, Russia
NORTH-HOLLAND AMSTERDAM LONDON NEW YORK -TOKYO
ELSEVIER SCIENCE PUBLISHERS B.V. SARA BURGERHARTSTRAAT 25 P.O. BOX 21 1,1000 AE AMSTERDAM, THE NETHERLANDS
Library of Congress Cataloging-In-Publication Data
Oleinik. 0. A. Mathematical problens In elasticity and homogenlzatlon / O.A. Oleinlk. A.S. Shamaev. G.A. Yoslflrn p. cn. (Studles in nathenatlcs and its applications ; v. 26 ) Includes blbllographical references. ISBN 0-444-88441-6 talk. paper) 1. Elasticlty. 2. Homogenlzatlon (Dlfferential equations) I. Shamaev. A. S. 11. Yosiflan. G. A. 111. Title. IV. Series. P A 9 3 1 .033 1992 6311.382--dc20 92-15390 CIP
--
ISBN: 0 444 88441 6
01992 O.A. Oleinik, A.S. Shamaev and G.A. Yosifian. 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, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science Publishers B.V., Copyright & Permissions Department, P.O. Box 521, 1000 AM Amsterdam, TheNetherlands. Special regulations for readers in the U.S.A. - This publication has been registered with the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of pans of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred to the publisher, Elsevier Science Publishers B.V. No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. This book is printed on acid-free paper. Printed in The Netherlands
CONTENTS
PREFACE CHAPTER I: SOME MATHEMATICAL PROBLEMS O F THE THEORY OF ELASTICITY
$1.Some Functional Spaces and Their Properties. Auxiliary Propositions
$2. Korn's Inequalities 2.1. The First Korn Inequality 2.2. The Second Korn lnequality in Lipschitz Domains 2.3. The Korn Inequalities for Periodic Functions 2.4. The Korn Inequality in Star-Shaped Domains 53. Boundary Value Problems o f Linear Elasticity 3.1. Some Properties of the Coefficients o f the Elasticity System 3.2. The Main Boundary Value Problems for the System o f Elasticity 3.3. The First Boundary Value Problem (The Dirichlet Problem) 3.4. The Second Boundary Value Problem (The Neumann Problem)
3.5. The Mixed Boundary Value Problem $4. Perforated Domains with a Periodic Structure. Extension Theorems 4.1. Some Classes o f Perforated Domains 4.2. Extension Theorems for Vector Valued Functions in Perforated Domains
vi
Contents 4.3. The Korn Inequalities in Perforated Domains
51
55. Estimates for Solutions of Boundary Value Problems of Elasticity in Perforated Domains
55
5.1. The Mixed Boundary Value Problem
55
5.2. Estimates for Solutions of the Neumann Problem in a Perforated Domain
56
56. Periodic Solutions of Boundary Value Problems for the System of Elasticity 6.1. Solutions Periodic in All Variables 6.2. Solutions of the Elasticity System Periodic in Some of the Variables 6.3. Elasticity Problems with Periodic Boundary Conditions in a Perforated Layer 57. Saint-Venant's Principle for Periodic Solutions of the Elasticity System
67
7.1. Generalized Momenta and Their Properties
67
7.2. Saint-Venant's Principle for Homogeneous Boundary Value Problems
71
7.3. Saint-Venant's Principle for Non-Homogeneous Boundary Value Problems
73
58. Estimates and Existence Theorems for Solutions of the Elasticity System in Unbounded Domains 8.1. Theorems of Phragmen-Lindelof's Type 8.2. Existence of Solutions in Unbounded Domains 8.3. Solutions Stabilizing to a Constant Vector at Infinity
59. Strong G-Convergence of Elasticity Operators
98
9.1. Necessary and Sufficient Conditions for the Strong
G-Convergence
98
9.2. Estimates for the rate of Convergence of Solutions of the Dirichlet Problem for Strongly G-Convergent Operators
111
Contents CHAPTER II: HOMOGENIZATION O F THE SYSTEM OF LINEAR ELASTICITY. COMPOSITES AND PERFORATED MATERIALS
119
51. The Mixed Problem in a Perforated Domain with the Dirichlet Boundary Conditions on the Outer Part of the Boundary and the Neumann Conditions on the Surface o f the Cavities
119
1.1. Setting of the Problem. Homogenized Equations
119
1.2. The Main Estimates and Their Applications
123
52. The Boundary Value Problem with Neumann Conditions in a Perforated Domain 2.1. Homogenization o f the Neumann Problem in a Domain
134
52
for a Second Order Elliptic Equation with Rapidly Oscillating Periodic Coefficients
134
2.2. Homogenization of the Neumann Problem for the System o f Elasticity in a Perforated Domain. Formulation of the Main Results 2.3. Some Auxiliary Propositions
140 142
2.4. Proof o f the Estimate for the Difference between a Solution o f the Neumann Problem in a Perforated Domain and a Solution o f the Homogenized Problem
149
2.5. Estimates for Energy Integrals and Stress Tensors
157
2.6. Some Generalizations
158
53. Asymptotic Expansions for Solutions o f Boundary Value Problems o f Elasticity in a Perforated Layer
163
3.1. Setting of the Problem
163
3.2. Formal Construction o f the Asymptotic Expansion
164
3.3. Justification o f the Asymptotic Expansion. Estimates for the Remainder
171
54. Asymptotic Expansions for Solutions of the Dirichlet Problem for the Elasticity System in a Perforated Domain
178
4.1. Setting o f the Problem. Auxiliary Results
178
Contents 4.2. Justification o f the Asymptotic Expansion 55. Asymptotic Expansions for Solutions of the Dirichlet Problem for the Biharrnonic Equation. Some Generalizations for the Case o f Perforated Domains with a Non-Periodic Structure 5.1. Setting o f the Problem. Auxiliary Propositions 5.2. Justification o f the Asymptotic Expansion for Solutions o f the Dirichlet Problem for the Biharmonic Equation 5.3. Perforated Domains with a Non-Periodic Structure 56. Homogenization of the System of Elasticity with Almost-Periodic Coefficients
6.1. Spaces of Almost-Periodic Functions 6.2. System o f Elasticity with Almost-Periodic CoefFicients. Almost-Solutions 6.3. Strong G-Convergence o f Elasticity Operators with Rapidly Oscillating Almost-Periodic CoefFicients 57. Homogenization of Stratified Structures 7.1. Formulas for the Coefficients o f the Homogenized Equations. Estimates of Solutions 7.2. Necessary and Sufficient Conditions for Strong G-Convergence o f operetors Describing Stratified Media 58. Estimates for the Rate of G-Convergence o f Higher-Order Elliptic Operators 8.1. G-Convergence o f Higher-Order Elliptic Operators (the n-dimensional case) 8.2. G-Convergence o f Ordinary Differential Operators
185
Contents CHAPTER Ill: SPECTRAL PROBLEMS
$1.Some Theorems from Functional Analysis. Spectral Problems for Abstract Operators
263
1.1. Approximation of Eigenvalues and Eigenvectors of Self-Adjoint Operators
263
1.2. Estimates for the Difference between Eigenvalues and Eigenvectors o f Two Operators Defined in Different Spaces
266
$2. Homogenization of Eigenvalues and Eigenfunctions o f Boundary Value Problems for Strongly Non-Homogeneous Elastic Bodies
275
2.1. The Dirichlet Problem for a Strongly G-Convergent Sequence o f Operators
275
2.2. The Neumann Problem for Elasticity Operators with Rapidly Oscillating Periodic Coefficients in a Perforated Domain
279
2.3. The Mixed Boundary Value Problem for the System o f Elasticity in a Perforated Domain
286
2.4. Free Vibrations o f Strongly Non-Homogeneous Stratified Bodies
290
$3. On the Behaviour o f Eigenvalues and Eigenfunctions o f the Dirichlet Problem for Second Order Elliptic Equations in Perforated Domains
294
3.1. Setting of the Problem. Formal Constructions
294
3.2. Weighted Sobolev Spaces. Weak Solutions o f a Second Order Equation with a Non-Negative Characteristic Form
296
3.3. Homogenization o f a Second Order Elliptic Equation Degenerate on the Boundary
308
3.4. Homogenization of Eigenvalues and Eigenfunctions of the Dirichlet Problem in a Perforated Domain
$4. Third Boundary Value Problem for Second Order Elliptic Equations in Domains with Rapidly Oscillating
313
Contents Boundary
4.1. Estimates for Solutions 4.2. Estimates for Eigenvalues and Eigenfunctions 95. Free Vibrations of Bodies with Concentrated Masses 5.1.Setting of the Problem 5.2.The case -oo < m < 2, n > 3 5.3. The case m > 2, n 2 3 5.4. The case m = 2, n > 3 96. On the Behaviour of Eigenvalues o f the Dirichlet Problem in Domains with Cavities Whose Concentration is Small
97. Homogenization of Eigenvalues o f Ordinary Differential Operators
98. Asymptotic Expansion o f Eigenvalues and Eigenfunctions o f the Sturm-Liouville Problem for Equations with Rapidly Oscillating Coefficients
§9. On the
356
Behaviour of the Eigenvalues and Eigenfunctions
o f a G-Convergent Sequence o f Non-Self-Adjoint Operators
REFERENCES
367 383
PREFACE
Homogenization o f partial differential operators is a new branch of the theory of differential equations and mathematical physics. It first appeared about two decades ago. The theory of homogenization had been developed much earlier for ordinary differential operators mainly in connection with problems o f non-linear mechanics. In the field o f partial differential equations the development of the homogenization theory was greatly stimulated by various problems arising in mechanics, physics, and modern technology, requiring asymptotic analysis based on the homogenization o f differential operators. The main part o f this book deals with homogenization problems in elasticity as well as some mathematical problems related t o composite and perforated elastic materials. The study of processes in strongly non-homogeneous media brings forth a large number o f purely mathematical problems which are very important for applications. The theory o f homogenization o f differential operators and its applications form the subject o f a vast literature. However, for the most part the material presented in this book cannot be found in other monographs on homogenization. The main purpose o f this book is t o study the homogenization problems arising in linear elastostatics. For the convenience o f the reader we collect in Chapter I most o f the necessary material concerning the mathematical theory o f linear stationary elasticity and some well-known results o f functional analysis, in particular, existence and uniqueness theorems for the main boundary value problems o f elasticity, Korn's inequalities and their generalizations, a
priori estimates for solutions, properties o f solutions in unbounded domains and Saint-Venant's principle, boundary value problems in so-called perforated domains. These results are widely used throughout the book and some o f them are new.
xii
Preface In Chapter II we study the homogenization of boundary value problems
for the system o f linear elasticity with rapidly oscillating periodic coefFicients and in particular homogenization of boundary value problems in perforated domains. We give formulas for the coefficients o f the homogenized system and prove estimates for the difference between the displacement vector, stress tensor and energy integral of a strongly non-homogeneous elastic body and the corresponding characteristics o f the body described by the homogenized system. For some elastic bodies with a periodic micro-structure characterized by a small parameter e we obtain a complete asymptotic expansion in
E
for
the displacement vector. A detailed consideration is given in Chapter II t o stratified structures which may be non-periodic. Some general questions o f G-convergence o f elliptic operators are also discussed. The theory o f free vibrations o f strongly non-homogeneous elastic bodies is the main subject o f Chapter Ill. These problems are not adequately represented in the existing monographs. In the first part of Chapter Ill we prove some general theorems on the spectra o f a family o f abstract operators depending on a parameter and defined in different spaces which also depend on that parameter. On the basis of these theorems we study the asymptotic behaviour of eigenvalues and eigenfunctions o f the boundary value problems considered in Chapter II and describing nonhomogeneous elastic bodies. This method is also applied t o some other similar problems.
We prove estimates for the difference between eigenvalues and
eigenfunctions o f the problem with a parameter and those o f the homogenized problem. Apart from the homogenization problems of Chapter II, the general method suggested in §I, Chapter Ill, is also used for the investigation of eigenvalues and eigenfunctions o f differential operators in domains with an oscillating boundary and of elliptic operators degenerate on a part of the boundary o f a perforated domain. This method is also applied in this book t o study free vibrations of systems with concentrated masses. The theorems of
51, Chapter Ill, about
spectral properties o f singularly
perturbed abstract operators depending on a parameter can be used for the in-
Preface
...
xl11
vestigation o f many other eigenvalue problems for self-adjoint operators. Some abstract results for non-selfadjoint operators and their applications are given in
58, Chapter Ill. Although the methods suggested in this book deal with stationary problems,
some of them can be extended t o non-stationary equations. With the exception o f some well-known facts from functional analysis and the theory o f partial differential equations, all results in this book are given detailed mathematical proof. This monograph is based on the research of the authors over the last ten years. We hope that the results and methods presented in this book will promote further investigation o f mathematical models for processes in composite and perforated media, heat-transfer, energy transfer by radiation, processes of diffusion and filtration in porous media, and that they will stimulate research in other problems o f mathematical physics, and the theory o f partial differential equations. Each chapter is provided with its own double numeration o f formulas and propositions, the first number denotes a section o f the given chapter.
In
references t o other chapters we always indicate the number o f the chapter where the formula or proposition referred t o occurs. When enumerating the propositions we do not distinguish between theorems, lemmas, etc. The authors express their profound gratitude t o W. Jager, J.-L. Lions,
G. Papanicolaou, and I. Sneddon, for their remarks, advice and many useful suggestions in relation t o this work.
This Page Intentionally Left Blank
CHAPTER l SOME MATHEMATICAL PROBLEMS O F T H E THEORY OF ELASTICITY
This chapter mostly contains the results concerning the system of linear elasticity, which are widely used throughout the book.
Here we introduce
functional spaces necessary t o define weak solutions o f the main boundary value problems o f elasticity as well as solutions of some special boundary value problems which are needed in Chapter II t o obtain homogenized equations and in Chapter Ill t o study the spectral properties of elasticity operators describing processes in strongly non-homogeneous media. Some results o f this chapter are very important for the mathematical theory o f elasticity. Among these are Korn's inequalities in bounded and perforated domains, strict mathematical proof o f the Saint-Venant Principle, asymptotic behaviour a t infinity o f solutions of the elasticity problems, etc. On the basis of the well-known Hilbert space methods we give here a thorough consideration t o the questions o f existence and uniqueness of solutions for boundary value problems of elasticity in bounded and unbounded domains, and we obtain estimates for these solutions.
$1. Some Functional Spaces and Their Properties. Auxiliary Propositions
In this section we define the principal functional spaces and formulate some theorems from Functional Anlysis t o be used below. The proof of these theorems can be found in various monographs and manuals (see e.g. [40], [106],
[107], [1171, [1081). Points o f the Euclidean space Rn are denoted by x = ( x l ,...,x,), y =
(yl,...,Y,), Let
R
= (tl, ...,tn)etc.; A stands for the closure in be a domain o f Rn,i.e.
IR" o f the set A.
R is a connected open set in Rn. If not
I. Some mathematical problems of the theory of elasticity
2
indicated otherwise we assume R t o be bounded. For the main functional spaces we use the following notations:
C,"(R) is the space of infinitely differentiable functions with a compact support belonging t o R;
ck(f=l) consists o f functions defined in f=l and possessing all partial deriva[k] which are continuous in 0 and satisfy the Holder condition with exponent k - [k],provided that k - [k]> 0; [k]stands for the maximum integer not larger than k. LP(R) ( 1 5 p 5 m) is the space o f measurable functions defined in R and
tives up t o the order
such that the corresponding norms
Ilf
ll~m(n) = ess
SUP
n
If I
are finite. For p = 2 we get the Hilbert space
( u ,v)o =
1
ifp=m
L 2 ( R )with a scalar product
u(x)v(x)dx;
n
H m ( R ) (for integer m
> 0) is the completion o f C m ( n )with respect t o the
norm
(1.1) where
Dau =
... + a,,
dlalu
ax:' ...ax;,
, a is a multi-index, a = ( a l ,...,an),la1 = a l
+
aj are non-negative integers.
H,"(R) is the completion o f C,"(R) with respect t o the norm (1.1). By dR we denote the boundary o f the domain R . Throughout the book we shall mostly deal with domains whose boundary is sufficiently smooth, in particular with Lipschitz domains and domains with the boundary of class C' which are defined as follows. Denote by
CR,Lthe cylinder
3
$1. Some functional spaces and their properties
L, R are positive constants, $ = ( y l , ...,yn-1). We call R a Lipschitz domain if for any point x0 E d R one can introduce orthogonal coordinates y = C ( x - xO),where C is a constant ( n x n ) matrix, is given by the such that in coordinates y the intersection of d R with equation yn = cp($),where p($) satisfies the Lipschitz condition in {$ : 161 < R ) with the Lipschitz constant not larger than L and where
cR,L
The numbers R a n d L are assumed t o be the same for any point xO E depend only on
d R and
R.
We say that the boundary d R of R belongs to the class CT if the functions cp($) defined above belong t o CT(I$I < R ) , 0 < r . Let 7 be a subset o f d R . Suppose that R is a Lipschitz domain and 7 has a positive Lebesgue measure on d R . For a set y of this type one can introduce the following spaces o f functions vanishing on 7 ,and spaces of trace functions:
H m ( R , y ) (for integer m > 0) is the completion with respect t o the norm (1.1) of the subspace o f C m ( f i )formed by all functions vanishing in a neighbourhood of y ; obviously H m ( R , d R ) = H r ( R ) ; ~ " + + ( yis)the factor space Hm+'(R)/Hm+'(R,y ) . We say that a function u E Hm+'(R) coincides on y with a function cp E Hm+'(R) together with its derivatives up t o the order m, if u - cp E Hm+'(fl,4. As usual the norm in ~ " + + ( yis)
= inf
{llv + v I I ~ m + l ~vn E~ ,~
' ~ ' y() 0 ) .,
V
y the space ~ ~ + f r is( ynon-trivial, ) since Hm+'(R) does not coincide with Hm+'(R, y ) . This fact is due t o Under the above assumptions on
Lemma 1.1 (The Friedrichs Inequality). Let
R be a bounded Lipschitz domain and let y be a subset of its boundary
I. Some mathematical problems o f the theory of elasticity
4
80.
Suppose that
y has a positive Lebesgue measure on d o . Then for any
cp E H 1 ( R , y ) the inequality
holds with a constant
C independent of cp; V p
-
d p ,..., -). acp (-6x1 ax,
If /. = a R , then (1.2) holds for any bounded domain R and any cp E HA(R). The proof of this lemma as well as some more general results o f this type can be found in [117], [62].
H 1 ( R ) and inequality (1.2) obviously does not hold for cp = const., we conclude that H1(R,7) # H 1 ( R ) . It follows that we also have Hmtl(R,y) # Hm+l(a). By H-'(R) is denoted the space dual t o H 1 ( R ,d R ) H,'(R). Since constant functions belong t o
Some properties of functions defined in Lipschitz domains are given in the next theorem. Results o f this kind in a much more general situation are proved in 1481, [117], [67]. Theorem 1.2. Let
R be a bounded Lipschitz domain. Then
1. The imbedding o f H 1 ( R ) in L 2 ( R )is compact. 2. If 0 C to
R0 and R0 is a domain of R n , then each v E H 1 ( R )can be extended R0 as a function 6 E H1(RO)such that
where
C is a constant depending on R only.
3. Each function w E H 1 ( R ) possesses a trace on longing t o
where
L2(aR)and such that
C1 is a constant depending on R only.
an (see [67],
[117]) be-
5
51. Some functional spaces and their properties
4. Functions w E H 1 ( R )such that
/
w dx = 0 satisfy the PoincarC inequality
n
with a constant C2 depending only on
R.
5. H 1 ( R )consists o f all functions which belong t o L 2 ( R )together with their first derivatives.
We assume that the domains considered henceforth at least have a Lipschitz boundary unless pointed otherwise. In order t o study homogenization problems for differential equations we shall also need the following spaces o f periodic functions. Let
Znbe the set of all vectors z
= ( z l ,...,2,) with integer components.
By s,(G) we denote the shift o f the set G by the vector z , i.e. s,(G) = z For the given G the set of all x such that
E - ~ XE
+ G.
G is denoted by EG.
We say that an unbounded domain w has a 1-periodic structure, if w is invariant with respect t o all the shifts s,, z E t o be an open connected set of
Zn. Note that w is also assumed
Rn.
The spaces o f periodic functions are defined as follows:
&(G)
is the space of infinitely differentiable functions in ij which are
1-periodic in x l , ...,x,;
w ~ ( w )is the completion o f &(G) with respect t o the norm in H 1 ( w n Q ) , Q = { x : 0 < xj < 1, j = 1 , ...,n); e r ( w ) is the space o f infinitely differentiable functions in w that are 1periodic in xl, ..., x,, and vanish in a neighbourhood o f dw; 0 W (w) is the completion o f 6 r ( w )with respect t o the norm in H1(wn Q ) . A function cp(x) is said t o be 1-periodic in x and belonging t o H1(w n Q ) , if cp is an element o f W ; ( W ) . Let w be an unbounded domain with a 1-periodic structure. Set
I. Some mathematical problems of the theory of elasticity
6
Denote by H' (w(a, b)) the completion with respect t o the norm in H1(&(a, b)) of the space o f infinitely differentiable functions in w(a, b) which are 1-periodic In
XI,
...,xn-l.
Elements o f H1(w(a, b)) can be referred t o as functions in H1(a ;, 1-periodic in xl,
b)).
..., x,-~.
Consider a set y on ~ w ( u ,b) such that y is invariant under the shift by any vector z = (2,O) E Zn. W e w r i t e u = v o n y f o r u , v E H1(w(a,b)), a&(a, b))
ifu-VE ~'(&(a,b),~n
.
Note that Hm(R), H,"(R) are Hilbert spaces with the scalar product
and W;(W), H1(w
H1(w(a,b))
n Q). H(;aI,
are also Hilbert spaces with the scalar product o f
b)) respectively.
Many problems considered in this book involve vector-valued and matrixvalued functions, whose components belong t o one o f the spaces defined above. For such cases we shall adopt the following conventions. For column vectors u = (ul, the sum uiv;, and as usual
...,un)*,
v = (vl,
...,vn)*
by (u, v) we denote
lul = (u,u)lI2. Here and in what follows summation
over repeated Latin indices from 1 t o n is assumed; the sign * denotes the transpose of a matrix, however in the case of column vectors this sign is sometimes omitted unless that leads t o a misunderstanding. For matrices A and B with elements ai, and bij respectively we set
,
(A, B) = aijbij
IAI = (A, A ) ' / ~
.
(1.8)
If vectors u, v or matrices A, B have elements belonging t o a Hilbert space
'Id with a scalar product
and write u , v E
(a,
we shall often use the following notation:
7-t; A, B E 'Id instead of
u,v E 7-tn; A, B E
'Idn2.
The proof o f uniqueness and existence theorems for solutions o f various boundary value problems considered below is based on the following well-
$1. Some functional spaces and their properties known Theorem 1.3 (Lax, Milgram). Let H be a Hilbert space and let a(u, v ) be a bilinear form on
H x H such
that
Then for any continuous linear functional 1on H (i.e. 1 E element u
H*)there is a unique
E H such that E(v) = a ( u , v )
for any v E H
(see [134]). The Sobolev imbedding theorem (see [117]) yields Lemma 1.4. Let R
c Rnbe a bounded
any u
E H 1 ( R ) the inequality
n 2
Lipschitz domain and 1 - -
+ -ns 2 0 .
Then for
holds with a constant C independent o f u. Denote by p ( x , A ) the distance in
Rno f the
point x E
Rnfrom
the set
A c lRn. Lemma 1.5. Let R be a bounded domain with a smooth boundary and
Bs = { x E
R , p ( x , a R ) < 61, 6 > 0. Then there exists b0 > 0 such that for every 6 E (0,6,,) and every v E H 1 ( R )we have
I. Some mathematical problems of the theory of elasticity
8
where c is a constant independent of 6 and v.
Proof.
Due t o the smoothness o f d R there is a sufFiciently small 60
a family o f smooth surfaces S,,
> 0 and
E [O,dO],such that S, is the boundary of a domain R, C 0 , R, 3 R,, if T' > T , R0 = 52, C,T 5 p(x,dR) 5 c27 if T
> B,.
x E S T , T E [O, bO],CI, c2 = const, R\R,
By virtue o f the imbedding theorem (see Theorem 1.2) we have
J
<
Iv12dS < c3 Ilv/lZ,i(nT1 cs llvllZ,l(n)3
T
E 10,601 7
S,
where cg is a constant independent o f T . Integrating this inequality with respect to
T
from 0 t o 6, we get
I I v I I L2~ ( B ~ ) 5 ~ 4 II~llLl(n, 6 . This inequality implies (1.12). Lemma 1.5 is proved. Let Cl be a bounded domain with a Lipschitz boundary. Denote by 2(IRn x
R ) the set o f all functions f ((, x ) which are bounded and measurable in (t,x ) E Rn x R , 1-periodic in and Lipschitz continuous with respect t o x uniformly in ( E Rn i.e.
<
I f ( ( , 2) - f (t,xO)I 5 Cf for any x,xO E
0, ( E I?,
Lemma 1.6. Let g(C,x) E
i ( R nx R ) ,
12
(1.13)
- xOI
where c, is a constant independent o f x, xO,(.
/ g((,z)d<
= 0 for any z E
a. Then the inequality
8
holds for every u , v E H 1 ( R ) ,where c is a constant independent o f e U,
E (O,l),
v. Moreover, if F ( ( , x ) E
i ( R nx R ) , then for any 1C, E L 1 ( R )we have
9
§1. Some functional spaces and their properties
where P ( x ) =
/ F((,x)d(, Q
=]0, I[.=
{C
: 0
<
< 1, j
= 1, ...,n).
Q
Proof. $I1 =
I' the set o f all z E Znsuch that s ( z + Q ) C R . Set e ( z + Q ) , G = R\nl. Let us consider the functions m ( x ) , C(x),
Denote by
U
zEP
~ ( xwhich ) are constant on every ~ ( +zQ ) and are given by the formulas
~ ( x=) E
- ~
J
u(x)dx
for x E E ( Z
+Q) .
++Q) Then we have
Let xO,x
E ~ ( +zQ ) . Since g ( ( , x ) satisfies the Lipschitz condition in x
and its mean value in ( vanishes for any fixed x , it follows that
Obviously, the estimate (1.17) holds for almost all x
+
The PoincarC inequality (1.5) in ~ ( zQ ) yields 110
- C I I L Z ( ~ I~ ) Cle IIVVIIL~(~~) ,
1 1 -~ ~ 1 1 ~ 2 (5n Cie ~ ) IIVUIIL~(~~, . By the definition o f ~ ( xwe ) get
E R1.
I. Some mathematical problems o f the theory o f elasticity
The set G belongs t o the Cza-neighbourhood of dR (C2 = const), and therefore according t o Lemma 1.5 we have
The last integral in (1.16) is equal t o zero. It follows from (1.16) by virtue of (1.18), (1.19), (1.20) and the Holder inequality that
where C5 is a constant independent o f
E.
These inequalities imply (1.14).
Let us prove (1.15). For any $ E C1(fi) the convergence (1.15) is obviously a direct consequence o f the inequality (1.14) for u = $, v = 1, g(E,x) = F ( t , x ) - fi(x). Approximating a given
4E
L1(R) by functions in C1(Q) and taking into
account the fact that F((,x) is bounded, we easily obtain (1.15) for any function $ E L1(R). Lemma 1.6 is proved. Corollary 1.7. Let w be an unbounded domain with a 1-periodic structure and let { $ c ) , (9,) be two sequences o f functions in LZ(Rn EW) such that
11
$1. Some functional spaces and their properties
l l ~e
where $, cp
~II~2(nncw) 0 +
E L 2 ( R ) . Then for any f ( < , x ) E
nncw
L(R"
x R ) we have
n
where
/
F ( x ) = m e 4 8 n w ) (f (., 1 . ) ) E Qnw
f (C,x ) d t -
(1.23)
Proof. It is easy t o see that
The last two integrals tend t o zero as
E
+ 0 due t o (1.21). Setting F ( < ,x ) =
f ( < , z ) x w ( ( )in Lemma 1.6, where x u ( ( ) is the characteristic function of the domain w , we get
This convergence and (1.24) imply (1.22) since
Lemma 1.8. Let a ( ( ) be a bounded function which is piecewise smooth and 1-periodic in
<. Let
/ a(<)d( =
0. Then there exist bounded piecewise smooth functions
0
a , ( < ) , i = 1, ...,n , which are 1-periodic in
aa40 < and such that a ( < ) = at;
h f . Let us use the induction with respect t o the number o f independent variables. For n = 1 the assertion of Lemma 1.8 is evident since one can take
I
a l ( t l ) = 0 a ( t ) d t . Assume that the lemma holds in the case of n - 1
I. Some mathematical problems of the theory of elasticity
12
independent variables. Let
=
(i,tn), ( E IRn-',
and let a ( < ) satisfy the
conditions of the lemma. Set 1
The functions b j ( t ) ,j = 1 , ...,n, are 1-periodic in ( and
+ ~ ( .i )
abj(t) a(()= atj
Obviously,
/ c(()d(
= 0 , where Q =
(1.25)
{i
: 0
< tj <
1 , j = I, ...,n - 1).
s
Since c ( i ) depends on n - 1 variables it follows from the above assumption n-1
ac. a&
-2.. Therefore, taking into account (1.25) we obtain the
that c ( [ ) =
j=1
needed representation for a ( ( ) . Lemma 1.8 is proved.
52. Korn 's inequalities 52. Korn's lnequalities lnequalities of Korn's type are essential for establishing the solvability of the main boundary value problems of elasticity as well as for getting estimates of their solutions. In this section we denote by u, v the vector valued functions u = (ul, ...,u,), v = (vl, ...,v,), and Vu, e(u) stand for matrices whose elements are
aui (VU);= ~ axj respectively. We obviously have
In the theory of elasticity u = (ul, ..., u,) is the displacement vector and
e(u) is the strain tensor.
2.1. The First Korn Inequality Theorem 2.1. Let R be a bounded domain o f H,'(R) satisfies the inequality
Proof.
Since C,"(R)
P.Then
is dense in
every vector valued function u E
H,'(R), it is sufficient t o prove (2.2) for
functions in C,"(R). By virtue o f the Green formula we get
I. Some mathematical problems o f the theory o f elasticity
14 for any u E
C r ( R ) . Therefore (2.2) is valid for
u since the second integral in
the right hand side o f the last equality is non-negative. Theorem 2.1 is proved.
Note that inequality (2.2) of Theorem 2.1 holds for any bounded domain
R even if its boundary d R is non-regular. 2.2. The Second K o r n Inequality in Lipschitz Domains The inequality
for any u =
( u l ,...,un) E H 1 ( R ) is called the Second Korn Inequality.'
In
contrast t o the First Korn Inequality the proof o f (2.3) is rather complicated and requires some additional conditions on
R. Inequality (2.3) as well as some
more general inequalities o f this type under various assumptions on the domain
R are proved in numerous papers (see e.g. the references in [42]). Using the method suggested in [42] we give here a simple proof for the Second Korn Inequality in a domain with a Lipxhitz boundary. This proof is essentially based on the next two lemmas. We assume R t o be a bounded Lipschitz domain o f the distance from the point
IRn. By p(x) is denoted x t o dR; we denote by A the Laplace operator.
Lemma 2.2. Let
v E C w ( R )n L 2 ( R ) , p2Av E L2(R). Then pVv E L 2 ( R ) and the
estimate
holds with a constant c independent o f
v.
Proof. The function p(x) satisfies the inequality p(x) - p(y) 5 lx - yl for any
x,y E 0. Indeed, denote by z, the point o f a R such that p(y) = ly
- zyl.
'One may omit the proof of (2.3) at first reading. A more simple proof of the Second Korn Inequality for star-shaped domains is given in $2.4.
$2. Korn's inequalities
15
5 )x-z,- y+z,\. Thus p(x) is Lipschitz p(x) possesses bounded weak derivatives o f first
Then p ( ~ ) - ~ ( yI ) )x-z,J-Jy-z,J continuous in R and therefore order in 0.
Taking into account these properties o f in
R(&)= 52 n { x
:
p(x) and using the Green formula
p(x) > 61, we obtain
I t follows that
where c2 is a constant independent of 6. Making 6 tend t o zero in this inequality and taking into account the fact that
p(x) > 6 in
we get
for any domain G such that G c
0 , where the constant c3 does not depend on G. Therefore (2.4) is satisfied and pVv E L Z ( R ) .Lemma 2.2 is pr0ved.n Lemma Let w
2.3.
E C W ( R )n L 2 ( R ) ,p
d2w E L 2 ( R ) . Then w az;dxj
E H 1 ( R )and
where the constant C does not depend on w .
Proof. It is easy t o see that for any scalar function f
E C1[O, b] we have
16
I. Some mathematical problems of the theory of elasticity
Using the mean value theorem let us choose
T
such that
This inequality together with (2.6) yields
where
Clis a constant
independent o f
f.
Let us cover R by the domains R i , i = 0 , 1 , ..., N , such that Ro = { x : p ( x , d R ) > 61, 6 = const > 0 , and Ri = { X : $ i ( ~ ' )< xki < Gi(x1)+ bi, xi = ( x l ,...,xki-l, xki+l, ...,I,,), x' E R : ) , i = 1 , ...,N , 1 5 ki 5 n , (possibly after an orthogonal transformation o f the variables x ) , where the functions $i are Lipschitz continuous and d R n d R i = { x : xki = $i(xi), x' E 52:). By virtue of Lemma 2.2 we find that
where
R t f 2 is the 612-neighbourhood o f Ro, the constant C2depends only on
6.
Ri is defined by the conditions: $(XI) < xk < aw $ ( x i ) + bi, x' E 0:.Setting b = bi, f = -, t = xk in (2.7) and considering a xj aw - as a function o f xk, we get from (2.7) a xj Suppose that the domain
§2. Korn 's inequalities
17
Since +(xl)satisfies the Lipschitz condition, it is easy t o see that I+(xl)+ a
-
xkl 5 Cp(x),where the constant C depends only on the Lipschitz constant for +(xl). Therefore integrating (2.9) over R: and making a tend t o zero we find
provided that 6 is chosen sufficiently small. Summing up these inequalities with respect t o
i from 1 t o N and using (2.8) we obtain
It follows that estimate (2.5) is valid since p(x) 2 6
> 0 in 0;.
Lemma 2.3 is
proved. Theorem 2.4 (The Second Korn Inequality). Let
R
be a bounded Lipschitz domain. Then each vector valued function
u E H1(n) satisfies the inequality (2.3) with a constant C depending only on 0.
Proof.
Obviously we can restrict ourselves t o the case of u E C m ( f i ) . By
a2~; a eij(u)- a ejj(u)
the definition of the matrix e(u) we have -= 2
8x3
(there is no summation over
i, j).
Consider the following equations
axj
ax;
I. Some mathematical problems o f the theory o f elasticity
18
Fj = 0 outside R , i, j = 1, ...,n. Let v; E H,'(Ro) be a solution o f the c Ro. According t o the well-known a priori estimate we have Set
equation (2.10) in a smooth domain Ro such that
This inequality can be easily obtained by virtue o f the Friedrichs inequality and the integral identity for solutions o f the Dirichlet problem for equation (2.10). Set v = ( v l ,...,vn)*, w = u
- V . Then
~ ( e i j ( w )= ) 0 in fl
,
e;,(w) E C m ( R ) ,
i, j
= 1, ...,n
.
Due t o (2.11) we get
where the constant C3 does not depend on u. Therefore using (2.4) we find that
It is easy t o see that
Therefore (2.13) yields the inequality
Combining this inequality with estimate (2.5) of Lemma 2.3 we establish
$2. Korn's inequalities Since w = u - v the above estimate implies
Therefore owing t o (2.11) we find that (2.3) is satisfied.
Theorem 2.4 is
proved. In applications it is often important t o have another version o f the Second Korn Inequality, namely the inequality
which holds for v belonging t o a subspace V of H1(R). Subspaces V o f that kind will often be dealt with below. Denote by
R
the linear space of rigid displacements o f Rn,i.e. the set
of all vector valued functions q = (ql, a = (al,
...,a),
...,vn)
such that 7 = a
+ A s , where
is a vector with constant real components, A is a skew-
symmetric (n x n)-matrix with real constant elements.
Here 7, a, x are
column vectors.
It is easy t o see that
R
is a linear space o f dimension n ( n - 1)/2
+ n.
Theorem 2.5. Let R be a bounded Lipschitz domain and let V be a closed subspace of vector valued functions in H1(R), such that V
nR
= {0), where
R
is the space o f
rigid displacements. Then every v E V satisfies the inequality (2.14).
Proof. Suppose that the assertion o f Theorem 2.5 does not hold.
Then there
is a sequence o f vectors urn E V such that
Since the imbedding H 1 ( R )
c L2(R) is
compact (see Theorem 1.2), it
follows that there is a subsequence mj + oo such that for some v E
L2(R)
we have vrnl + v in L2(R). According t o Theorem 2.4 the Second Korn Inequality (2.3) is valid in R, and therefore
I. Some mathematical problems of the theory of elasticity
This estimate and (2.15) show that is a closed subspace of
urn)-+ v in H 1 ( R )as mj
+ co. Since
V
H 1 ( R ) ,by virtue of (2.15) we conclude that
-
The last equality implies that
avi avh
-+-=O, ax,, axi
i,h=l,
...,n .
R.
Les us show that any v satisfying (2.16) belongs t o Consider the mollifiers for v :
where v = 0 outside R ,
p ( ( ) E C r ( R n ) .p ( ( ) 2 0 ,
p(()d( = 1. ~
( 6=) 0
m n
> 1. One can easily verify (see e.g. (1171, [311) that 'v
C W ( G )and vc -t v in H 1 ( G )as E -+ 0 for any subdomain G such that G C 52.
forl(1
It follows from (2.16) that for sufficiently small
E
E
Since the vc are smooth in G these equations imply
a 2 ~ ;dxkaxh Therefore v f = a t x j
a2~;,
+
a2vi -- -- a 2 q in G . d x i a x h axiaxk dxhdxk bf, where a t j , bf are constants such that atj = - a f i .
Due t o the convergence o f vc t o v in Thus v E V
n R,I I v I I ~ I ( ~ = ) 1, which
H 1 ( G ) as
E
+ 0 we have v E
72.
is in contradiction with the condition
V fl R = (0). Theorem 2.5 is proved. Corollary 2.6. In Theorem 2.5 one can take as V one of the spaces
V = { V E H1( R ) : ( v ,V ) ~(n)I = 0
Vq E R} ,
$2. Kern 's inequalities
21
We shall now give some other examples of spaces V whose elements satisfy the inequality (2.14). Spaces of this type are often used below t o establish the existence of solutions o f boundary value problems for the elasticity system and t o obtain estimates for these solutions. Theorem 2.7. Let R be a bounded domain with a Lipschitz boundary. Suppose that the set
y
c dR can
be represented in the form x, = cp(i), where 3 =
(XI,
..., 1,-1)
varies over an open subset of Rn-', ~ ( 3 is) a Lipschitz continuous function. Then each vector valued function v E H1(R,y) satisfies the inequality (2.14).
H1(R, y) n R = {0), then in order t o obtain (2.14) we can use Theorem 2.5 with V = H1(R, y). Let r] E H1(R, y) n R. Therefore 77 = 0 on y. Every rigid displacement has the form r] = b A x , where A is a skew-symmetric matrix with constant b = 0 is linear, elements, and b is a constant vector. Since the system A x
Proof. If we show that
+
+
it is obvious that the ( n - 1)-dimensional surface y = { x : x, = cp(3)) must belong t o a hyperplane, provided that A # 0. Therefore the dimension b = 0 is not less than of the space formed by all solutions o f system A x n - 1, and consequently this system can have a t most one linearly independent
+
equation. Thus any two equations of the system are linearly dependent, and therefore since all elements on the main diagonal o f A vanish, the coefficients by xl,
..., x,
vanish, too. Hence r]
= 0. Theorem 2.7 is proved.
2.3. The K o r n Inequalities for Periodic Functions Here we establish the Korn inequalities similar t o (2.14) for 1-periodic vector valued functions. Theorem 2.8. Let w be an unbounded domain with a 1-periodic structure and let w a domain with a Lipschitz boundary. Then for any v E
W;(W)
n Q be
such that
I. Some mathematical problems o f the theory o f elasticity
the inequality
holds with a constant C independent of v.
Proof. Denote by V the linear space consisting of all restrictions t o w n Q o f vector valued functions in W ; ( W ) satisfying the conditions (2.17). It is easy t o see that V is a closed subspace o f H 1 ( w fl Q) and that any rigid displacement 1-periodic in x is a constant vector. Therefore if v E V (2.17) we have v
= 0.
Now Theorem 2.5 for
n R then
by virtue of
R = w n Q yields the inequality
(2.18). Theorem 2.8 is proved. The Second Korn inquality of type (2.14) for functions 1-periodic in 2 =
( x l , ...,x , - ~ ) is the result of Theorem 2.9. Let w be an unbounded domain with a 1-periodic structure and let the domains w ( a , b ) , Lj(a,b) ( 0
< a < b <
m) be defined by (1.6).
Suppose
that ;(a, b) has a Lipschitz boundary. Then for any vector valued function v E
H' (,(a, b)) such that
/
v d x = 0 the following inequality holds
&(ah)
where c is a constant independent of
v.
The proof of Theorem 2.9 is almost exactly a repetition o f that o f Theorem 2.8. It should only be noted that a rigid displacement 1-periodic in constant vector.
i is also a
$2. Korn's inequalities 2.4. The Icorn Inequality in Star-Shaped Domains In many applications it is important t o know the nature o f the dependence of the constants in Korn's inequalities on the geometric properties of the domain. This dependence can be characterized on the basis o f the elementary proof of the Korn inequality in a star-shaped domain, which is given in this section. Korn's inequalities in unbounded domains and some more general inequalities of that type for the norms in LP(R) and in weighted spaces were considered in [42], [43], [68], [46]. to
A domain R is said t o be star-shaped with respect t o a ball G belonging any two points x E G, y E 51 lies in R.
R, if the segment connecting
Theorem 2.10. Suppose that R is a bounded domain o f diameter R and respect t o the ball
QR1 = { x
: 1x1
< R 1 ) . Then
R
is star-shaped with
for any u = ( u l,...,u,) E
H 1 ( R )we have the inequality
where C1, Cz are constants depending only on n.
Proof.
Obviously it is sufficient t o prove (2.20) for smooth vector valued
functions u ( x ) . Let R1 = 1. By C j we denote here constants which can depend only on n . Let v = ( v l ,..., v,) be a solution o f the system
At); =
a eik(u) 2 (2 -
k=l
axk
a
- -ekk(u)) in
ax;
with the boundary conditions
Multiplying (2.21) by v, and integrating by parts in R the resulting equality, we find that
I. Some mathematical problems o f the theory o f elasticity
J
(2.23)
IVv12dx 5 C3 I l e ( u ) l k ( n ) .
n
Set
w
= u - v. For any smooth V = (Vl,
..., V),
the following identities are
valid
Therefore due t o (2.24), (2.21) we have
It follows from (2.23) that
Therefore by virtue of (2.26) and Lemma 2.2 we get
where p = p ( x ) is the distance from x E R t o
dR.It follows from
(2.28) and
(2.24) that
Let us apply the following inequality
where C is a constant independent of a and f. The proof of (2.30) follows immediately from (2.6).
dw;/dxjand the segment AP P is any point on aR,0 is the origin.
Let us apply (2.30) t o the function f = belonging t o the segment O P , where Considering P as the origin, we obtain
52. Korn's inequalities
Let us choose the point A such that A E
QR,. I A l = A E
[f , l ] ,
where dw is the area element on the unit sphere. Such a choice o f A is possible due t o the mean value theorem. Obviously (2.31) implies
Let us integrate (2.33) over the unit sphere. Since the domain shaped with respect t o
QRl with R1 = 1 it follows that IP - XI
Therefore (2.32), (2.33) yield
R is star-
< p(x)R.
I. Some mathematical problems of the theory of elasticity
26
R1 = 1 follows from (2.23), (2.29), (2.34), since v. The inequality (2.20) with any R1 > 0 can be obtained from (2.20) with R1 = 1, if one passes t o the variables y = x/R1. Estimate (2.20) with
w =
u-
Remark 2.11. The coefficient by the second term in the right-hand side of (2.20) is asymptotically exact and cannot be improved in the following sense. Let u = Ax
+ B,
where A is a skew-symmetrical matrix with constant elements, B is a constant vector. Then (2.20) holds (in the form of an equality) with the coefficient
C2(R/R1)", provided that R has the volume o f order Rn. Remark 2.12. The inequality o f type (2.20) holds for any bounded smooth domain 0 (and even for a Lipschitz domain), since such a domain is a union of a finite number of star-shaped domains. Remark 2.13. Using a slightly more detailed analysis in the proof of Theorem 2.10 we can find a more exact coefFicient by the first integral in the right-hand side o f (2.20). Namely, under the assumptions of Theorem 2.10 the following inequalities of Korn's type are valid
In order t o prove (2.35) we should use the inequality
32. Korn's inequalities
where
C is a constant independent o f
a and f . This inequality can be easily
obtained from the Hardy inequality (see e.g. (421, [44]). For the proof of (2.36) the inequality (2.37) should be replaced by the following one
where C is a constant independent o f a and f (see [152]). Estimate (2.35) cannot be improved in the following sense. Consider a vector valued function u = $(Ax
+ B ) , where A is a constant skew-symmetrical
B is a constant vector, $ E C m ( R n ) ,G(z) = 0 in QR,, $(x) = 1 outside of QzR1= {x : 1x1 < 2 R 1 } ,QzR, C 0 . Then (2.36) (in the form of an equality) holds for u(x) with the coefficient C1(R/R1)"by the first integral in the right-hand side, provided that R has the volume of order R". matrix,
Theorem 2.14. Suppose that
R satisfies the conditions o f Theorem 2.10 and
u E
H1(R).
Then
where y is the distance of QR, from 6'0.
Proof. Let cp E C F ( R ) ,cp = 1 in QR,, 0 5 9 5 Theorem 2.1 we have
1 in R. Then according t o
I. Some mathematical problems of the theory of elasticity It follows that
\(Vu\(12(pRl) 5 2 \le(u)l/t(o)+ C3Y211~112qn).
(2.39)
Estimates (2.20), (2.39) imply (2.38). Theorem 2.14 is proved. Theorems 2.10, 2.14 can be applied to study homogenization problems in domains having the form of lattices, carcasses, frames, etc.
53. Boundary value problems o f linear elas ticity
53. Boundary Value Problems of Linear Elasticity
3.1. Some Properties of the Coeficients of the Elasticity System In a domain
R c Rnconsider the differential operator o f linear elasticity
...,u,)
Here u = (ul,
is a column vector with components ul,
..., u,,Ahk(x)
are (n x n)-matrices whose elements af/(x) are bounded measurable functions such that
where {qih) tcl,
~2
is an arbitrary symmetric matrix with real elements, x E
= const
R,
> 0.
We say that a family of matrices Ahk, h , k = 1, ...,n, belongs t o class E(rcl, n2), if their elements a ? ' are bounded measurable functions satisfying conditions (3.2), (3.3). In this case we also say that the corresponding elasticity operator
t belongs t o class E ( K nz). ~,
The operator L defined by (3.1) can also be written in coordinate form as follows
a
-(
( u )
axh
a x )
auj
-)axk ,
i = 1, ...,n .
In the classical theory of linear elasticity for a homogeneous isotropic body the coefficients o f operators (3.4) are given by the formulas
where X
>
6,j = 0 for i
0, p
>
0 are the Lam6 constants, bij is the Kronecker symbol:
# j, 6ij = 1 for i = j .
for any symmetric matrix {qih).
In this case we have
Moreover, the family of the matrices Ahk,
h , k = 1, ..., n , belongs t o the class E(2p,2p +nX). Indeed, i t is obvious that nl = 2p, and the estimate
KZ
5 2 p + nX follows from
(3.5), since
I. Some mathematical problems o f the theory o f elasticity
Thus the elasticity operator corresponding t o a homogeneous isotropic body has the form
where Ul,hk =
aZul axhaxk
In order t o study the boundary value problems for the system o f elasticity we briefly describe some simple properties of the elasticity coefficients. These properties are easily obtained from the relations (3.2), (3.3) and will be frequently used below. With each family o f matrices Ahk(x) o f class E(nl, n2) for any fixed x we associate a linear transformation maps a matrix ( with elements
M
of the space of (n x n)-matrices, which
tjkinto the matrix M(
with the elements
Then according t o (1.8) we have
Denote by
€* the transpose
o f the matrix
€.
Lemma 3.1. . for Let Ahk, h , k = 1, ...,n, be a family of matrices o f class E(nl, K ~ ) Then any ( n x n)-matrices
€
= {&), 71 = { v ; ~ )with real elements the following
conditions are satisfied
Proof. By virtue of the first
inequality in (3.2) we obtain that
$3. Boundary value problems of linear elas ticity
Due t o (3.3) and (3.6) the bilinear form ( M ( , n )
can be considered as a
scalar product in the space of symmetric (n x n)-matrices. Therefore by (3.2), (3.3) and the Cauchy inequality we get
(Mt,n)
=
1
4 (M(t+ t * ) , n + v*) 5
It follows from (3.2) and (3.3) for 7 = ( t KI
It + <*I2 I (M(t+ E*),E
It + t * I In + s * l .
+ t * ) that
+ E*) = 4 ( M t , t )
.
Lemma 3.1 is proved. Lemma 3.2. Each operator (3.1) o f class E(nl, n2) ( K ~tc2 ,
det lla!:thtkll
#0
Proof.Consider the following
for a fixed
t # 0.
for
161 # 0
..., tn).
quadratic form
Jivh and summing with respect t o i, h Itirli12+ 1t12191=20. Therefore 77 = 0. Thus J(7) > 0
Multiplying each of these equations by
# 0.
< = ([I,
is elliptic, i.e.
If J(q) = 0 it follows from (3.9) that
from 1t o n we obtain for 9
> 0)
Lemma 3.2 is proved.
I. Some mathematical problems o f the theory o f elasticity
32
3.2. The Main Boundary Value Problems for the System of Elasticity
L: be an elasticity operator o f type (3.1) belonging t o class E ( n l , n z ) , n l , n2 > 0 , and let R be a bounded domain o f Rn occupied by an elastic body. The displacement vector is denoted by u = ( u l ,...,tin)*. Let
The following boundary value problems are most frequently considered in the theory of linear elasticity.
The first boundary value problem (the Dirichlet problem)
involves finding the displacement vector u at the interior points of the elastic body for the given displacements u = @ at the boundary and the external forces f = ( f i ,...,f,) applied t o the body.
The second boundary value problem (the Neumann problem)
i.e. a t the points of the boundary the stresses u ( u ) = cp are given.
Here
v = ( 4 ,...,vn) is the unit outward normal t o dR. The third boundary value problem (the mixed problem)
It is assumed here that the boundary d R of R is a union o f two sets such that r n S = 0.
and S
In order t o prove existence and uniqueness of solutions of these problems, it is necessary t o impose certain restrictions on below.
d R , r, S , which will be specified
33
$3. Boundary value problems o f linear elasticity
In $6 we shall also consider some other boundary value problems for the system of elasticity, in particular problems with the conditions o f periodicity in some of the independent variables. Let u = ( u 1 ,...,u,) be the displacement vector and let e ( u ) be the corresponding strain tensor, i.e. e ( u ) is a matrix with elements eij(u) =
1 dui (2
= -
duj
axj + -). axi
Set
Then taking into account (3.7), (3.8) for
= V u , [* = ( V u ) ' , we find
3.3. The First Boundary Value Problem (The Dirichlet Problem)
R be a bounded domain o f Rn(not necessarily with a Lipschitz boundary), f j E L 2 ( R ) ,j = 0,1, ...,vz, cp E H 1 ( R ) . We say that u ( x ) is a weak solution o f the problem Let
a?
L ( u ) = f " + - ax, in 0 , u=cp o n 8 0 ,
(3.14)
if u - 9 E HA(R) and the integral identity
holds for any v 6 H,'(R). Theorem 3.3. There exists a weak solution u ( x ) o f problem (3.14), which is unique and satisfies the estimate
where the constant
%(a)depends only on nl, nz in (3.3)
and the constant in
the Friedrichs inequality (1.2) for 7 = 30.
Proof. It follows from
(3.15) that w = u - cp must satisfy the integral identity
I. Some mathematical problems of the theory o f elasticity
for any v E H,1(R). Note that due t o the Friedrichs inequality (1.2), the First Korn inequality (2.2) and estimates (3.13) the quadratic form
satisfies the conditions of Theorem 1.3, if we take as H the space o f all vector valued functions with components in H,'(fl). Obviously the right-hand side of (3.17) defines a continuous linear functional on v E H t ( R ) . Therefore by Theorem 1.3 there is a unique element w E H i ( R ) satisfying the integral identity (3.17). Setting u = w cp we obtain the solution of the problem (3.14). Let us prove the estimate (3.16). Set w = u - cp, v = u - cp in (3.17). Then by virtue of the Friedrichs inequality (1.2), the First Korn inequality (2.2)
+
and estimate (3.13) we find
where the constant C3 depends only on KI, ~2 and the constant in (1.2). Since I llull - llcpll 1 5 IIu - (pll, the estimate (3.18) implies (3.16). Theorem 3.3 is proved.
53. Boundary value problems of linear elas ticity
35
The details, concerning the smoothness o f the solutions obtained in Theorem 3.3, are given a thorough consideration in the article [17]which contains in particular the proof o f the fact that the smoothness o f
f",
cp and the coefficients of
d R , the data functions
L guarantee the smoothness of the weak solution
u ( x ) o f problem (3.14). Denote by H - ' ( 0 ) the space o f continuous linear functionals on the space o f vector valued functions with components in H,'(R). As usual the norm in H - ' ( 0 ) is defined by the formula
It follows from the proof of Theorem 3.3 that
defines a continuous linear functional on
fc.1
=
J
[ ( f O , v)
n for any v E
H,'(R), namely
-31
(f', Xi
dx
H,'(R). We obviously have n
l l f ll~-l(nI ) C
lIfmll~2(n) , C = const . m=O
On the other hand, for any
f E H - ' ( R ) there exist functions f m E L 2 ( R ) ,
m = 0, ...,n, such that
in the sense o f the integral identity (3.19), and
Indeed, by the Riesz theorem (see [107]), every continuous linear functional
f ( v ) on H i ( R ) can be represented as a scalar product in H,'(R), i.e. there is a unique element u E Ht(R) such that
I. Some mathematical problems of the theory of elasticity
36
Setting v = u in (3.22) and taking into consideration the definition o f the norm in
H-'(a), we find that
Setting
f0
= u,
fi
=
-
e,
by virtue o f (3.22), (3.23) we obtain the repre-
sentation (3.20) and the estimate (3.21). Remark 3.4. In the special case when ip = 0 in (3.14), we can consider the problem
for any
f
E H-'(R), since
f can be represented in the form (3.20). Then by
Theorem 3.3, due t o (3.21) we have
where the constant C depends only on
61,rc2,
and the constant in the
Friedrichs inequality (1.2) for y = do.
3.4. The Second Boundary Value Problem (The Neumann Problem) In this section we assume R t o be a bounded domain with a Lipschitz boundary.
Let S1 be a subset o f dR with a positive ( n - 1)-dimensional
Lebesgue measure on dR. Set
We say that u ( x ) is a weak solution of the problem
where
fj
E L2(R),j = 0, ...,n , ip E L2(S1), if the integral identity
$3. Boundary value problems o f linear elas ticity
holds for any v
E H1(R). d R , fj, cp, Ahk are not smooth, the boundary conditions in
Note that if
(3.27) are satisfied only in a weak sense, namely in the sense of the integral identity (3.28). The integral over
S1 in the right-hand side of (3.28) exists
due t o the estimate
I I v I I ~ 2 ( ~ ~ 5) C ( R )IIvIIH1(n) for any v
(3.29)
E H 1 ( R ) ,which follows from Proposition 3 o f Theorem 1.2.
Theorem 3.5. Suppose that
for any rigid displacement q E
R. Then there exists a weak
solution u(x)of
problem (3.27). This solution is unique (to within an additive rigid displacement) and satisfies the inequality
Here the constant
~ ~ ( $depends 2) only on n l , n2, the constants in (3.29) and
in (2.14) when V is a closed subspace o f H 1 ( R )orthogonal t o R with respect t o the scalar product in Proof. Let
L 2 ( R )or H 1 ( R ) .
H = V in Theorem 1.3, where V is either of the spaces defined in
Corollary 2.6. Since inequality (3.29) is valid for the elements o f V, it is easy t o see that the right-hand side o f the integral identity (3.28) is a continuous linear functional on v E
H . By the same argument that has been used in the
proof of Theorem 3.3, due t o the Second Korn inequality and the estimate (3.13), we find that the bilinear form in the left-hand side o f (3.28) satisfies
I. Some mathematical problems o f the theory of elasticity
38
the conditions of Theorem 1.3. Thus there is a unique element u E H such that the integral identity (3.28) holds for all v E H . For v E
R the left-hand
side o f (3.28) is equal t o zero due t o the fact that L ( v ) = 0 in R , o ( v ) = 0 on dR; the right-hand side o f (3.28) is also equal t o zero for v E
R,since
we have assumed that conditions (3.30) are satisfied. Therefore the integral identity (3.28) holds for all v
E H 1 ( R ) ,which means that u ( x ) is a solution
of problem (3.27). Estimate (3.31) can be obtained from (3.28) for v = u , the Second Korn inequality and (3.13), (3.29). Theorem 3.5 is proved. Remark 3.6. In Theorem 3.5 we can choose a solution u(x)orthogonal in L 2 ( R )or H 1 ( R ) to the space of rigid displacements
R. For such u ( x ) we
have the following
estimate
where the constant C2(R)depends on the same parameters as the constant
C l ( R ) in (3.31). This fact is due t o the Second Korn inequality (2.14) (see Theorem 2.5). Remark 3.7. Similarly t o the case of the Dirichlet problem one can prove the smoothness of weak solutions o f the Neumann problem, provided that the coefficients a f / ( s ) , the boundary of R, and the data cp,
f",
i = 0, ..., n , in (3.27) are smooth (see
[I711. 3.5. The Mixed Boundary Value Problem In a bounded domain R C Rn we consider the following boundary value problem for the operator
C of class E ( n l ,n 2 ) ,n l , n2 > 0:
53. Boundary value problems o f linear elas ticity
where
fj
E
...,n , cp E LZ(S1),
L Z ( R ) ,j = 0,1,
is the unit outward normal to
E H 1 f 2 ( y ) v, = ( v l , ..., v,)
dR.
Before giving a definition of a solution of the mixed problem we impose the following restrictions on
aR, y, S1, SZ.
1. d R = 7 U $ U S2 and y, S1, S2 are mutually disjoint subsets of d R .
2. R is a domain with a Lipschitz boundary d o , y contains a subset satisfying the conditions of Theorem 2.7. Note that all further results are also valid under weaker assumptions on and
dR
y which guarantee the inequalities (1.2), (2.14).
We define a weak solution of problem (3.33) as a vector valued function
u E H 1 ( R )satisfying the integral identity
for any v E
H 1 ( R , y ) ,and such that u = iP on y (i.e. u - E H 1 ( R , y ) ) . ~ l / ~ (wey can ) consider @ as an element of
Note that by the definition o f
H1(R). Theorem 3.8. There exists a weak solution and satisfies the estimate
u ( x ) of problem (3.33). This solution is unique
I. Some mathematical problems of the theory of elasticity
40
where the constant C ( 0 ) depends only on 6 1 , K ~ the , constant in (3.29) and the constants in the Korn inequality (2.14) for vector valued functions in
H 1 ( R ,-y) (see Theorem 2.7).
Proof.
From (3.34) we conclude that w = u - @ must satisfy the integral
identity
for any v E H 1 ( R , y ) . Due t o Proposition 3 of Theorem 1.2 the inequality (3.29) holds for a l l v E H 1 ( R ,-y), and according t o Theorem 2.7 the inequality (2.14) is also valid for such v. Inequalities (2.14) and (3.13) show that the bilinear form in the left-hand side of (3.36) satisfies all assumptions of Theorem 1.3 with H = H 1 ( R ,-y). By virtue of (3.29) the right-hand side of (3.36) defines a continuous linear functional on H ' ( O , y ) . It follows from Theorem 1.3 that there is a unique element w E H1(O,-y) satisfying the integral identity (3.36). Obviously u = w+@ is the solution of problem (3.33). Let us prove estimate (3.35). Setting v = w in (3.36) by virtue of (2.14) and (3.13), we have
Therefore taking into account (3.29) for v = w ,we find that
Therefore
$3. Boundary value problems of linear elasticity
41
since w = u - 9 . Note that in the proof of the last estimate we can replace 9 by any that 9 -
& such
6 E H1(R,y), and this would not affect t h e constant C3 which does
not depend on 9. Thus by the definition of the norm in Theorem
3.8 is proved.
H ' / ' ( ~ we ) obtain (3.35) from (3.37). •
I. Some mathematical problems of the theory of elasticity
42
$4. Perforated Domains with a Periodic Structure. Extension Theorems
4.1. Some Classes of Perforated Domains
Rn with a 1-periodic structure, i.e. w is invariant under the shifts by any z = (zl, ..., z,) E ZZn. Let w be an unbounded domain of Here we also use the notation:
Q={x
:
O < x j < l , j = 1 , ..., n ) ,
p(A, B ) is the distance in Rn between the sets A and B, E is a small positive parameter. In what follows we shall mainly deal with domains w satisfying Condition B (see Fig. 1): B1 w
- is a smooth
unbounded domain of
Rn with a 1-periodic structure.
B2 The cell o f periodicity w n Q is a domain with a Lipschitz boundary.
B3 Theset Q\G and the intersection o f Q\w with the 6-neighbourhood (6
< i)
o f dQ consist of a finite number of Lipschitz domains separated from each other and from the edges of the cube Q by a positive distance.
Fig..
$4. Perforated domains with a periodic structure We shall consider two types of bounded perforated domains
43
Re with a pe-
riodic structure characterized by a small parameter e.
A domain Re of t y p e I has the form (see Figs. 1, 2, 3):
R is a bounded smooth domain o f Rn,w is a domain with a 1-periodic structure satisfying the Condition B; Re is assumed t o have a Lipschitz bound-
where
ary.
R
Fig.
Fig.. The boundary of a domain Re of type I can be represented as dRc = I',US,, where
r, = d R n ew, Se = (dRe) n R.
A domain Re of t y p e II has the form (see Figs. 4, 5a, 5b):
I. Some mathematical problems of the theory of elasticity
44
where
R
is a bounded smooth domain.
TEis the subset of Znconsisting of all z such that
E
is a small parameter.
Fig.. 0;
Fig. 5a. Q1
Fig. 5b.
$4. Perforated domains with a periodic structure We assume that
45
R1, R;, RE (the sets of interior points o f
nl, n;, a')
are
bounded Lipschitz domains. The boundary 8Rc of a domain REo f type II is the union o f d f l and the surface SE c R of the cavities, S, = ( d V )
4.2.
n R.
Extension Theorems for Vector Valued Functions in Perforated Domains
In order t o estimate the solutions o f the above boundary value problems for the system of elasticity in perforated domains
RE we shall construct
exten-
sions t o R of vector valued functions defined in REand prove some inequalities (uniform in
E)
for these extensions.
Lemma 4.1. Let G c 2)
c Rn and let each of the sets G , V , V\G
be a non-empty bounded
Lipschitz domain (see Fig. 6). Suppose that y = ( 8 G ) n V is non-empty. Then for vector valued functions in
P
:
H~(v\G) there is a linear extension operator
H'(D\G) + H 1 ( V ) such that
where the constants cl, ..., c4 do not depend on w E H ~ ( D \ G ) .
Fig..
I. Some mathematical problems of the theory of elasticity
46
Proof.
Let us first show that each w E H~('D\G) can be extended as a
function
6 E H 1 ( V ) satisfying the inequality
with a constant c independent o f w. Indeed, consider the ball
B c Rn containing
a neighbourhood o f the
set V . According t o Proposition 2 of Theorem 1.2 the function w can be extended from 'D\G t o the entire ball B as a function w1 E H 1 ( B ) . Taking the restriction of w1 on V we get a function 6 which satisfies the inequality (4.9). Denote by W the weak solution o f the following boundary value problem for the system of elasticity
where
C is an arbitrary operator o f class E(n1, K Z ) with
constant coefficients.
Note that the last boundary condition in (4.10) should be omitted if d G n d V =
0.
By Theorem 3.8 W exists and satisfies the inequality
Therefore due t o (4.9) we obtain
Set
P(w)=
w(x)
for x E V\G ,
W(x)
for x E G .
It is easy t o see that P ( w ) is a vector valued function in H 1 ( V ) . By virtue of (4.10) we have Pv = 17 for any q E
R. Taking into
account (4.11) and the
Korn inequality (2.3) in DIG (see Theorem 2.4) we conclude that estimates (4.5), (4.6) hold with constants cl, cz depending only on G and 'D. Let us prove the estimate (4.8) for Pw. Suppose that (4.8) does not hold. Then there is a sequence o f vector valued functions vN E H~(v\G) such that
47
54. Perforated domains w i t h a periodic structure
I I P v N I I ~ ~5 ( vci) l l v N I I ~ l ( v \ ,~ )
(4.13)
Ile(PvN)II~2(v) 2 N lle(vN)II~2(v\~) ,
(4.14)
but
Without loss o f generality we can assume that
I
(vN,r])dx= 0 for any rigid
V\G
displacement q, since P ( v
t r])
= Pv
+ r] due t o (4.10),
any bounded domain wo and any v E H1(wo)we have
J
(4.12), and for
le(v
+ q)I2dx =
wo
1le(~)(~dx.
By (4.15) and the Second Korn inequality (2.14) in D \ G (see
wo
Corollary 2.6) we get
Thus vN
N
-+ CCJ
-+
0 as N
-+
m in
H'(D\G), and therefore I I P v ~ I I ~ --, I ( 0~ )as
due t o (4.13). On theother hand, (4.14) implies that I l e ( P ~ ~ ) 1 1 2 ~2(~)
1. This contradiction establishes the inequality (4.8).
1
To prove (4.7) we choose a constant vector C such that
P ( w + C)dx = 0. Because of the Poincarh inequality (1.5) in D\G
it
V\G
follows from (4.5) that
Therefore (4.7) is valid since V C = 0, PC = C . Lemma 4.1 is proved.
Theorem 4.2 (Extension o f functions in perforated domains of type 11). Let Re be a perforated domain of type II. Then for vector valued functions in
H 1 ( R c )there is a linear extension operator P, : H 1 ( R c )-+ H 1 ( R )such that
I. Some mathematical problems of the theory of elasticity
48
for any v E H 1 ( R c ) where , the constants q ,..., c4 do not depend on E ,v .
P,,,f.Let v ( x ) E H1(Rc). Set V ( J )= v ( E [ )and
fix z E
T,, where Tc is
the index set in the definition o f a perforated domain Rc o f type II (see (4.3)). Consider the function V ( [ )in the Lipschitz domain w
n ( z + Q ) . By
4.1 one can extend V ( J )as a vector valued function PIV E H 1 ( z
Lemma
+ Q ) such
that
Extending V ( [ )in this way for every z E T, we get a vector valued function
PIV which satisfies the inequalities (4.21) for any z E Tc with constants 1 6 , ..., IC3 independent o f z. If the distance between Q\G and dQ is positive (i.e. Q\G lies in the interior of cube Q ) , then the function ( P ~ v ) ( is~the ) extension whose existence is x asserted by Theorem 4.2, and therefore we can take ( P c v ) ( s )= ( P , v ) ( - ) . E where V ( J )= v ( E [ ) . However, if Q\L;) has a non-empty intersection with dQ (as in Fig. I), the function P I V ( J ) may not belong t o HI(&-'R), since its traces on the
49
$4. Perforated domains with a periodic structure adjacent faces o f the cubes z
+ Q, z E T,, do not necessarily coincide.
In a
neighbourhood o f such faces we shall change PIV as follows. For 1 = 0,1 set &Q =
U
{ t E dQ,
tk= I}.
k=l
Due t o Condition 63 on w the intersection of the 6-neighbourhood of d Q with Q\W
consists o f a finite number o f Lipschitz domains separated from
each other and from the edges o f Q by a positive distance larger than some
61 E (0,1/4). For 1 = 0 and 1 = 1 denote by
those o f the domains
just mentioned whose closure has a non-empty intersection with d,Q (see Fig.
7). Therefore each ~f lies in the 6-neighbourhood of dQ and is adjacent t o a face o f Q lying on the hyperplane tk= Ifor some Ic.
1 -.p: d
.__I
Fig. 7. -
T, E Znbe the same as in the definition of a perforated domain Re of type II (see Figs. 4, 5a, 5b). Denote by T,' the set of z E T, such that (T,! + z) n d ( ~ - l O ~#) 0 for some j = 1, ...,ml. The extension PIV(t) constructed above is such that P I V E H1(g) for + z, any open g C e-'R which has no intersection with any of the domains z E T,, yj+ z , z E T,'. Let us change P I V in these domains so as t o obtain a function in HI(&-'a). Let the domain R1 and the set
Fig.. The domains G1, ...,GN are shaded
pale.
I. Some mathematical problems of the theory of elasticity
50
GI, ..., GN all mutually non-intersecting domains having the $ + z , z E T, or yi + z , z E T,' (see Fig. 8). Obviously p(G,, Gt) > 61 for s # t. The number N tends t o infinity as E t 0, however, GI, ..., GN are the shifts o f a finite number o f bounded Lipschitz domains. Denote by
form either
PIV(e). We have constructed the sets GI, ..., GN in dG1 U ... U dGN contains all those parts o f the faces of the cubes z + Q, z E T,, where the traces o f PIV(J) may differ. Set Go = G1 U ... U GN. Then one clearly has PIV E H'(E-'R\Go). Denote by G~ the 61/2-neighbourhood o f Gj. By virtue of Lemma 4.1 let us extend PIV t o each of the sets G j as a vector valued function P2V satisfying the following inequalities Consider the extension
such a way that the set
IIVCP~VIIL~(G,) I M3 IIVCP~VIIL~(G,\G,~ , Ilec(P~V)11~2(c,) 5 M4 I ~ ~ c ( P ~ V ) I I L Z ( C 7 ,\G~) P2q = q if q E R,where the constants Ml, ...,M4 do not V, j . Set U(J) = (PIV)(J) for J E (c-'R)\G0, U(J) = (P2V)(J) for J E GO.
and such that depend on
Applying the estimates (4.21). (4.22) we finally conclude that taken as the extension
u(-)2&
can be
(Pev)(x)satisfying the conditions (4.21). Theorem 4.2
is proved. Theorem 4.3 (Extension of vector valued functions in perforated domains of type 1). Let Re be a perforated domain o f type I and let that
Ro be a bounded domain such
fi C 00,p(dRO,R)> 1. Then for every sufficiently
a linear extension operator PC :
small
E
there exists
H1(RE,rE)-+ H,'(Ro) such that
$4. Perforated domains with a periodic structure
for any u
E H1(Rc,re),
C1, C2,C3 do not depend on E , u . Moreover, (P,u)l, = 0 for any open g such that g C Ro\R, if E is suffi-
where the constant ciently small.
Let
Znsuch that ~ ( +zQ n w ) n 52 # 0. ~ (+zQ n w), and let fil be the interior of
Denote by Tc the set of all z E
Proof.'
@
be the interior o f
U
zETe
U
E(Z
+ Q ) . For each u E H1(RC,r,)we introduce the following vector
PET*
valued function
u(x)
,
XERC,
0
,
XE~;\R,
0
,
x E Ro\fil
.
It is easy t o see that U ( x ) E ~ ' ( f i f )According . t o Theorem 4.2 one can extend U ( x ) t o the domain Ro.
P,u =
Denote this extension by P ~ U and , set
Feu. Obviously the conditions (4.23)-(4.25)
are satisfied. The last
statement of the theorem holds since Pcu = 0 in Ro\fil.
Theorem 4.3 is
proved.
•
4.3. The Kern Inequalities in Perforated Domains In this section we prove the Korn inequalities (with constants independent of E ) for perforated domains Re of types I and II. These results are widely used in Chapter II for the homogenization of various elasticity problems. Theorem 4.4 (Korn's inequalities in perforated domains of type 11). Let Rc be a perforated domain o f type II. Then for any vector valued function u E H1(Rc)the inequality
he proof is based on the extension of a function u from H1(Qc,r,) by u = 0 outside il and the subsequent application of Theorem 4.2 in a new perforated domain which is different from that of Theorem 4.2 but is also of type 11.
I. Some mathematical problems o f the theory o f elasticity
holds with a constant C independent o f u, E . Moreover, if one of the following conditions is satisfied
(u,~)Hl(n.)=O, V q E R , or
(u,71)~2(nr) =0
, Vq E R ,
IIuIIHl(ne)5 Cl
~ ~ ~ ( u ) I I L ~ ( S 3Z ~ )
then
where the constant Cl does not depend on u , E .
Proof. The estimate (4.26) in
R
immediately follows from the Korn inequality (2.3)
(see Theorem 2.4) and the extension Theorem 4.2. Indeed, let P, be the
extension operator constructed in Theorem 4.2. Then
Suppose now that u ( x ) satisfies (4.27). Then
E R . Let P,u E H 1 ( R ) be the extension of u constructed in Theorem 4.2. Denote by qo the orthogonal projection of P,u on R with respect t o the scalar product in H 1 ( R ) . Then
for any rigid displacement q
Due t o the Corollary 2.6 we have
~ zvirtue ( ~ ) .o f (4.30) and Theorem since I(e(Pcu- q0)llL2(n)= I ~ ~ ( P , u ) ~ (By 4.2 the last inequality yields
54. Perforated domains with a periodic structure
L Cq lle(Pcu)Ili2(n) 5 C5 lle(u)112L2(n*). Suppose that (4.28) is satisfied. Then
I I u I I ~ ~ (5~ IIu c ) - r111iz(n*) ,
V7) 'rl 7 2 .
Choosing 7 = q0 such that (4.27) holds for u
(4.32)
- qo, we
obtain by (4.29) for
u - qo, that
Therefore, IIuIIZ2(n*)L C6 Ile(u)II22(n*) by virtue o f (4.32). This inequality together with (4.26) implies (4.29) for vector valued functions u(x) satisfying (4.28).
Theorem 4.4 is proved.
Let us now prove the Korn inequality i n a perforated domain
I for vector valued functions in H1(Rc) vanishing on
0' o f type
re.Note that
Theorem
2.7 provides an inequality of this kind with a constant which may depend on
E,
however, in what follows we need the inequality with a constant independent of
E.
Theorem 4.5. Let
W
be a perforated domain of type I. Then for any vector valued function
v E H'(Rc, r c ) the inequality
is valid, where C is a constant independent of
Proof. Let v E H1(Rc,I',)
E
and v.
and denote by P,v E H i ( n o ) the extension o f v t o
the domain Ro constructed in Theorem 4.3. Due t o Theorem 2.1 the vector valued function Pcv satisfies the Korn inequality of type (2.2) in n o . Therefore by (4.25) we have
I. Some mathematical problems of the theory o f elasticity
IIvIIHl(n*)5 IIPcvII~l(no)5 cl Ile(Pcv)IIL2(no)5
5 C2 lle(v)Il~2(n~) , where the constants
C1, Cz do
not depend on
E,
v. Theorem 4.5 is proved.0
Directly from Theorem 4.2 and Proposition 3 of Theorem 1.2 we obtain Lemma 4.6. Let
Re
for any
be a perforated domain of type II. T h e n
v E H 1 ( R c ) ,where C
is a constant independent o f E ,
v.
$5. Estimates for solutions of boundary value problems of elasticity
55
$5. Estimates for Solutions o f Boundary Value Problems o f Elasticity in Perforated Domains
In $3 existence and uniqueness o f solutions for the main boundary value problems of linear elasticity were established together with the estimates of these solutions through the norms o f the given functions. If the domain occupied by the elastic body or the coefficients o f the system depend on a parameter E,
the constants in these estimates may depend on
E.
In this section we show
that for perforated domains Rc defined in $4 the constants in estimates of type (3.31), (3.35) can be chosen independent o f
E,
provided that the coefficient
matrices o f the elasticity system belong t o the class independent o f
E ( n l , n 2 ) with n l ,
KZ
E.
5.1. The Mixed Boundary Value Problem
R" be a perforated domain of type I (see (4.1)), dRc = Sc U rC,where S, is the surface o f the cavities, S, = R n d R c , I?, = d R n d R c . Let
Consider the following boundary value problem
L 2 ( R c ) ,j = 0, ...,n , E H 1 ( R ' ) , L is an elasticity operator of type (3.1) belonging t o the class E ( n l , n 2 ) . where f j E
In the general situation this problem was considered in $3 (see Theorem 3.8). The next theorem represents a more precise version o f Theorem 3.8 for perforated domains
RE.
Theorem 5.1.
RE be a perforated domain o f type I and let the coefficient matrices o f the L belong t o the class E ( n 1 ,n 2 ) with constants n l , n2 > 0 independent o f E . Then there exists a weak solution u ( x ) of problem (5.1), which is Let
operator
unique and satisfies the inequality
I. Some mathematical problems o f the theory of elasticity
where C is a constant independent o f
E.
m. Existence and uniqueness o f the solution o f problem (5.1) mediately from Theorem 3.8 with S1 =
0, S2 = Sc, y
follow im-
= r e . As stated in
Theorem 3.8, the constant C in (5.2) depends only on tcl, K Z ,and the constant in the Korn inequality (4.33) for vector valued functions in H1(R',
re).
According t o Theorem 4.5 the last constant can be chosen independent of
E,
C which is also independent o f
E.
and therefore (5.2) holds with a constant Theorem 5.1 is proved. Remark 5.2.
Every vector valued function f 0 E L2(Rc)defines a continuous linear functional I(v) on H1(RC,I',)by the formula l(v) =
( f O , ~ ) ~ z ( ~ . ) Denote .
by
11 fOII* the norm o f this functional in the dual space ( ~ ' ( f l 're))*. , Then
11
11
Obviously fO)l* 5 fO1lL~(ne).It follows from the proof of Theorem 3.3 that we can replace the estimate (5.2) by
5.2. Estimates for Solutions of the Neumann Problem in a Perforated Domain In a perforated domain Re o f type II consider the second boundary value problem o f elasticity
afi
L ( u ) = p + - in Rc
,
o ( u ) = (P + V;fi o n aR
, a ( u ) = u;f' on Sc ,
dxi
(5.5)
§5. Estimates for solutions of boundary value problems of elasticity
57
where
In contrast t o Theorem 3.5 the next theorem establishes estimates uniform in
for the solutions of problem (5.5).
E.
Theorem 5.3. Let RE be a perforated domain o f type 11, and
for any rigid displacement 7 E the operator E.
L
R. Suppose
that the coefficient matrices of
belong t o the class E(lcl, K Z ) with
~
1~2 ,
> 0 independent
of
Then problem (5.5) has a unique solution u ( x ) such that
( u , ~ ) ~ l ( n . ) = O ,V v E R , and
where C is a constant independent o f e.
Proof. Existence and
uniqueness o f a solution of problem (5.5) follow from
Theorem 3.5 and Remark 3.6. We also have the estimate (5.9) for u ( x ) with a constant C depending only on ~ 1nz, and the constant in the Second Korn inequality (4.29), which does not depend on constant independent of
E.
E.
Therefore (5.9) holds with a
Theorem 5.3 is proved.
In order t o study the spectral properties o f the Neumann problem o f type (5.5) (see Ch. Ill) we shall need the following auxiliary boundary value problem in the domain R" of type II:
I. Some mathematical problems of the theory of elasticity
58
L 2 ( R c ) ,j = 0, ...,n, cp E L 2 ( a R ) ,the matrices A h k ( x ) belong t o E ( / c l ,/ c 2 ) , p(x) is a bounded measurable function in Rc such that
where f j E the c l a n
We say that
u ( x ) is a weak solution of problem (5.10) if u ( x ) E H'(Rc)
and the integral identity
H'(Rc). a ( u , w)the bilinear form in the left-hand side o f (5.12). This form satisfies all conditions o f Theorem 1.3 for H = H 1 ( R ' ) with constants c l , holds for any w E Denote by
c2 independent o f E. This fact is due t o the Korn inequality (4.26). Therefore
existence, uniqueness and estimates of solutions o f problem (5.10) are proved on the basis o f (5.12) in the same way as Theorems 3.5, 3.8. We have thus established Theorem 5.4. Let Rc be a perforated domain o f type II, and let the family of matrices A h k ( x ) ,
h, k = 1, ...,n , belong t o the class E ( n l , K ~ ) .Suppose that conditions (5.11) are satisfied and the constants Q , c , , n l , nz do not depend on E. Then problem (5.10) has a unique solution u ( x ) , and this solution satisfies the estimate
where C is a constant independent o f
E.
59
$6. Periodic solutions o f boundary value problems $6. Periodic Solutions o f Boundary Value Problems for the System of Elasticitv
To study homogenization problems for the system of elasticity we need existence theorems for some special boundary value problems.
6.1. Solutions Periodic in All Variables Let w be an unbounded domain with a 1-periodic structure, which satisfies Condition B of $4, Ch. I. Consider the following boundary value problem
w is 1-periodic in
x
,
I
wdx=O,
Qnw
I
where the vector valued functions F j ( x ) are I-periodic in x , F j E L2(w n Q ) ,
j = 0, ...,n , the family of matrices A ~ ~ (belongs x) t o the class E ( K , ,rcz) and x.
their elements a f i ( x ) are 1-periodic in
W e define a weak solution of problem (6.1) as a vector valued function w
E w ~ ( w )such that
I
w d z = 0 , and the integral identity
Qh
=
/
dv [ ( F m , -)
Qnw
ax,
- (PO,v)] d.
holds for any v E W ; ( W ) . Theorem 6.1.
F O d x = 0 . Then problem (6.1) has a unique solution, and this solu-
Let J
Qn w
tion satisfies the estimate
I. Some mathematical problems of the theory of elasticity
60
where the constant C depends only on nl, K * , w. The proof o f this theorem rests upon Theorem 1.3 and is quite similar t o the proof o f Theorem 3.5. In this case one should take as H the space o f vector valued functions v E
W;(W)
furnished by Theorem 2.8.
such that
J
v d x = 0; the Korn inequality is
Qnw
In what follows we shall often use the fact that solutions o f problem (6.1) are piece-wise smooth, provided that the coefficients of the system (6.1) and the functions F j , j = 0, ...,n , are piece-wise smooth and may loose their smoothness only on surfaces which do not intersect dw. Let us consider these questions more closely. We assume that there are mutually non-intersecting open sets Go,..., G , with a 1-periodic structure and such that G j
c w,j
= 0,1, ...,m; G j n d w = 0 ,
j = 1, ..., m ; Go = w\(G1 U ... U G,); GI, ..., G , have a smooth boundary. We say that a function cp which is 1-periodic in x belongs t o class 6 (cp is called piece-wise smooth in w and smooth in a neighbourhood o f d w ) if cp has bounded derivatives of any order in G j , j = 0,1, ...,m. Theorem 6.2. Let w ( x ) E W ; ( W ) be a weak solution o f problem (6.1), and suppose that the
6. Then w also belongs t o w is piece-wise smooth in w and smooth in a neighbourhood of dw.
elements of A h k ( x ) ,F j ( x ) belong t o class
Proof.The smoothness of w
6, i.e.
in a neighbourhood o f dw follows from the gen-
eral results on the smoothness of solutions of the elasticity system near the boundary (see [17]). Let x0 E d G j , xO
6 dw, and consider the set G j n { x
:
lx - xO1< 6 ) =
q;(xo). It is shown in [17], Section 13, Part I, that for sufFiciently small 6 the function w has bounded derivatives of any order in qj6(x0). The smoothness of w at the interior points of w , which do not belong t o d G j , is also proved in (17). Therefore w E d .
$6. Periodic solutions o f boundary value problems
6.2. Solutions of the Elasticity System Periodic in Some of the Variables Let the coefficient matrices A h k ( x )of the differential operator t o the class E ( K , ,K (51,
~ ) and ,
C
belong
suppose that their elements are 1-periodic in ? =
..., xn-1).
In this section w is an unbounded domain with a 1-periodic structure, which satisfies the Condition B of $4 (see Fig. I), the domains w ( a , b) and ;(a, b) are defined by (1.6). Set
Let gt be a non-empty open set belonging t o f i t and invariant with respect t o the shifts by any vector z = ( z l ,...,
0 ) E Z n . Set
Fig.. Consider the following boundary value problem
w is 1-periodic in
P,
wdx=O, O(5.b)
where $. $I,,
F are vector valved functions 1-periodic in 2 . Fj E L2 (;(a, b ) ) ,
j = 0 ,..., n ; $, E L 2 ( g , ) , $ b E L2(gb), 0 I a < b < m, un = -1 on g,, vn = 1 on gb. The domain &(a,b) is assumed t o have a Lipschitz boundary.
I. Some mathematical problems o f the theory o f elasticity
62
We define a weak solution of problem (6.6) as a vector valued function
w E H 1 ( u ( a ,b)) such that for any v E ~ l ( u ( ab ,) ) the following integral identity is valid:
Theorem 6.3. Let
Then there exists a weak solution w o f problem (6.6), which is unique, and w satisfies the estimate
where C is a constant depending only on w, a , b, nl, n2. This theorem is proved in a similar way t o Theorem 3.5. In this case we take as H the subspace o f v such that
/
B1( u ( a , b ) ) formed
by all vector valued functions
vds = 0. Then the Second Korn inequality follows from
Theorem 2.9. To estimate the right-hand side of (6.7) we should use the inequality
§6. Periodic solutions of boundary value problems
63
which holds due to Proposition 3 of Theorem 1.3 and the Korn inequality (2.19). Let us also establish the existence and uniqueness of the solution of the following mixed boundary value problem:
a(?),$a(?).
F j ( x ) . j = 0 , ...,n,are 1-periodic in 2, F j E L ~ ( sb )() , ~ , +b E L2(ib), E ~ ' / ~ ( d . ) . A vector valued function w is called a weak solution of problem (6.11), if w E ~ l ( w ( a6 ,) ) . w = Q on d., and the integral identity where
is satisfied for any v E
H'(@(a,6)) n H 1 ( B ( a ,b), d.).
Theorem 6.4. There exists a weak solution w ( x ) o f problem (6.11), which is unique and satisfies the inequality
where C is a constant depending only on w , K , , ~ 2a,, b.
I. Some mathematical problems of the theory of elasticity
64
Proof.
By virtue o f Theorem 2.7 the Korn inequality (2.19) holds for any (w(a,b)) n ~ ' ( 3 ( ab), , i.) (i.e u = 0 on 9.). Moreover, it follows from Proposition 3 of Theorem 1.2 and the Korn inequality, that v
E
~l
Taking into account the inequalities (2.9), (6.14) and following the proof of Theorem 3.8, we establish the existence o f the solution of problem (6.11) and the validity o f the estimate (6.13).
6.3. Elasticity Problems with Periodic Boundary Conditions in a
Perforated Layer In this section
Re denotes the perforated layer
where w is an unbounded domain with a 1-periodic structure, w satisfies the Condition B of 54, d = const
2
1 is a parameter,
C-'
is a positive integer.
Set
In
Re consider the following boundary value problem:
The coefficient matrices of operator
L are assumed t o be o f class E(n1, nz),
their elements are functions 1-periodic in 2 , f j , iP1, iP2 are also 1-periodic in
$6. Periodic solutions of boundary value problems
We define a weak solution o f problem (6.15) as a vector valued function
u E ~ ' ( f l ' )such that u = (P1 on
ro, u = (P2 on rd and u satisfies the integral
identity
v = o on for any v E H1(nC),
rou rd(i.e.
v E H1(nc) n~
l ( f i ~ , F uO Fd)).
Theorem 6.5. There exists a weak solution u ( x ) o f problem (6.15) which is unique. Moreover, u(x) satisfies the inequality
where C is a constant independent o f
E.
This theorem can be proved similarly t o Theorems 3.8 and 6.4 by virtue of the following Lemma 6.6. Every vector valued function v E
H1(ne) vanishing on r o U r d satisfies the
inequalities
where the constants C1 and C2 do not depend on
E,
d, v.
Proof. This lemma is established by the same argument as Theorems 4.2,
4.3
and is also based on the construction o f suitable extensions o f vector valued functions defined in RE. Let v E H'(s~'), v = 0 on
roU rd.We extend v t o ~w as follows
I. Some mathematical problems of the theory of elasticity
Set
By analogy with the proof o f Theorem 4.2 we can extend 6 t o the entire layer
B as a function p v E H ' ( B ) such that I?v = 0 for
rc,
= -1, z, = d
+ 1,
and
It is shown below that
Inequalities (6.19)-(6.21)
imply (6.17), (6.18).
To complete the proof o f Lemma 6.6 let us outline the method t o obtain (6.20), (6.21). Obviously (6.20) is a kind o f Friedrichs' inequality, which holds since
for any w E
C ~ Bsuch ) that w(i,-1) = 0.
The estimate (6.21) is similar t o the First Korn Inequality. It can be proved in the same way as (2.2) in Theorem 2.1. To this end we approximate P v by a
w m ,which are 1-periodic in i and vanish in a neighbourhood of the hyperplanes x, = -1, x, = d + 1. Then, similarly t o the proof of Theorem 2.1, we integrate by parts over B taking into account the 1-periodicity of w m in i. sequence o f smooth vector valued functions
§ 7. Saint - Venant 's principle for periodic solutions 57. Saint-Venant's Principle for Periodic Solutions o f the Elasticity System Initially formulated in 1851 Saint-Venant's Principle has ever since been widely used t o study various theoretical as well as practical problems in mechanics. The mathematical expression o f Saint-Venant's Principle, its applicability and formal justification were and still are the subject o f intensive research (see e.g. [94], [133], [126], [37], [153]). Roughly speaking St. Venant's Principle asserts that if the forces statically equivalent t o zero are applied t o a part V of the body contained in a subdomain V' o f
R
R,
then the energy
is small, provided that the distance between
V' and V is sufficiently large.
Fig. 10. In the case of an elastic cylinder St. Venant's Principle implies that if the applied forces are nonvanishing only on an end-face of the cylinder and the mean values of these forces and of their moments are equal t o zero, then the solution of the corresponding boundary value problem has the form of a boundary layer near the end-face. In this book the asymptotic properties of solutions o f the elasticity system, which are closely related t o Saint-Venant's Principle, will be used t o construct boundary layers for the asymptotic expansions of solutions o f the elasticity system with rapidly oscillating periodic coefficients.
7.1. Generalized Momenta and Their Properties In this section w is an unbounded domain with a 1-periodic structure satisfying the Condition B of $4. We introduce the following notation
I. Some mathematical problems of the theory of elasticity
S ( a , b) = (dw) fl { x : a
< x , < b} ,
The coefficient matrices A h k ( x )o f operator class E ( n l , n 2 ) , n l , nz = const
> 0 , and their
C
are assumed t o belong t o
elements a;hjk(x)are functions
P = ( z l ,...,x,-1). A vector valued function u ( x ) is called a 1-periodic in i solution of the
1-periodic in system
with the boundary conditions
if u E ~ l ( w ( t ~ , tand ~ ) for ) any u E ~ ' ( w ( t , , t ~ such ) ) that v = 0 on
r,, U rt, the following integral identity holds:
! (A"
au av
-, -)dx ax,
ax,
=
oi(t1,tz)
A vector valued function u ( x ) is called a weak 1-periodic in 2 solution of system (7.2) in w ( 0 , m ) with the boundary conditions (7.3) on S ( O , m ) , if u ( x ) is a weak 1-periodic in P solution of (7.2) with the boundary conditions (7.3) for every t l , t2 such that 0 5 tl < tz < m. It is assumed that fl E ~ ~ ( L j ( t ~ ,j t = ~ 0) ,).., , n , 0 5 tl < t 2 < m , and the vector valued functions f j are 1-periodic i n P. Note, that if t 2 = rn then the functions f j may not belong t o L2 ( L j ( t 1 , m ) ).
5 7. Saint- Venant 's principle for periodic solutions
69
u ( x ) of system (7.2) in w(tl,t2)with the boundary conditions (7.3) on S ( t l ,t 2 ) we introduce the vectors P ( t , u ) , For a weak 1-periodic in 2 solution
which are called generalized momenta, setting
Existence of
P(t, u ) follows from
Lemma 7.1. Suppose that the vector valued function
f" is such that
u ( x ) is a weak 1-periodic in f solution o f system (7.2) in w ( t l , t z )with S ( t 1 , t ~ ) Then . the generalized momenta P ( t ,u ) satisfy the following conditions
and
the boundary conditions (7.3) on
P ( t ,U ) = slim s-' -++~
J
au
&(t,t+s)
P(tl',u ) - P(tl,u ) =
/
Ank axk dx =
f 0 dx =
3(t1,t11)
where
J r,,
f" d i +
/
fn
di
,
(7.8)
rill
tl < t' < t" < t2.
Proof. If the coefficients of system
(7.2), the functions f j , j = 0, ..., n , and
u ( x ) are sufficiently smooth, the relations (7.7) are obvious, and integration by parts directly results in (7.8). Consider now a weak solution u ( x ) . Let e l , ...,en be the standard basis o f Rn. Take v = t9(xn)erin the integral identity (7.4), where 29(xn)is a continuous scalar function such that 6 ( t ) = 1,
70
I. Some mathematical problems o f the theory o f elasticity
29(xn) = 0 for t l < xn < t - hl and for t + h2 < xn < t 2 , d ( x n ) is linear + h 2 ] ,h l , h2 being sufficiently small
on each of the segments [t - h l , t ] , [t,t
positive constants. Then due t o (7.4) we have
( f O , v ) d x- h;'
= 3 ( t - h 1 ,t+hz)
J
( f " , e r ) d x + h;'
&(t-hl , t )
J
( f n , e r ) d x,
G(t,tthz)
It follows due t o (7.6) that the first and the second integrals in the left-hand side of this equality have finite limits as hl + $0 or h2 + +O respectively. Making hl tend t o zero in (7.9) and then making h2 tend to zero, we obtain (7.7). Let us prove (7.8). Set v = dl(xn)erin the integral identity (7.4), where 29' is a continuous function such that d l ( t l )= dl(tll) = 0, 29 = 1 on (tl+h, t"-h),
d(x,) is linear on [t', t'
+ h] and on [t" - h , t ] , h > 0 is sufficiently small.
It
thus follows from (7.4) that
This relation together with (7.6) yields (7.8). Lemma 7.1 is proved.
If the functions f j , j = 0 , ..., n, and u as well as the elements o f matrices Ahk are sufficiently smooth, it is easy t o see that
In the rest of Chapter I it is assumed that for systems o f type (7.2) conditions (7.6) are always satisfied for every t E ( t l ,t 2 ) .
$7. Saint-Venant 's principle for periodic solutions 7.2. Saint- Venant 's Principle for Homogeneous Boundary Value Problems O f primary importance in Continuum Mechanics is Saint-Venant's Principle for bodies of cylindrical type with the conditions u(u) = 0 on the lateral part of the boundary. The details concerning this case can be found in [94]. In applications t o the theory of homogenization it is necessary t o have estimates which express Saint-Venant's Principle for various boundary value problems with periodic boundary conditions. Theorem 7.2 (Saint-Venant's Principle). Let s, h be integers such that s
> h > 0,
and let u(x) be a weak 1-periodic
in 2 solution of the system
with the boundary conditions
Let P ( s
+ 1,u) = 0. Then
where A is a positive constant independent o f u, s h; A depends only on 2 - hk bui duj h ( 0 , l ) and the coefficients of (7.10); I&(u)l - a . . - -. " dxh dxk
Proof. S e t g = 3 ( s - h , s + l + h ) , g l
=3(s-h,s),g2
=L(s+l,s+l+h).
Let {urn) be a sequence of vector valued functions in ~ ' ( u ( 0 , m)) 1-periodic in 2 and such that urn -+ function a(),.
E [s + 1,s
m. We define the scalar
setting @(xn) = exp [A(x. - (s - h))] for x,
@(xn) = exp(Ah) for x, x,
u in H1(g) as m
E [s, s + 11, @(xn) = exp[A(s
t [s - h, s],
+ 1 + h - x,)]
for
+ 1 + h], where A is a positive constant t o be chosen later.
Taking v = ( a - l)um in the integral identity for u(x) in g, we obtain
I. Some mathematical problems o f the theory of elasticity
t=O Let us fix t and choose a constant vector C which satisfies the condition
Then by virtue of the PoincarC inequality (2.3) in
R =
w : , the Second
Korn inequality (2.19) and (3.13) we get
where Mo is a constant independent o f t and rn. Taking into consideration (7.14) and the fact that P ( x n , u )= 0 for x, E
1F1
-h,s), for x E w:, (S
= A@ for xn E ( s - h, s). exp(At) 5 l ( x n )
< exp [ A ( t+ I ) ]
5n
we obtain
<
(I
112
G M ~ A ~ ~ ( ~q+u )~I 2)d x )
< &M0AeA -
JE(u)12l dx w
+ om ,
(
112
m
X
)
5 (7.15)
:
where C 2 is a constant independent o f s , t , h and E , -+ 0 as m + m. We deduce from (7.15) that
$7. Saint-Venant's principle for periodic solutions
A similar inequality holds for
g2, and
can be proved in the same way as (7.16).
Making m tend t o infinity we find from (7.16) and (7.13) that
J
jE(u)12(@- 1)dx 5 C M 0 A e A
IE(u)12 @ d x
.
g1ug2
9
Estimate (7.12) follows from this inequality if we choose the constant A such as t o satisfy the condition C M o A e A = 1. Theorem 7.2 is proved. Another version of Saint-Venant's Principle is given by Theorem 7.3. Let w(x) be a weak 1-periodic in ? solution o f the system L ( w ) = 0 in w(0, k + N) , where k
> 0, N > 0 are integers,
Let P(t,w) = 0 for t E (0, k
J
IE(w)12dx
and
+ N). Then
5 e-AN
J
IE(w)12dx
,
b(O,k+N)
W,k)
where A is the constant from Theorem 7.2.
7.3. Saint- Venant 's Principle for Non-Homogeneous Boundary Value Problems Consider ( n - 1)-dimensional open sets g j C gj
#
Q,g, = g o + ( O
,...,O , j ) ,
rj,j = 0,1,2, ... , such that
g j + r = g j for all z = (21, ..., 2,-,,0)
E En.
Existence of such g, is guaranteed by the Condition B of 54 on the domain w. Set
I. Some mathematical problems o f the theory o f elasticity
Let us first prove some auxiliary results. Lemma 7.4. Let cp E L2(&),$ E L2(4N)and
for some integer N > 0. Then there exists a weak 1-periodic in d solution of the problem
where v = ( 4 ,...,v,) is the unit outward normal t o aw(0, N ) . Moreover, U ( x ) satisfies the inequality
where C is a constant independent o f N, and
Qm
= (mes Go)-'
(&(b,J fOdx-
$2)
BN
m=1,
..., N-1,
$o=p,
Proof. Existence of the solution
$J~=--$.
(7.21)
U ( x ) of problem (7.19) follows directly from
Theorem 6.3, since (6.8) holds with a = 0, b = N due t o (7.18). Let us prove (7.20). Setting v = U in the integral identity (6.7), we obtain
§ 7. Saint-Venant 's principle for periodic solutions
75
Denote by Vm, m = 1, ..., N , weak 1-periodic in i solutions of the following boundary value problems
where do,..., d N are vector valued functions defined by (7.21), ( u l , ..., u,) is the unit outward normal t o d w ( m - 1 , m ) . Let us check the solvability conditions of type (6.8) for problems (7.23). For m = 1 using (7.18), (7.21), we find
J
dodi-J
Bo
=
/
pdi+
J BN
J
J
pdi-
Bo
81
Po
=
$ldi=/
G(1,N)
$di-/
lpdlPo
fOdx.
G(0,l)
For m = N it follows from (7.21) that
/ BN
fOdx+J d d i = BN
ddi+
J 40,l)
fOdx =
I. Some mathematical problems of the theory of elasticity
76 If
m = 2, ..., N - 1, relations (7.21)
J
=
yield
fOdx.
G(m-1,m)
(7.23), and therefore according to Theorem 6.3 the solutions Vm, m = 1, ..., N , exist and satisfy Thus the solvability conditions hold for problems
the inequalities
where C is a constant independent o f m, N . I t follows from the integral identity for
+ J ($,, u)d?Brn
/
Vm that
($m-1,
u)d?.
(7.25)
gm-1
Summing up these equalities with respect t o m from 1 t o
N , we find
5 7.
Saint-Venant's principle for periodic solutions Comparing this relation with (7.22) we conclude that
This inequality together with (7.24) yields (7.20). Lemma 7.4 is proved.
By the same argument we establish Lemma 7.5. Let
U(x)be a weak
Then
1-periodic in
U(x)satisfies the following
P
solution of the problem
inequality
where C is a constant independent of N , and
$
J
= (mes ijo)-'
~
l
- ( mJ, N )
m=0,1,
...,N - 1 , $ J ~ = - I I , .
I. Some mathematical problems of the theory of elasticjty
78
M. Consider w=
a vector valued function w such that w E H' (w(0, N ) ) ,
on Yo, w = 0 in w(1/2,
N ) . It follows from the integral identity for U
that
=
/ ,?(I
a(u- " ) )
- (lo, U - w ) ] dx t
axi
3(O,N)
Denote by Vm 1-periodic in
i weak
solutions o f problems (7.23) with
$m
given by the formulas (7.28). The solvability o f these problems is established similarly t o the solvability o f the corresponding problems in the proof of Lemma 7.4. The functions Vm satisfy the inequalities (7.24), where 40,$1,
..., $N
are
defined by (7.28). The integral identity for Vm implies
+J
m
u-
Bm
-
J
(+m-l,
u - w)d? .
Sm-1
Summing up these equalities with respect t o m from 1 t o N , and taking into consideration the fact that
U - w = 0 on go, we obtain
From this relation and (7.29) we conclude that
§ 7. Saint- Venant 's principle for periodic solutions
This inequality and (7.24) imply (7.27). Lemma 7.5 is proved. Lemma 7.6. Let u E
where
H' (w(0,N)),u = 0 on
Mo is a
ro. Then
constant independent of
Proof. Consider
N
and u.
a vector valued function w which is a weak 1-periodic in
solution of the problem
1
L(w)= u in w(0,N ) , u(w)= -U
o n gp, ,
u ( w ) = O o n dw(O,N)\(roUgN), w = O on By virtue o f Lemma 7.5 w satisfies the inequality
Fo.
J
2
I. Some mathematical problems of the theory of elasticity
80
Setting v = u in the integral identity for w we obtain
This inequality and (7.32) yield (7.30). Lemma 7.6 is proved. For some applications it is important t o have an extension of Theorem 7.2 to a more general situation, namely, t o the case o f non-zero boundary conditions, external forces and generalized momenta. Saint-Venant's Principle for solutions of a non-homogeneous system of elasticity is expressed by Theorem 7.7 (Generalized Saint-Venant's Principle). Let
u ( x ) be a weak 1-periodic in 2 solution of the system C ( u )= p
afi +in ax;
w(tl,t2)
with the boundary conditions
where
t 2 > tl
+ 2, t l ,
t 2 are positive integers, and for any t E ( t l , t 2 ) let
conditions (7.6) be satisfied. Then for any integer s, h inequality
> 0 such
that s - h
> t l , s + 1 + h < t 2 the
5 7.
81
Saint-Venant 's principle for periodic solutions
holds for u ( x ) . Here C is a constant independent o f s , h; A is the constant from Theorem 7.2.
Proof.
Consider a vector valued function U ( x ) which is a 1-periodic in 3
solution of the problem
o ( U ) = vif' on dw(s - h,s
+ h + l)\(gs-h
J
U gs+h+l) ,
where cp, $ are constant vectors chosen in such a way that
P ( s - h, U ) = P ( s - h, U ) , P(s+h+l,u)=P(s+h+l,U). We have
P(s-h,U)= -
J
o(U)di.= -
ra-h
Now we can find
1C, and cp from
J is-h
ydi+
J
fnd3.
rr-h
(7.37):
Let us show that the solvability conditions for problem (7.36) with the above chosen $, cp are satisfied. Indeed by virtue o f (7.8) we obtain
I. Some mathematical problems of the theory of elasticity
- r d i + J f " d i = ra+h+l
ra-h
J
fOdx.
&(a-h,s+h+l)
Therefore according to Lemma 7.4 a solution of problem (7.36) exists and satisfies the inequality
/
Ie(U)12dx 5
G(s-h,a+h+l)
where
Since
it follows from (7.37) that
$m
= (mes
i0)-' G(8-h+m,s+h+l)
Therefore
f O d x - P(s - h,u) -
§ 7. Saint- Venant 's principle for periodic solutions
83
It is easy to see that u - U is a weak 1-periodic in 2 solution of system (7.10) with the boundary conditions (7.11). Moreover, P ( u - U,s - h ) = 0. Then by Theorem 7.2 we have for u - U :
J
IB(U
- u)12dx 5 e
G(s,s+l)
-
~
J
~
JE(U
- u)12dx .
G(s-h,s+l+h)
This inequality and (3.13) imply
Estimate (7.35) follows from this one and (7.39), (7.40). Theorem 7.7 is proved.
I. Some mathematical problems of the theory of elasticity
84
$8. Estimates and Existence Theorems for Solutions o f the Elasticity System in Unbounded Domains
In this section we use the notation of $7.
8.1. Theorems of Phragmen-Lindelof
's
Type3
The classical Phragmen-Lindelof's theorem for the Laplace equation has been the subject o f various generalizations for elliptic equations and systems (see the review [49]). The next theorem is closely related to the generalized Saint-Venant Principle (see Theorem 7.7) and can be considered as a theorem of PhragmenLindelof's type. Theorem 8.1. Let the vector valued functions f j , j = 0,
...,n, satisfy
the inequalities
where cl, al are positive constants; and let u ( x ) be a weak 1- periodic in 2 solution of the system
such that
P ( O ,U ) = -
/
f"dx+Jfnd2,
4 0 , ~ )
fa
3Theorems of Phragmen-Lindelof's type describe the behaviour of solutions of elliptic boundary value problems in unbounded domains. There are many results of this kind. Of particular interest here are theorems which give sufficient conditions for the decay at infinity of solutions belonging to classes of functions whose growth at infinity is not too rapid.
$8. Estimates and existence theorems
85
where c is a constant independent o f s , 60 = const, 0
< 60 5 A, A
is
the constant from Theorem 7.2. Then there exist constants c 2 , c3, a 2 , a3 independent o f s and a constant vector w,
Proof. By virtue o f the formulas
- -
such that
(7.8), (8.3) we have
fOdx+J r d ? .
P.
G(8,oo)
Therefore, taking into account inequalities (8.1), we get
Setting h = [s/2] in (7.35) and using (8.4), (8.7), (8.1) we establish the inequality (8.5). Let us prove estimate (8.6). For every s = 0,1,2, ... set
w, = ( m e s ~ ( 0I))-' ,
/
u(x)dx .
(8.8)
G(s,s+l)
In the domain w ( s , s
+ 2 ) consider a weak 1-periodic in i solution o f the
problem
where X , is the characteristic function of the set g. It follows from (8.8) and the integral identity for the solution of problem (8.9) that
I. Some mathematical problems of the theory of elasticity
By virtue of Theorem 6.3 we have \ \ E ( V ) \ \ ~ .~
(w(s,9+2))
5 C,
where C is
a constant independent o f s. Therefore due t o the inequalities (8.5) proved above we find
Jw, - w , + ~ 5 J cexp(-aos) Therefore, there is a vector w,
,
a0 = const
>0 .
= lim w,. Moreover, a-w
where the constants K1,
do not depend on s , t. Making t tend t o infinity
in this inequality we obtain
In order t o prove the estimate (8.6) we apply the Korn inequality (2.19) in L(s,s
where
+ 1). We have
is a constant independent of s . Now we obtain estimate (8.6) from
this one and (8.5). Theorem 8.1 is proved.
87
$8. Estimates and existence theorems Remark 8.2. Suppose that under the assumptions of Theorem 8.1 we have
f" = 0, i
=
1 , ...,n. I f f0 and the coefFicients of system (8.2) are sufficiently smooth for large x , it follows from the a prion' estimates for solutions o f elliptic systems (see [I], [17]) that for large s we have
Moreover, Theorem 8.1 and the imbedding theorem (see [117]) imply for
m > n / 2 - 2 the inequality
max lu - wmI 5 J(s,s+l)
c [~XP(-QS) + llfOllHm
(G(s-l,s+2))
I
holds with constants C , a3 independent of s.
8.2. Existence of Solutions in Unbounded Domains In this section we consider existence o f solutions for the following boundary value problem
L(u) = f0
a? in +8x1
u = @ on T o ,
w(0, co) ,
o(u)= v i f i on S ( O , c o ) ,
u is 1-periodic in 2 .
I
(8.11)
Solutions of similar problems are used in Chapter II for the construction of boundary layers in the homogenization theory.
It is assumed in (8.11) that @ E ~ 1 / 2 ( i ' ~is) 1-periodic in 2, fj are 1periodic in 2 and belong t o ~ ~ ( h ( t , , t , for ) ) any t l . t 2 such that 0 5 t1 <
t 2 < c o , j = 0 , 1 ,..., 72. We say that u ( x ) is a weak solutiaon of problem (8.11) i f u = @ on rO, u ( x ) belongs t o 8 l ( u ( t , , t 2 ) ) for any t l , t 2 such that 0 5 t1 < t2 < m , and u ( x ) satisfies the integral identity (7.4).
I. Some mathematical problems of the theory of elasticity
88
Estimates o f Saint-Venant's type (see Theorems 7.2, 7.3, 7.4) make it possible t o prove existence and uniqueness of solutions for problem (8.11) in classes o f functions growing at infinity. Theorem 8.3. Suppose that
where
M,6= const, A
is the constant from Theorem 7.2, 0
< 6 5 A.
Then
for any constant vector q = ( q l , ...,q,) there is a unique weak solution u ( x ) of problem (8.11) such that P ( 0 , u) = q and the following estimate is satisfied
r
where C is a constant independent o f k ; 61 is an arbitrary constant from the interval ( 0 , 6 ) .
Proof.
Denote by v N a weak 1-periodic in 2 solution o f the problem (7.26)
with
It is easy t o see that
Indeed, due t o (7.8) we have
Therefore taking into account (8.14) and the formula
58. Estimates and existence theorems
p ( N , v N )=
/ (1di.+ / f ' d i BN
,
PN
we find that
, in (7.28) are Since 11, is given by (8.14), therefore the functions $
/
1 1 , , . = ( r n e ~ i j ~ ) - ' ( - ~&(o,m) fOdi+/fndi). Po
(8.11)
I t thus follows from (7.27) and Lemma 7.5 that
where C is a constant independent o f N . The function vk+N+l- v ~ satisfies + ~ all the conditions o f Theorem 7.3, since from (8.15) we have
P(0,Vk+N+' - V k + N ) = 0 . Therefore from (7.17) we get
Taking into account (8.17) we conclude from the above inequality that
I. Some mathematical problems of the theory of elastjcity
90
Let us estimate the last sum in the right-hand side o f (8.18). W e have
where 61 is any constant from the interval (0,6), C depends on b1 and does not depend on N , k. T o obtain the last inequality we also used the conditions (8.12) and the fact that mesG(0, m ) 5 c l m . Thus (8.12), (8.18), (8.19) yield
Therefore
$8. Estimates and existence theorems
where
Mz is a constant independent of k and N
I t follows from (8.20) that
Therefore
where
is any constant from (0,6), the constants
M, do not depend on k,
s, t. Inequality (8.21) implies that
Note that vkt"
vktstt = 0 on
vkts - vkt'+t we deduce that
rO. Therefore applying Theorem 2.7 to
I. Some mathematical problems of the theory of elasticity
where the constant C1 depends on k but does not depend on s , t.
It follows from (8.22), (8.23) that for any k the sequence vS converges in B1(w(O,k)) as s + m t o a vector valued function u. Making s tend t o infinity in the integral identity for v h e see that u is a solution of problem (8.11). Making t tend t o infinity in (8.21) for s = 0 and using (8.17) for N =
k,
we obtain
2
+ \l@(l~lll(i.o) where M8 is a constant independent of
1
,
k. This proves inequality (8.13).
To complete the proof o f Theorem 8.3 we need t o show that P(0, u ) = q. According t o (7.8) we have for s
5 m:
P(s,vm)-P(O,vm)=
/
fodx+/
~(0,s)
fndi-1
f,
fndi.
fo
Integrating both sides of this equality from 0 t o t we get
Passing here t o the limit as m
4
m we obtain the above equality with
vm
replaced by u . Passing t o the limit as t + +O in the equality for u we find that
93
58. Estimates and existence theorems
P ( 0 , u ) = q. The uniqueness of u ( x ) follows from Theorem 7.3. Theorem 8.3 is proved.
8.3. Solutions Stabilizing to a Constant Vector at Infinity Existence of solutions for problem (8.11) and their estimates in the case of external forces, which rapidly decay at infinity, are established by Theorem 8.4. Suppose that inequalities (8.1) are satisfied. Then there exists a unique solution of problem (8.11), such that
Moreover there is a constant vector C , such that
a0 are positive constants independent of s. where M I , M2,
Proof. I t is obvious that conditions (8.1) q=
J Po
fndi-
J
imply (8.12). Set
fOdz
(8.26)
G(O,w)
in Theorem 8.3. Let u ( x ) be the solution of problem (8.11) whose existence is asserted by Theorem 8.3 with P ( 0 , u ) = q. Our aim is t o show that u ( x ) satisfies inequalities (8.24), (8.25). We first check that estimates (8.4) hold for u ( x ) . Indeed, for f j , j = 0 , ...,n , inequalities (8.12) are valid with 6 = A. Therefore inequalities (8.13) hold for u ( x ) with 61 = 3A/4 < 6. I t follows from (8.13) that estimates (8.4) hold with 60 = A/4. Thus we can use Theorem 8.1, which implies the estimates (8.5), (8.6). By virtue o f the Korn inequality of type (2.3) in G ( s ,s I ) , we have
+
I. Some mathematical problems o f the theory o f elasticity
Therefore (8.5), (8.6) yield (8.25). Let us prove estimate (8.24).
Consider the vector valued functions v N
constructed in the proof o f Theorem 8.3. These functions satisfy inequalities (8.17). Taking as q in (8.17) the vector given by (8.26), and passing t o the limit as N + m we get estimate (8.24). Here we also used estimates (8.1) and the convergence of vN t o u in
H' ( ~ ( 0k ), ) as N
-+ m for any fixed k .
This convergence was established in the proof of Theorem 8.3. Theorem 8.4 is proved. For the vector C, presses C,
in (8.25) we can obtain an explicit formula which ex-
in terms of
fj,
j = 0 , ..., n , and the boundary values o f u ( x ) on
ro . To this end we shall need some auxiliary functions v', r = 1, ...,n , whose existence is guaranteed by Theorem 8.3. By v', r = 1 , ...,n , we denote weak solutions o f the following boundary value problems \
L ( v r ) = 0 in w(0, m) , = 0 on I'o
,
P ( 0 , v r ) = -er
,
V'
a ( v P )= 0 on S ( 0 , m ) , >
v' is 1-periodic in ?
,
where e l , ...,en is the standard basis of
J
Rn.
According t o Theorem 8.3 v' can be chosen so as t o satisfy the inequalities
By Lemma 7.6 we also have
58. Estimates a n d existence theorems
95
Theorem 8.5. Suppose that all conditions o f Theorem 8.4 are satisfied. Then the constant vector C , = ( c k , ...,c&) in (8.25) is given by the formulas
/
CL =
dvr
[(fO.v')-
(fi, g xi) ] d ~ +
~(O,OO)
where
V'
are the solutions of problems (8.27) satisfying the inequalities (8.28),
(8.29). Note that if v'
and the coefficients of the operator
L:
are sufficiently
smooth, then the integral
is defined in an obvious way. Let us give a meaning t o this integral when v r ,
r = 1, ...,n , are weak solutions of problem (8.27).
It is easy t o see that
for smooth v' and any scalar function qjs E C 1 ( u ( O , m ) ) such that qj6 is 1-periodic in 5,
$6
= 1 in w ( 0 , 6 ) , $6 = 0 in w ( 2 S , m ) , 6 = const
> 0.
It
follows from the integral identity for 'v that the integral on the right-hand side o f (8.31) does not depend on integral as
/
$6
and 6. That is why we can consider this
( u ( v T ) , u ) d i :in the case of weak solutions v'.
Po Proof of Theorem 8.5. Fix an integer s > 1 and consider a scalar function cp(x,) E C O ( R 1 )such that cp(x,) = 1 for x, E (O,s), cp(x,) is linear for x, E [s,s 11, cp(x,) = 0 for x, E [s 1, m ) . Set v = cpv' in the integral 1. Taking into account (8.31) and identity (7.4) for u with t l = 0, t 2 = s the integral identity for v' we find
+
+ +
96
J G(O,s+l)
I. Some mathematical problems o f the theory of elasticity dyvT
[(fO,u v r )- (f',
dx = -
J G(s,s+l)
du dyvT (A" -, -)dx dxk dxh
=
It is easy t o see that the second and the fourth integrals in the right-hand side of the last equality are bounded by
and therefore these integrals decay exponentially as s + co due t o (8.25), (8.29), (8.28). Consider the third integral in the right-hand side of (8.32). Using the definition of P ( t , v T )and the fact that P ( t , v T )= P(O,vT)= -eT, we obtain
Therefore, if we make s tend t o infinity, the formula (8.32) reduces to (8.30). Theorem 8.5 is proved.
97
$8. Estimates and existence theorems Remark
8.6.
Theorem 8.5 implies that under the assumptions of Theorem 8.4 conditions of decay for a solution of problem (8.11) read
where
are the boundary values of
u(x)on
ro;r = 1, ...,n.
I. Some mathematical problems o f the theory o f elasticity
98
$9. Strong G-Convergence o f Elasticity Operators Homogenization of differential operators considered in the next chapter is closely associated with the notion o f strong G-convergence. The theory o f
G-
convergence and strong G-convergence was developed by many authors (see
[22]-[24] and the review [148]). The initial works on the subject date back t o the 60's and belong t o S. Spagnolo, ([118], (1191).
9.1. Necessary and Suficient
Conditions for the Strong G-Convergence
Consider a sequence of the elasticity operators
where a E ( 0 , l ) is a small parameter; A';j(x),i, j = 1, ..., n , is a family of matrices of class E(rc1,rc2); rcl, n2 are positive constants independent of a ; R is a bounded Lipschitz domain o f
IRn.
We also consider another elasticity operator
of class E ( i l , i 2 ) where , 21,
i2are positive constants which may differ from
K1, K2.
A sequence o f operators { L , ) is called strongly G-convergent t o operator k ) , if for any f E H - ' ( 0 ) the sequence uc E H i ( R )
2 as a + 0 ( L ,
a
of solutions of the problems
converges t o u0 E H i ( R ) weakly in H,'(R) as
a -+ 0, where u0 is the solution
of the problem
moreover,
7i(x)
A:J
..
... . auo auc t 9 i ( 2 )E AZ3 - weakly in L2(R) axj
axj
59. Strong G-convergence of elasticity operators as E -+ 0, i = 1,2,
...,n (see [148]).
Remark 9.1. In the above definition o f the strong G-convergence it is sufficient t o require that
uc -+ u0 and y: for any
f
-+
ji as
belonging t o a subspace V
E -+
c H-'(0)
us show that in such a case uc -+ u0 and yf
0 dense in H - ' ( 0 ) . Indeed, let
jifor any f E H - ' ( 0 ) . Consider a sequence f m E V, such that f m -+ f in the norm of H - ' ( 0 ) as m -+ m. Denote by u',, 6 , solutions o f the following problems -+
.-av ax
Let us introduce matrices r c ( v ) and r ( v ) whose columns are A:3 -, -..
av axj
1, ..., n , and AZ3-,
.
,
2
=
J
z = 1, ...,n , respectively. Then for any vector valued
function v E HA(S2) and any matrix valued function w E L2(S2) we have
It is easy t o see that the right-hand sides of these equalities converge t o zero as
E -+
0, since by Theorem 3.3 and Remark 3.4 (see (3.25))
with a constant C independent of e , m , and
I. Some mathematical problems of the theory of elasticity
100 as
E
+ 0 for a fixed m due t o the definition o f strong G-convergence with
f = frn E V . The matrices r C ( u c ) ,f'(uO) with columns yf,
+', i = 1 , ..., n , are some-
times called weak gradients. O f great importance for the theory of strong G-convergence is the following Condition N (see [148]). We say that a sequence o f the elasticity operators
{LC)satisfies the Con-
dition N, if there exist matrices * j ( x ) , i , j = 1, ..., n , and matrices N,"(x) E
H 1 ( R ) ,s = 1, ...,n , such that for N1.
E +
0 we have
N,"+O weaklyin H 1 ( R ) , -..
s = 1 , ...,n ;
a~,j
+
Ai3 f A! - A: + A i j ( x ) weakly in L 2 ( R ) , 8x1
N3.
a (A: - A i j ) -+0 axi
-
in the norm of H - ' ( 0 )
,
Note that in the Condition N, the family o f matrices k j ( x ) i, j = 1, ...,n , is not assumed t o define the coefficients o f an elasticity system, i.e. relations of type (3.2), (3.3) are not imposed on k j ( x ) . Obviously it only follows from the Condition N that the elements of the matrices A i j ( x ) belong t o L 2 ( R ) . However, as it is shown below (see Theorem 9.1), the Condition N actually implies relations (3.2), (3.3) for
Aij,
and therefore their elements are bounded
measurable functions. Theorem 9.1. Suppose that the Condition N holds for the sequence of operators class E ( n l , n 2 ) and n l , n2 are positive constants independent of any cp E C r ( R ) we have
E.
{LC)of Then for
101
59. Strong G-convergence of elasticity operators
where the matrix A* is the transpose o f A;
6ij,
E is the unit matrix with elements
bpk is the Kronecker symbol.
Moreover, the family of matrices
A ~ Pq ,, p = 1, ...,n , belongs t o the class
E(IE~, I E ~and ) therefore defines a system of linear elasticity.
Proof. Let us first
establish formula (9.4). Denote by
J,4p
the integral in the
right-hand side of (9.4). Then
where J,;
..., Ji successively stand for
the integrals on the left-hand side of
the last equality. Let us estimate these integrals. Taking into consideration the fact that a weakly convergent sequence in a Hilbert space is bounded and that the imbedding H1(R) c L2(R) is a compact one, we deduce from the Condition N1 that
I. Some mathematical problems of the theory of elasticity N,S
-+
0 strongly in L2(R) ,
aNi
--+ 0 weakly in L ~ ( R ), dxj
as
E -+
0, s,j = 1,..., n , where C = const and does not depend on
E.
It is
easy t o see that
Therefore
Jf -+ 0 as
E
-t
0 by virtue of (9.7) and the Condition N3.
Using the Holder inequality and the fact that the elements of matrices A: are bounded uniformly in
Ji
E,
we conclude that
0 due t o (9.6), (9.7). From (9.6) we get Jg -+ 0 as E -+ 0, and the Condition N2 implies that J,' converges t o the left-hand side of (9.4) as E + 0. Thus formula (9.4) is proved. Now let us show that the family of matrices Apq, p, q = 1, ...,n , belongs t o the class E(rcl, K ~ ) i.e. , that their elements iif,P(x) satisfy the relations (3.2),
Thus
-+
0 as
E
-+
(3.3). The equality 6fX
= i i i follows directly from the Condition N2 and relations
(3.2) for the elements of Azq. In order t o prove that iifX
= iijh(;let us note that these relations are equiv-
= (29~)'. The last equality follows from (9.4) and the equality A: = (A:')' which holds due t o (3.2) for the elements of matrices A:. Now let us prove the inequalities (3.3) for kj(x). First we obtain the lower bound. Let {qih) be a symmetric (n x n)-matrix with constant elements. Denote by gk the column vector whose components are qlk, ...,q,k, and by gh* the line ( g h l ,...,ghn). By virtue of (9.4) we have
alent t o
A P ~
$9. Strong G-convergence o f elasticity operators for any cp E
Cr(R),cp > 0.
It is easy t o see that
Set
Ck(c,x) is a column vector with
components
N,Pis are the elements of matrices N:. Denote by J, the integral in (9.9) after the limit sign. Then
where
JE =
1
cp ai:h(x)C;,(r, 2) hi(€, x)dx
n
According t o Lemma 3.1 we have
It is easy t o see that
where 11f: = NA, qsq. Therefore
Let us multiply this equality by cp(x) due t o (9.11) and the relation
we get
20
and integrate it over
R. Then
I. Some mathematical problems of the theory of elasticity
where p, -, 0 as
E -+
0 owing t o (9.6), (9.7).
Since the second and the third integrals in the right-hand side of the last inequality are non-negative, it follows from (9.10) that
Passing here t o the limit as e
--t
0, by virtue of (9.9) we obtain the inequality
Since ,(x) is an arbitrary non-negative function in C,""(R) the last inequality yields the lower bound in (3.3). Let us establish the upper bound in (3.3) for the elements of matrices Fix a symmetric ( n x n)-matrix r] =
{vih)with constant elements.
just shown (see (9.4), (9.5)) that for any cp E C,"(fl), cp relations are valid
Therefore
It follows that
2 0,
AP~.
We have
the following
$9. Strong G-convergence o f elasticity operators
for a subsequence E'
0, since according t o Lemma 3.1 we have
--t
Due t o the conditions (A?)* = A:j we get
Therefore
Since
779*
inequality
AZP f 5
T5
n2 q i h q i h
n2qihqih
Theorem 9.1 is proved.
J
by virtue of the Condition N2 we obtain the
cpdx, which implies the upper bound in (3.3).
n
Theorem 9.2. Suppose that Condition N is satisfied for the sequence o f elasticity operators
{LC)of class E ( n l , n Z ) , and K I , n2 are positive constants independent of E . {LC)is strongly G-convergent t o an elasticity operator 2 as E + 0, and
Then
the coefficient matrices
$j(x)
of
belong t o the class E ( n l , n 2 ) .
Proof. We have already established in Theorem 9.1 that define a system o f elasticity and belong t o the class same as for operators
the matrices k j ( x )
E ( K , ,K
~ with )
n l , n2 the
L,. Let us prove the strong G-convergence of L, t o c
as e + 0. By virtue of (3.21) and the representation (3.20) for the elements of
H-'(a)
the Condition N3 can be rewritten in the form
I. Some mathematical problems of the theory of elasticity
106
+ 0, j = 0, ..., n , s = 1, ..., n. Here we have
FjS + 0 strongly in L 2 ( R )as E also used the relations
(see the proof of Theorem 9.1). Consider the vector valued function cpuc, where cp is an arbitrary scalar function in
C,"(R), and uc is a weak solution of the problem
It follows from (9.13) that
=
/ [F;cpuc
1-d,uC ax;
- F:~
dx .
n By the definition of a weak solution o f problem (9.15) we have
Subtracting (9.16) from (9.17) we get
Theorem 3.3 implies
$9. Strong G-convergence of elasticity operators where
. auc
yj = Azk - and C1,Cz are constants independent of E . ax k
Due to the weak compactness of a ball in a separable Hilbert space and the compactness of the imbedding H 1 ( R )c L 2 ( R ) ,the inequalities (9.19) imply that there exist vector valued functions U E H,'(R), ?j E L 2 ( R )such that
uc' --t U weakly in H i ( R ) and strongly in L 2 ( R ) ,
Yj
jj weakly in L 2 ( R ),
j = 1, ...,n
,
I
(9.20)
for a subsequence E' -t 0. Note that by virtue of (9.6), (9.7), (9.13), (9.19) the first integral in the left-hand side of (9.18) and the integral in the right-hand side of (9.18) converge to zero as E -t 0. Therefore we deduce from (9.18) that
where p,, -t 0 as E' -+ 0. Since uc' - U -+ 0 strongly in L 2 ( R )as E -+ 0, we can pass t o the limit in (9.21) as E' -+ 0. Then taking into account (9.14), (9.20) and the Condition N2 we see that the first integral in the left-hand side of (9.21) is infinitely small as E' -t 0, and the second integral converges to
Therefore
since cp is an arbitrary function in C,"(R). Let us show that U ( x ) is a weak solution of the problem
By the definition of a weak solution of problem (9.15) we have
I. Some mathematical problems of the theory of elasticity
for any matrix
M ( x ) E H,'(R).
Passing t o the limit in this integral identity as E' -+ 0, by virtue o f (9.20), (9.22) we obtain
Therefore
U(x)is indeed a weak solution o f problem (9.23).
The above considerations show that from any sequence (u", $,, can always extract a subsequence such that
y:,, -+ 7' weakly in L2(R) as E" {L,) is strongly G-convergent t o
u"'
-+
U
weakly in
...,y,",) we
H t ( R ) and
0. Therefore the sequence o f operators as e -+ 0. Theorem 9.2 is proved.
+
2
Theorem 9.3 (On the uniqueness of the strong G-limit). Let
2 as
L % 2 and L,
elasticity operators o f class dent of
E,
E
-+ 0 , where {L,) is a sequence o f the
E ( K ~K ~, ) K , ~K~ , are positive constants indepen-
2, 2 are elasticity operators with bounded measurable coefficients. 2 and i? coincide almost everywhere in R.
Then the coefficients o f operators
Proof. Let 6 be any vector-valued function with components i n C,"(R). Set
f = 2 6 and consider a sequence us E H,'(R) o f the solutions o f the following problems
By virtue o f the strong G-convergence o f
uc +
auS + A" 8.; dz, ax .. auc -. 86 A' a,+ A - axj ..
A:J
....
*..
\'3
C, t o
J!
and
we have
weakly in H,'(R) , weakly in L 2 ( R ) , i = 1,..., n weakly in L 2 ( R )as e + O ,
,
i = 1,...,n
,
are respectively the coefficient matrices o f the operators 2, i . a. ri 86 Therefore A'j - = A axj axj almost everywhere in R for any fi E C,"(O). It
where At3, A
.. .
:ij
.
109
$9. Strong G-convergence o f elasticity operators :ij
follows that A'j = A
almost everywhere in
R. Theorem 9.3 is proved.
Theorem 9.4. Let {C,)
be a sequence of elasticity operators belonging t o class
with positive constants operator. Then C,
K,,
tc2 independent o f E , and let
E ( K * K,
~ )
i be an elasticity
E as E + 0 , if and only if the Condition N is satisfied LC and E .
for the coefficient matrices of the operators
Proof. SufFiciency o f the Condition
2
N for the strong G-convergence o f L, t o
is established in Theorem 9.2. Let us prove the necessity. Suppose that G
LC==+ k as E + 0. Consider a sequence of matrices B!, j = 1 , ..., n , such that Bj are weak solutions of the problems
It is easy t o see that IIBjllH;(n,5 C with C = const independent of since the elements of matrices
E,
A:(x) are bounded uniformly in E and we can
apply Theorem 3.3. Due t o the weak compactness o f a ball in a separable Hilbert space there is a subsequence E' + 0 such that
j = 1 ,...,n .
B$ + B{ weakly in H 1 ( R ) as E ' + O , Let us define the matrices
fi:, as weak solutions of the following boundary
value problems
Set N:, =
-B::+ Mil. Since LCis strongly G-convergent t o L?, it follows that M:,-+B{
weaklyin
j = l ) ...)n )
Ht(R),
dM2: *.dB: A$ -+ A:/ - weakly in L 2 ( R ) , 8x1 8x1
i) j = 1 , ..., n .
Therefore the Condition N1 is satisfied for the sequence weakly in
H1(R).
Since the elements o f the matrices are bounded in the norm of
1
E A:
L 2 ( R ) uniformly in
a
-N! 8x1 E,
E'
(9.25)
+ 0, i.e. N;' + 0
+ A?, i, j
= 1, ...,n ,
it follows that there is a
subsequence E" + 0 o f the sequence E' + 0 such that
I. Some mathematical problems o f the theory o f elasticity
a
A:,, - N:, 8x1 where A:
+ A,:
,
weakly in L 2 ( R )
+ A:(X?
(9.26)
are matrices with elements in L2(R).
Let us consider the Condition N3 for the sequence E" + 0 and the matrices
N!,, :
a ax, a .. . . a - -A'3 = L B ; - -A:' a ax;
a,,
..
-..
ax; *
a +ax, A;?,
..
-)8x1
- (A:?, - A:J) = - (A;:, ..
a *.a&
= - (A" - - A ' )
6's;
ax;
..
.
8x1
(9.27)
The integral identity for problem (9.24) yields
where M is any (n x n)-matrix with elements in HJ(R). Passing t o the limit in this equality as
E" +
0 and using (9.26), we obtain
a -.a~i ax, axl
..
- ( A -- A ) = 0
Therefore,
and
by
(9.27)
we
a - A?) = 0. Thus the Condition N with the matrices A: ax: (e;,
-
find
that
is satisfied
f o r t h e subsequence E" -+0.
It follows from Theorem 9.2 and the uniqueness of the strong G-limit (Theorem 9.3) that A ? = h'j almost everywhere in R. Let us show that the Condition N holds for the entire sequence E + 0. Define matrices N j as weak solutions of the problems
a
-(
ax,
a ~ , j= a (
A -) ax1
ax;
A -A )
,
N: E H$)
It follows from (9.27) that these relations hold for
E
(9.28)
.
= a",
~2
= N:.
Therefore, from any sequence N: defined by (9.28) we can extract a subsequence which satisfies the Conditions N1-N3 with matrices A'j(x). Hence the whole sequence N j satisfies the Condition N. Theorem 9.4 is proved.
111
$9. Strong G-convergence o f elasticity operators Corollarv 9.5. Let { L C )be a sequence o f operators of class pendent of
E,
the operator
and let
I? as E + 0.
L,
E ( n l ,K
~ with )
n l , nz > 0 inde-
Then the coefFicient matrices of
I? also belong t o the class E ( K ~ ,2 ) .
9.2. Estimates for the Rate of Convergence of Solutions of the Dirichlet Problem for Strongly G-Convergent Operators It was shown in the previous section that the Condition N guarantees only weak convergence in
HA(R) of solutions u' of problems (9.2) t o a solution of
problem (9.3). However, if one imposes some additional restrictions on the convergence of the functions in the Condition N, it becomes possible t o obtain estimates for the difference u0 - uc - v, in the norm of
H 1 ( R ) ,where v, is
the so-called corrector. We assume here that the boundary of the domain of the G-limit operator
R and the coefficients
are smooth.
To characterize the degree of deviation of the coefficients of of
L, from those
I? we introduce the following functional spaces. Denote by
H - m l W ( R ) ,( m 2 0 is an integer) the space whose elements
are distributions o f the form
where
f , E L W ( R ) . The
norm in
H-"vW(R) is defined as
where the infimum is taken over all representations of f in the form (9.29). Lemma 9.6. Let g =
V " g, E H-mlW ( R ) ,g, E L m ( R ) , u E H m ( 0 ) . Then one can
I4Sm define an element ug E
H - m ( 0 ) by the formula
I. Some mathematical problems o f the theory of elasticity Moreover
Proof. Let us show that (9.30)
correctly defines a continuous linear functional
Dag; be another representation of the
on H r ( R ) . Indeed, let g = lalSm
element g E H-"tm(R), g; E L M ( R ) .Then for any
1C, E C F ( R )the following
identity holds in the sense of distributions
(-l)Ial
=
1
g,~1C,dx .
(9.32)
n
lallm
Since gk, g, have bounded norms in L M ( R ) the , last equality is valid for all such that DalC, E L 1 ( n ) ,la1
5 m, and in particular for 1C, = up.
II,
The inequal-
ity (9.31) follows from (9.30) and the definition o f the norms in H-"vM(R) and H-"(R). Lemma 9.6 is proved. We say that a sequence o f the elasticity operators
{L,)of class E ( n l ,K
~ )
> 0 ) satisfies the Condition N', if there exist matrices a i j ( z ) , i, j = 1, ..., n , N,d(x) E H 1 ( R )n L M ( R ) ,s = 1 , ..., n ,such that ( K ~ K,Z
= const
N'2.
aNj Aal = A: A: + Aij(x) in the norm o f C -
N'3.
-
A , .
as
E
ax r
a
ax;
(a: -
+
aij)
+ 0 in the norm o f
H-lvM(R)
+ 0.
It is easy t o check that Condition N' implies Condition N. Therefore the matrices
aij
define an elasticity operator
k
which also belongs t o the class
E ( K I K, Z ) . Let us introduce the following parameters t o characterize the rate o f convergence in Conditions N'l-N'3:
59. Strong G-convergence of elasticity operators
,8,
max
=
-
a,j=l,...,n
kjllH-l,m(n) ,
max
yc =
j=l,
...,n
Theorem 9.7. Suppose that the operators
t , ,E satisfy the Condition N', and the coefficients
i'hjk(x)of the operator k are smooth functions. Then the solutions of problems (9.2), (9.3) with f E H 1 ( R )satisfy the inequalities
where the constants
K 1 , K 2 do not depend on
E,
v' is the solution of the
Dirichlet Problem
auO Applying the operator LCt o uc--6+vc we obtain
Proof. Set 6 = uO+N,d-. ax,
the following equalities wh~chare understood in the sense o f distributions
I. Some mathematical problems o f the theory of elasticity
114
According t o (9.39) F;,
F,E E H - ' ( R ) and
where ye, P, are defined by (9.35), (9.34), the constant c is independent of
E.
It is easy t o see that F,' also belongs t o H - ' ( R ) and
where cl is a constant independent o f Since uc - 6
+ 'v
is defined by (9.33).
E , a,
E H,'(R), it follows from (9.39)-(9.42) and Remark 3.4
that
where c2 is a constant independent of
E.
Since
k
is an elliptic operator with
smooth coefficients, the well-known a pm'ori estimates for solutions o f elliptic boundary value problems (see [ I ] , [17]) yield
IIu011~m+2(n) 5 cm
Ilf ~ I H ~ ( R ) ,
m = 0,1,2, ... .
(9.44)
These inequalities and (9.43) imply (9.36), (9.37). Theorem 9.7 is proved.
Thus i t is evident that in order t o estimate the difference between uc and
uO it suffices t o construct matrices N,J satisfying the Conditions N'l-N'3 and then estimate a,,PC, r c , IIvcIIHl(C2)* 11vC11~2(~). Let us give the simplest example in which the Condition N' is satisfied.
115
$9. Strong G-convergence of elasticity operators Example 9.8. Let A y ( x ) -+ a ' j ( x ) in the norm of
L m ( R ) as
E
4
0, i, j = 1, ...,n. Set
N,"(x) 0 in 0 , s = 1 , ..., n. Then the Conditions N'l-N'3 are satisfied with a, = 0, P,,Y, 5 SUP I I A ~ - A i j l l L r n c n ) . Therefore i,j=l,...,n
C = const. In fact, according t o Theorem 9.7 we should have placed stead o f
11 f llLz(n)
11
in the right-hand side of the last inequality.
fllHlcn)
in-
Neverthe-
less in this situation, as one can see from the proof of estimate (9.41), we have
((F,'((H-i(n) 5 C sup ((A: - A i j l l L m ( n )
I I u O I I ~ Z ( ~ ) .Therefore
estimate
i,j
(9.45) is valid. Now we consider a less trivial example, when the Condition N' is satisfied (see also Chapter 11, $8).
Let the coefficient matrices A?($) o f the operators L, have the form A ' ; ~ ( x= ) .. x o f the matrices A i j ( ( ) be A t 3 ( - ) ,i, j = 1, ..., n , and let the elements E
smooth functions 1-periodic in (. Operators of this kind in a much more general situation will be studied in Chapter II, where another approach is suggested in relation t o such problems. Let us define the matrices
x N,"(x) setting N,b(x) = E N ' ( - ) , where N s ( ( ) E
are 1-periodic in ( solutions of the system
As it was shown in $6.1, this system possesses a solution in the class o f smooth functions 1-periodic in
5.
Let us define the coefficient matrices Aij for the operator strong G-limit of the sequence
{LC)as E
-+
0. Set
k , which is the
I. Some mathematical problems o f the theory of elasticity
116 where
(f)= J f (()d(, Q = {(
: 0
< tj < 1, J'
= 1 , ..., n } .
Q Let us show that the matrices A?, i i j , N," satisfy the Condition N1. The Condition N'1 holds since N S ( t )are smooth. Moreover a,
< CE,
C = const. Equations (9.46) show that the Condition N13 is also satisfied. It is easy t o see that y, = 0. Consider now the Condition N12. Obviously & ( x )
- k j ( x ) r Bij(4), E
and B " ( ( ) are matrices whose elements are smooth functions 1-periodic in
.
Moreover
1~ ' ~ ( ( ) d (
= 0, by virtue of (9.47). According t o Lemma 1.8
Q .. x d B t J ( - )= E - q i j ( e , x ) , where the elements o f the matrices
ax,
E
smooth
1
functions
x B1~(--)IIH ..
uniformly
bounded
in
E,
x.
F;'~(E,x ) are
It follows that
< C E . Hence PC 5 C E ,C = const.
In order t o obtain an effective estimate for uC-u0 we must have an estimate
for IIvCIIHlcfl).Let cp,(x) be a truncating function such that
It is easy t o see that v" is a solution of the problem
and
a
ayc
auO
~ N duo P aZu0 . (9.50) -+~cp~NpdEj dxp dtPdx dNP Since the elements of the matrices NP, -, p, j = 1, ...,n , are bounded -=~-N~-+cp,axj dxj dx,
functions we have
atj
Kc is the set of all x E R such that cp,(x) # 0. It is obvious that Ii', lies in the 2~-neighbourhoodof 8 0 . Therefore by Lemma 1.5 we have IIVUOII~?(~.) c ~ E ~~ ~I U~ O I IThus ~(~).
where cl is a constant independent of E ,
59. Strong G-convergence of elasticity operators
Il*cll~l(n)
I c3&'J2 IIuOIIp(n) ,
~3
= const
.
Applying Theorem 3.3 to the solution of problem (9.49) we get IIvCIIHl(n)
5 c ~ E " IlfllLP(CI) ~ .
Therefore we can deduce from Theorem 9.7 that
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CHAPTER ll HOMOGENIZATION OF T H E SYSTEM OF LINEAR ELASTICITY. COMPOSITES AND PERFORATED MATERIALS
This chapter deals with homogenization problems in the mechanics of strongly non-homogeneous media. Most of the results are obtained for the system of linear elastostatics with rapidly oscillating periodic coefficients in domains which may contain small cavities distributed periodically with period E.
In mechanics, domains of this type are referred t o as perforated. The
main problem consists in constructing an effective medium, i.e. in defining the so-called homogenized system with slowly varying coefficients and finding its solutions which approximate the solutions of the given system describing a strongly non-homogeneous medium. In Chapter II we give estimates for the closeness between the displacement vector, the strain and stress tensors, and the energy o f a strongly nonhomogeneous elastic body and the corresponding properties o f the body characterized by the homogenized system under various boundary conditions. Homogenization problems for partial differential equations were studied by many authors, (see e.g. [5],
[3],[110],[148],[82],[83]and the bibliography
given there as well as at the end of the present book).
$1. The Mixed Problem in a Perforated Domain with the Dirichlet Boundary Conditions on the Outer Part o f the Boundary and the Neumann Conditions on the Surface of the Cavities
1.1. Setting of the Problem. Homogenized Equations Let R' = R In
n EW
be a perforated domain o f type I, defined in
RE we consider the following boundary value problem
$4,Ch. I.
11. Homogenization of the system of linear elasticity
E ( K ~K Z,) , K I , ~2 = const > 0 whose elements aihjk(()are functions 1-periodic in J . It is also assumed that a : / ( f ) are piece-wise smooth in w and the surfaces across which they or their derivatives may loose continuity do not intersect dw i.e. the functions a:! where A h k ( J )are ( n X n)-matrices o f class
belong t o the class
6' defined in $6.1, Ch. I.
Existence and uniqueness of solutions o f problem (1.1) for QC
f"
E L2(nE),
E H 1 ( R E )are guaranteed by Theorem 5.1, Ch. I.
Our aim is t o study the behaviour of a solution u' o f problem (1.1) as E + 0 and t o estimate the closeness of uE t o uO,which is a solution o f a boundary value problem in the domain R for the homogenized system o f elasticity with constant coefficients. Using the approximate solutions thus obtained we shall calculate effective characteristics such as energy, stress tensor, frequencies of free vibrations, etc., of a perforated strongly non-homogeneous elastic body, whose elastic properties can be described in terms o f problem (1.1). The homogenized system corresponding t o problem (1.1) has the form
where the coefficient matrices
( P ,= ~ 1 , ...,n ) are given by the formula
and matrices N * ( J )are solutions o f the following boundary value problems for the system of elasticity
4 N 9 ) = -ukAkq on dw N q ( J ) is 1-periodic in J
,
,
/
N q ( W=0
Qnw
Q = { J , O < ( j < l , j = l ,...,n ) .
,
1
(1.4)
121
51. Mixed problem in a perforated domain Existence of the matrices
Nq
follows directly from Theorem 6.1, Ch. I.
According t o Theorem 6.2, Ch. I, the elements o f the matrices Nq are piecewise smooth functions in w belonging t o the class
6.
System (1.2) can also be derived by the method of multi-scale asymptotic expansions which is thoroughly described in numerous sources (see e.g. [3], [5], [110]). We shall not reproduce here this well-known procedure since for the system of linear elasticity it is essentially the same as for second order elliptic equations (see e.g. [5]). Theorem 1.1. The homogenized system (1.2) is a system o f linear elasticity, i.e. the elements of the matrices
Ak' satisfy the conditions
for any symmetric matrix 7 = { y i h ) , where it1, k2 are positive constants. In other words the operator
Proof.
k
belongs t o the class E ( k l , k 2 ) .
In the special case of w = Rn,i.e.
R' = R , the relations (1.5), (1.6) 2
can be obtained from Theorem 9.2, Ch. I, since the matrices N,9(x) and
ENq(-) E
ak'satisfy the Condition N which can be easily verified on account of (1.3),
(1.4). In the general case when w may not coincide with Rn, i.e.
RE may be
a perforated domain in the proper sense, Theorem 9.2, Ch. I is not applicable, and we shall use another method t o prove the relations (1.5), (1.6).
Let C be a column vector with components el, ..., en. Denote by C* the line (el,...,en).By A' we denote the transpose of the matrix A. Thus A( = y is
a column vector with components yj = a&,
j = 1, ...,n , and y* = ( * A is a
line with components yj = ciaij,j = 1 , ..., r ~ .
It is easy t o see that the second equality in'(1.5) follows directly from (1.3) and the properties of the elements of the matrices APq(t), since
11. Homogenization of the system of linear elasticity
122
where NA, are the elements of the matrices Nq. Let us establish the first equality in (1.5), which is equivalent t o ( A ~ ' J ) *=
A~P. It follows from the integral identity for solutions o f problem (1.4) that for any matrix M ( J )E W ; ( U ) we have -
dM J 6
dNq
-d t
~ ~ j ( t )
Qnw
%
dM
J
=
.
Qnw
Making use of the relations ( A k j ( t ) ) *= A j k ( t ) , ( A B ) * = B*A* for matrices A, B , we obtain from (1.8) that
-
J
dN9* dM* =Ajk(0,dt=
Qnw
J
~
~
dM* ~ (
o
~ (1.9) d
Qnw
Setting M = NP* in (1.9) and taking into account (1.3) and the relation
(Apj)* = Ajp we find
I t follows that the coefFicient matrices of the homogenized system can be written in the form
t
.
51. Mixed problem in a perforated domain
123
Replacing p by q and q by p in this formula and taking the transpose of the equality obtained, we see that
h'q
= (A~P)'.
In order t o prove the inequalities (1.6) let us note that iiP/qihvjk = qh*AhkVk where
sk is a column with components
qlkr...,qnk, and qh* =
(vlh,
qnh).
...?
For any symmetric matrix 71 with constant elements q;h we obtain due to (1.10) that
Let w = (Nq
+ tqE)qq be a vector
valued function with components
w l , ..., w,. It then follows from (1.11)that 8gq,pl)jq= (mes Q n w)-I
/
dw* -
Qnw = (mes Q n w)-'
dw
-d t = X j
-d(
.
Qnw Suppose that for a symmetric matrix q we have iiY;qipqJq= 0. It then
2 [0.~ ~ ~ ) follows from (1.12) and the estimate (3.13), Ch. I, that I l e ( ~ ) 1 I ~ = Therefore w is a rigid displacement (see the proof o f Theorem 2.5, Ch. I). On
+
the other hand w ( t ) = ( N Q (,E)qq. Therefore due t o the periodicity of
Nq(6) the vector valued function Nqqq must be constant, and the matrix q must be a skew-symmetrical one. It follows that q = 0. Thus i i ~ ~ q i , q > jq 0 for 7 # 0, which proves the lower bound in (1.6). The upper bound in (1.6) holds because o f the formula (1.7) for 2:;. Theorem 1.1is proved.
1.2. The Main Estimates and Their Applications Let us take as an approximation t o the solution of problem (1.1) the following vector-valued function
124
II. Homogenization of the system of linear elasticity
where NP(E) are the matrices defined by (1.4) and u O ( x )is the solution of the problem
Theorem 1.2. Suppose that u c ( x ) is a weak solution of problem (1.1) in
W ,f' E L 2 ( R c ) ,
iPc E H'(Rc), f0 E H1(R), iPO E H 3 ( R ) and u O ( x )is a weak solution of the homogenized problem (1.14). Then
where C is a constant independent o f
E,
the norm
11 . 1,
is defined by (5.3),
Ch. I.
Proof. Applying the operator LC t o uc - ii we obtain the following equalities which hold in the sense o f distributions
Since the matrices N 8 satisfy the equations (1.4), it follows that
Lc(uC- ii) =
125
§1. M i x e d problem in a perforated domain
Define the matrices NPq(<) ( p , q = 1, ...,n ) as weak solutions o f the boundary value problems
NPq is 1-periodic in ( ,
/
N p q ( ( ) d <= 0
Qnw
.
1
The existence of NPq(<)follows from Theorem 6.1, Ch. I and the equalities (1.3). Thus we deduce from
(1.16), (1.17) that
Therefore
L C ( U ' - ~=) fE - f0
a
+EFO+ +- F k , dxk
(1.18)
where
Let us consider now the boundary conditions on Sc for uc - 12. We have
11. Homogenization of the system of linear elasticity
By virtue of the boundary conditions on a w for
Nq
and
NPq
it follows that
On the outer part o f the boundary o f Re we have
Let us show that
where c is a constant independent o f
E.
To this end it suffices t o find a vector
valued function \kc E H 1 ( O c )such that V!,
+ EN'
E H1(R",I',),
We define Q e ( x ) as follows. Let cp, be a scalar function in Cw(Q)such that cpc(x) = 1 if p ( z , a R )
IVcpl
< c2&-l.
5 E . cp.(x)
Set
It is easy t o see that Q , E H 1 ( R c )and
=
o if p ( x , 8 0 ) 2 2&, o 5 cp,(x) 5 1,
81. M i x e d problem in a perforated domain aQc = - & - acp, N S - - E cduo pc-
axj
dxj
dN"uO- dxj d x ,
ax.
Therefore taking into account the properties of cp, and the fact that the matrices N s ( [ ) and dNB(E)/dEjhave bounded elements, we obtain the inequality
~ I P C I I H ~5( O ~3 ~ ( I) I ~ ~ I I H + ' ( KI.I)U ~ I I H ~ ( K.C , )
(1.24)
By virtue of Lemma 1.5, Ch. I, we get IIuOIIH~(K.)
Ic4&lI211~011~2(n) 3
where c4 is a constant independent of E . This inequality together with (1.24) yields (1.23). Therefore estimate (1.22) is valid. On the basis o f (1.18), (1.20), (1.21), (1.22) we conclude that u' - ii is a weak solution of the following mixed boundary value problem studied in 55, Ch. I:
Here 4, satisfies the inequality (1.22) and
where the constant cg does not depend on E , since the elements of the matrices Ahk, NP, Npq are piecewise smooth functions (see Theorem 6.2, Ch. I). It follows by virtue of Theorem 5.1, Ch. I, and Remark 5.2 that
This inequality implies (1.15) since due to the a priori estimates for solutions of elliptic systems (see [I]) we have:
11. Homogenization of the system of linear elasticity
128
Theorem 1.2 is proved. We now prove some important results which follow from Theorem 1.2. Formula (1.13) for an approximate solution of problem (1.1) allows us t o estimate some effective characteristics o f strongly non-homogeneous bodies, in particular the stress tensor and the energy. Let
R' be a subdomain o f R with a smooth boundary. Set
The integrals Ec(ue), Eo(uO) represent the energy contained in and
R' respectively.
Theorem 1.3 (On the Convergence o f the Energy). Suppose that all conditions o f Theorem 1.2 are satisfied. Then
where c is a constant independent of
E.
Proof. It follows from Theorem 1.2 that
II~f(x)ll~2(n*) 5
5s
+ IIfO - f ' l l * +' , 11
+ ~ l @ ~ l l ~ ~ n+( s n ) ) - @'llx~t~~ , r.~]
where q,is a constant independent o f
E.
Therefore
R' n R'
129
$1. Mixed problem in a perforated domain
where
Since the elements of the matrices
dN" Aij, - are bounded (see Theorem Xi
6.2, Ch. I) we get
This inequality together with (1.29) yields
Therefore
Let us introduce the matrices
a
Hat(<) r - (N" X i
+ t S ~ ) ~ 'ja ( (
- (mes Q n w ) k t .
11. Homogenization of the system of linear elasticity
130
In the rest o f the proof it is assumed that the matrices A i j ( ( ) , N s ( ( ) ,
d N " / d ( i are defined in Rnand are equal t o zero in Q\w.
Thus we obtain
from (1.10)
Note that due t o our assumptions we can replace the domain o f integration
0' n R' in (1.30) by 0'. Therefore after a suitable transformation, (1.30) becomes
From this equality and (1.32) we conclude that
Note that due t o Theorem 6.2, Ch. I, the elements o f the matrices Hst are bounded functions. Denote by Jc the set of all vectors z E Znsuch that E ( Q by J: denote the set of all z E
Znsuch that E ( Q
+ Z ) c 0' and
+ z) n dR' # 0. Then
Ec ( u c )- (mes Q n W )E0(u0) = =
x,
'EJ*
-auO* Hat(-)
dx, ~(~+g)nn~
duo
- dx
e dxt
+
It is clear that the first sum in the right-hand side o f (1.36) can be represented in the form
131
$1. Mixed problem in a perforated domain where
G, is an open set lying in the 6-neighbourhood of dR1 and 6 is o f order
E . Therefore using Lemma 1.5, Ch. I, we deduce that
Consider now K 2 , i.e. the second sum in the right-hand side of (1.36). Denote by
R1' the set formed by the cubes ~ ( fz Q ) , when z takes values in
J,. Set 7t(x) = -
mes EQ
J
auO
-dzforz~a(z+Q). t
c(z+Q)
+
The vectors y t ( x ) are constant on each o f the sets ~ ( z Q). We have
It is easy t o see that
'' zEJ.
duo 2
mes EQ
( m e s ~~ ~ ~ ) S~
J lzl ++Q)
E
Q
Taking into account the Paincark inequality in
Let us estimate the sum since
IG.
duo 2
dx =
J lzl
dz
0''
E(Z
.
+ Q ) we obtain
Due t o (1.33) the last integral in (1.38) vanishes
rt are constant on each o f the sets E ( Z + Q) for z E J,.
It follows that
11. Homogenization o f the system o f linear elasticity
Therefore taking into account (1.39), (1.40) we obtain
The inequality
IIU0 IIH~(*)5 C ~ ( I I P I I ~ ( ~ ) + I I ~ O I I H ~ , ~ ~ ~ ~ ) ) and (1.31), (1.36), (1.37), (1.41) imply estimate (1.27).
Theorem 1.3 is
proved. Let us consider now the convergence of the stress tensors, i.e. o f matrices whose columns are
where 'u is the solution o f problem (1.1). The stress tensor corresponding t o the homogenized problem (1.14) has the form
In the homogenization theory the matrices with columns u,P,u,P are also referred t o as weak gradients or flows. In the next theorem it is assumed that
and that
Theorem 1.4. Suppose that the conditions o f Theorem 1.3 are satisfied. Then
$1. Mixed problem in a perforated domain
1 0,'
-1ax,
x auO
- (mes Q n w)u,P - G"(-)
a
where c is a constant independent o f
L2(n)
'
and the matrices GP'J(J) are defined
E,
by the formulas
aN8 GP"J) = AP8(<) Api -- ( m e s ~ n w ) Afor ~ ~J E Q
+
a
~
GP" (() = -(mes Q n w),&'~
for J E Q\w
Moreover, if
fc
,W
(1.45)
.
= f O , @' = @O then
u,"(x) --+ (mesQ n w ) ~ , P ( x )weakly in ~ ~ (as0 E )+ 0
Proof.
Let us make use of the relations (1.28), (1.29) which hold due t o
Theorem 1.2. Then according t o (1.42) we have
-
(apa+ A~ -)d N a ati
duo axs
-+
(x)
.
Therefore taking into account (1.43) we get
u,P(x) - (mes Q n w)ag(x) =
This equality and (1.29) imply (1.44). The weak convergence of u,P(x) t o (mes Q ~ w ) u , P ( x ) for follows from (1.44) and the fact that convergence Gpa(-)
auO ax,
J
= Q0
GP"(J)d( = 0 which implies the weak
Q
- -t 0 in L2(R)as E
I. Theorem 1.4 is proved. .
fE = f O ,
+ 0 by virtue o f Lemma 1.6, Ch.
11. Homogenization o f the system o f linear elasticity
134
$2. The Boundary Value Problem with Neumann Conditions in a Perforated Domain Results similar t o those of $1for the mixed problem can also be proved for the Neumann problem in a perforated domain
Rc o f type II (see $4, Ch. I).
However, in the last case some difficulties arise in obtaining estimates for the boundary values o f the conormal derivative of a rapidly oscillating corrector
x duo E ax,
which has the form E N ' ( - ) - outside a neighbourhood of
do. Therefore
in order t o clarify the main ideas used in the proof o f an analogue t o Theorem 1.2 we shall first consider the Neumann problem in a domain of
E
R
independent
for a single second order elliptic equation with rapidly oscillating periodic
coefficients. It should be noted that the absence of cavities makes the proof much simpler.
2.1. Homogenization of the Neumann Problem in a Domain R for a Second Order Elliptic Equation with Rapidly Oscillating Periodic Coefficients In a bounded smooth domain
R
consider the following Neumann problem
where (y,...,un) is the unit outward normal t o
dR. It is assumed that a i j ( [ )
are smooth functions in R n , 1-periodic in [, and such that
where
61,K~
= const
> 0.
The functions f and cp are sufficiently smooth and
satisfy the condition of solvability of problem (2.1)
J n
fdx=
J
cpd~.
(2.2)
an
We define the functions N P ( t ) , p = 1 , ..., n , as solutions of the problems
$2. Boundary value problem with Neumann conditions
Set
Q={x : O<x3
,...,n ) .
As an approximation to the solution of problem
(2.1)we take the function
x duO(x) 6 = u O ( x ) & N d ( - )-,
+
where
&
ax,
u0 is the solution of the homogenized Neumann problem
duo = 6'3.. duo v; = cp
auA
In analogy with
ax
.
(1.16)simple calculations show
Let us define the functions
Then
on dR
that
N i s ( J )as solutions of the problems
11. Homogenization of the system of linear elasticity
136 where
dNia d2u0 - akh -
FO
ath
k -
k = 1, ..., n
dXSdx; '
dNi"uO Fl = -akh -
-a
a& axsaxiaxk
i j ~ s
,
(2.8)
d3u0
axsaxiaxj
(2.9) a
Consider the boundary conditions
a
-6)
..
-(uC- 6 )
a''
dxj
~ Y A ,
- y - a t J.v. i -duo +
v; =
d N s duo
d2u0
(axj atj ax,
-axsaxj )
=
.. aNs duo .. d2u0 .. auO - -a'J -vi - - &a'3ViNs-+ (iY3 - at3)ui- .
dfj
dxa
dx,dxj
dzj
Therefore
d -
-6)
= [;is
E
~ V A .
-
1-atj
x .. x d N S duo - ass(-) (-1 vi - E
dx,
a2u0 &at3viNS-. dx,dxj
(2.10)
W e introduce the following notation & l'a
ij=
- a i a ( f )- a''(() a N s ( < ),
ati
(El,..+,Ej-l,fj+l,
$={(I
:
..., En) E Rn-l
&=t,o
2.1. Functions a i d ( t )defined by (2.11) satisfy the relations
Lemma
a Qi a ( f ) = 0 at;
J
s = 1,...,n ;
,
d S ( [ ) d i j= o ,
s; (there is no summation over
j).
4 E R' , s,j
= 1, ..., n
,
137
92. Boundary value problem with Neumann conditions
Proof. The equalities
(2.12), (2.13) follow directly from (2.3) and the definition of hhk. Let us prove (2.14). Denote by Q;,,, the set
Multiplying (2.13) by
tj - t l and integrating over
/ aia(0(t2
- tl)d& -
=
g2
/ Q:,
we get for each j
d S ( ( ) d (.
t,
Setting tl = t 2 - 1 and taking into account (2.12) we obtain (2.14) for t = t 2 . Lemma 2.1 is proved. Lemma 2.2. Let a'"(() be functions in H 1 ( Q ) 1-periodic in ( and satisfying the conditions (2.12)-(2.14). Then for any v E H 1 ( R ) the following inequalities are valid
where c is a constant independent of
E,
v.
Proof. Denote by I: P(E(Z
+
the set of all indices z E Znsuch that ~ ( $zQ) C R , (EZ Q ) . It is easy to see that Q ) ,d o ) 2 E . Set R1 =
+
U
~€1:
d ax;
n\nl
-
J an,
Therefore
x
- (aik(--))vdx =
0 =
J
aikvivdS -
an
aiku;vd~-
J
R\RI
dv aik -dx dzi
,
IC = 1, ..., n
11. Homogenization of the system o f linear elasticity
We clearly have
Let us estimate the first integral in the right-hand side of (2.16).
It is easy t o see that
an1consists o f ( n- 1)-dimensional faces o f the cubes
I + Q ) for some z E If. Denote by uj',..., uj) the ( n - 1)-dimensional faces + Q) for z E Ifsuch that ujk is parallel t o the hyperplane of the cubes E(Z
E(Z
xj
= 0 and lies on
an1,j = 1,...,n. We thus have
+
Denote by qj" the cube ~ ( z &) whose surface contains the set u,9. It is obvious that among the cubes q;, j = 1, ..., n; s = 1, ...,l j , there cannot be more than 2n identical ones. Due t o the condition (2.14) for any u; we have
since ui = 0 for i # j (there is no summation over j).Set
) constant on each surface a,". Thus the function ~ ( x is Taking into account (2.18) and the fact that not more than 2n cubes q," can have a non-empty intersection we obtain
$2. Boundary value problem with Neumann conditions
Here we have also used the inequality
which can be proved i f we pass t o the variables [ = e-'x and apply Propositions 3 and 4 of Theorem 1.2, Ch. I, in the domain R = E-'qj". The relations (2.16), (2.17), (2.19) imply (2.15). Lemma 2.2 is proved. Therefore due t o (2.7)-(2.10)
we have
where Fl, F t , F2 are bounded uniformly with respect t o Setting w = uc - 6
+ vc where 7'
E.
is a constant such that
/w
d x = 0, we
n
obtain from (2.21)
auO
Applying Lemma 2.2 t o v = -w and using the Poincare inequality (1.5), Ch. I, we find that
ax,
140
II. Homogenization of the system of linear elasticity
Thus we have actually proved Theorem 2.3. Suppose that uc, u0 are solutions o f problems (2.1), (2.5) respectively, and f , cp are smooth functions satisfying the solvability conditions for problems (2.1), (2.5). Then there is a constant qc such that
where c is a constant independent o f
E.
In the same way as it was done in $1we can obtain estimates for the closeness o f energy integrals and weak gradients related t o problems (2.1), (2.5). In this section we omit the consideration of these questions. However, estimates of this kind for the system o f elasticity in a perforated domain are established below.
2.2. Homogenization of the Neumann Problem for the System of Elasticity in a Perforated Domain. Formulation of the Main Results In the rest o f this section Rc denotes a perforated domain of type II defined in $4, Ch. I. The boundary dRc is a union o f d R and the surface of the cavities
S, c R . In Rc we consider the boundary value problem o f Neumann type for the system o f linear elasticity:
It is assumed that the elements of the coefficient matrices Ahk satisfy the same conditions as the coefficients of the system (1.1),
fC
E L2(Rc),
@ E L 2 ( d R )and satisfy the conditions of solvability for problem (2.22), i.e.
'$2. Boundary value problem with Neumann conditions where
R is the space of
rigid displacements.
Existence and uniqueness (to within a rigid displacement) of a solution of this problem follow from Theorem 5.3, Ch. I. Consider also the Neumann problem for the homogenized elasticity operator
(mes Q n w)6(u0)= Go
J
on 8 0 ,
ahk
duo vhahk-, the matrices are defined by the formulas (1.3), axk 11,' E L 2 ( d R ) ,f 0 E L 2 ( R ) ,satisfy the solvability conditions where 6(u0)
I (Go,
(mes Q n w ) - I
l)dS =
an
/ ( f Ol,) d x
Vq E R .
R
It is important t o note that the factor mes Q n w appears in the Neumann d R . This factor is equal t o 1 if RE coincides with R (see $2.1,
conditions on
formula (2.5)). To characterize the closeness between functions f O ,
11,' and
fc,
1,' we
introduce the following notation.
L2(Rc),11, E L 2 ( d R )the scalar prod1define ( ~ continuous ~ ) linear functionals on H1(R'),
For any vector valued functions f E ucts
( f ,v ) ~ z ( ~($, . ) ,~
)
~
and therefore f and 1C, can be considered as elements of the dual space
H1(Rc)*.Let us denote the norms of the respective functionals as 11 f ~
~
i.e.
~
~
~
H
~
*
llH1.,
~
IIHI*
Note that I l f 5 I l f l l ~ z ( n *l l)1 ~1 , l l ~ l - I c I I ~ I I L Z ~ ~ ~ ) . We seek an approximate solution of problem (2.22) in the form
x duo a ( x ) = uo + + E ~ ~ ( X ) N~ ( -. ) E dx, Here (1.4),
u0 is the solution o f problem (2.23), N Y t ) are solutions of problems cp(x) is a truncating function which satisfies the following conditions:
11. Homogenization of the system o f linear elasticity
cp E C c ( R ) , IVql lCE-' , cp = 0 in R\R1
,
cp(x) = 1 for x E R1 such that p(x,dRl) 2 CIE , where cl, c are constants independent of
E;
(2.25)
R1 is defined by formula (4.3),
Ch. I. In contrast t o the case, considered in 52.1, o f a single second order elliptic equation in a non-perforated domain, here the truncating function cp enters the expression for C (cf. (2.4)) since the solution u ' is considered in the per-
x Rc but the matrices N S ( - ) are in general not defined in a & neighbourhood o f dR.
forated domain
The main result of this section is Theorem 2.4. Suppose that f c E
L2(Re),f0 E H 1 ( R ) ,@ E L 2 ( d R ) ,6 ' E H~/~(~R).
Then the solutions u', u0 of problems (2.22), (2.23) respectively satisfy the following inequality
where c is a constant independent o f depend on
E;
7" is a rigid displacement which may
E.
The proof o f this theorem is given in Section 2.4 and is based on the lemmas established in the next section.
2.3. Some Auxiliary Propositions Let us introduce the notation
dNS
a''(<)= 2' - A"(<)- A i j ( t ) - ,
atj
i, s = 1, ...,n
$2. Boundary value problem with Neumann conditions
Lemma 2.5. The matrices a i s ( ( )satisfy the following conditions
J
d s ( ( ) d i j= (mes 2 : - mes Q n w ) i j S ,
5; (there is no summation over j).
Proof. Equalities (2.27),
(2.28) follow directly from (1.3), (1.4).
Let us prove (2.29). Multiplying the system (1.4) by ((j - t l ) E , where is the unit matrix, and integrating over Qj,,,
Each integral in (2.30) over d(Qi,,,
E
n w, ( t l < t z ) ,we obtain
n w ) can be represented as a sum of
integrals over the sets
Q:,naw,
S:,u$,,
(j $ u s ; . r=l +I
Since
and the integrands are 1-periodic in (,, r (2.30) that
#
j , r = 1, ...,n , it follows from
II. Homogenization of the system of linear elasticity
(there is no summation over j). Setting t, = t2 - 1 in this equality and taking into account (1.3) we find that
It follows from the definition o f a'" that
A
.
A
.
= (mes S,3)A3' - (mes Q n w ) k s = = (mes j;j - mes Q n w ) A j s
.
Lemma 2.5 is proved. Remark 2.6. If the domain
/
Re is not perforated, A
i.e. w = Rn,Rc = R , then
.
a j S ( t ) d i j = 0, since mes S,3 = rnes Q n w = 1 (see Lemma 2.1).
9; Lemma 2.7. Let al, ..., a2, be (n - 1)-dimensional faces of the cube EQ = {x : 0 E,
j = 1, ..., n). Then each u E H1(eQ) satisfies the inequality
<xj <
$2. Boundary value problem with Neumann conditions
where c is a constant independent of
M.
Set o1 = { x l = 0)
i, j, E .
n EQ, o2 = { x 2 = O} n E Q , S1 = & - l o l , = (0, y2, y3, ...,y,), G2 = ( y 2 ,0 , y,, ..., y,)
s2 -- a -1 a2. Consider the points 6'
on the faces Sl, S2 o f cube Q. The segment g ( t , y2, y3, ...,y,) = tjjl + ( l -t)jj2 for
t E [O,1] belongs t o Q . It is easy t o see that for any v E H 1 ( Q ) we have
Integrating this equality with respect t o y2, ..., y, from 0 t o 1 we obtain the estimates
&-(,-I)
J
01
v2
do - & - ( n - l )
/ v 2 d o 4 CE-"E / I v I
02
1VrvI d x
EQ
which imply (2.31). Estimate (2.31) for other faces can be proved in a similar way. In the next two lemmas we establish some inequalities, uniform in E , for functions defined on the set d R 1 which is the boundary of the domain O 1 given by formula (4.3), Ch. I. The domain R 1 depends on a and its boundary 6 ' 0 1
+
consists of the ( n - 1)-dimensional faces of the cubes ~ ( zQ), z E T,. Denote by
a,!, ...,o) the faces o f the cubes a ( z + Q) for r E T,
the hyperplanes x j = 0 , j = 1, ...,n , and laying on d o 1 . Then
parallel t o
II. Homogenization o f the system o f linear elasticity
146
+
The cube ~ ( z Q ) , z E
T,, on whose boundary lies the set ojS is denoted by
qjJ It is easy t o see that among gj, j = 1, ...,n , s = 1, ..., lj, the number of
the identical cubes is not greater than 2n. Lemma 2.8. Let u E H1(R). Then
where c is a constant independent of
Proof. According
E.
t o (2.32) dR1 consists of the sets ufi,and each ufiis an
(n - 1)-dimensional face o f the cube
qg.
The boundary dR is a smooth
surface, therefore each cube qj" possesses a face u,j such that u,,jis parallel to the hyperplane xm(j,,) = 0 and u,,jis the orthogonal projection along the of a surface SaVjC dfl which is given by the equation
axis
and cle
5
Ix - yl
5 c2c for
x E o,,j, y E
where constants cl, cz, M do not depend on Denote by
Q S jthe
Q S jt o EQ
s, j.
set formed by the segments orthogonal t o u,j and
connecting the points o f a,,. and mapping
E,
Ss,j,
SaVj.Then
using a suitable diffeomorphism
and taking into account Lemma 2.7 we find that
2 IIullr2cS,,) 5 ~ ( I I ~ I I ~+ ( sI I,~,I ,I ~ ~ ( Q , , ).)
Therefore by Lemma 2.7 we get 2 IIuIIL~(~;, 5 c l ( l l ~ l l t ( ~ ,+ , ) llullBl(Q.,,)) .
Summing these equalities with respect t o s, j we obtain estimate (2.33), since due t o the smoothness o f dR there is an integer k independent of
E
and such that each Q,, can have a non-empty intersection only with a finite number o f QI,$which is not greater than
k.
Lemma 2.8 is proved.
Lemma 2.9. Let the matrices yhk(x) E Lm(dfll) be such that
$2. Boundary value problem with Neumann conditions
[ yhk(x)ds= 0
(there is no summation over h )
,
where u r are the same as in (2.32), y = const. Then for any vector valued functions uOE
H3(SZ), w E H1(LR) the inequality
holds with a constant c independent o f
Proof. Consider a function
E.
r ( x ) defined almost everywhere on dR1 by the
formula
r ( x ) = (mes a;")-'
J ~i"
duo uhyhk-dS for x E
Obviously r ( x ) is constant on each
ax,
0;". Therefore
07
setting
&(x) = (mes oY)-'
i" and taking into account (2.34) and the Poincarh inequality in a;" we obtain
5C '1
=
j=1
mes 0;
s=l (rnesq12
II. Homogenization of the system of linear elasticity
148
It follows from Lemma 2.8 that
J
Ir12dS
<
~
IIuoIIZx3(n) ~ ~
2
7
~
anl where c2 is a constant independent o f
E.
It is easy t o see that
Due t o (2.36) and Lemma 2.8 we have
Let us estimate the second integral in the right-hand side of (2.37). Define
) dR1 by the formula the vector valued function ~ ( x on r)(x) = (rnesd,)-'
/
w d x for x E 0;
s!"
Therefore
Here we have used the fact that among q; the number of identical cubes is not greater than 2n; and we have applied the inequalities (2.20) for v = w. By the definition of
r we have
$2. Boundary value problem with Neumann conditions
Therefore since q ( x ) is constant on each at,we find by virtue o f (2.39), (2.36) and Lemma 2.9 that -
This inequality together with (2.37), (2.38) yields (2.35). Lemma 2.9 is proved. 2.4. Proof of the Estimate for the Digerence between a Solution of
the Neumann Problem in a Perforated Domain and a Solution of the Homogenized Problem In this section we give proof o f Theorem 2.4. Let us apply the operator ii is given by (2.24). Then
a (A") ,auc r.(~€ - ii) = axh xk -
C, t o the vector valued function uc - 6, where
d (A" -) due - d (Ah* -
a
auO + EPN' -)) ax,
axh 8 (Ahk duo -)duo + -
-
axk
-&ah3
axh
dxk
dxk
dxj
=
duo
-)axk
axk axh axk d(cpNa) duo a2u0 -a [Ahk(& --+ E ~ N -)] ' = dxh ax/; dx, dxkdxs a duo auO ~ ( P N auO ~ ) = f'- f ' + - (A" axh
axh
a
- - (ahk- (,O
-) dxk
-
-
150
11. Homogenization of the system of linear elasticity
Taking into account the equations (1.4) for
- 6) = f' - f 0
a ((1- $,)(A" +dzh
+
[Ahk- Ah* - &Ah;
-f
8 9 [Ahk- Ahk - E A
d -
d axh
- E(P -( -
1-d x j
+
dxk
-E*AhkNs
dxk
axh
d2uO dxkdx,
d2u0 A ~ ~N ~ )axkdx, d3u0 = I' - f0 dxkdxhdxs dNk dx;
+
[A"
-
d3u0 E ~ A ~ ~ N ~ dxkdxhdx, '
-~
duo
-axk 1
duo ~ hd lN k -
!&AhjNk!f]
B x ~ axj
-
+
axh
-
N" we obtain
h * & ~ hj
T
a [ ( 1 - $,)(A" - A") -1duo + +dxh dxk d A S h ~ ' aZu0 a$, auO +I ax, a ~ ,
Let us define the matrices N h k ( J )as weak solutions of the following bound-
ary value problem
151
$2. Boundary value problem with Neumann conditions
a
dNhk
a
a~~ atj
) ~ hj - Ahk - - -( A " N ~ ats
a ( N h k )= -vsAshNk on
,
I~W
+ Ahk in
w
(2.40)
,
N h k ( t ) is 1-periodic
in
[ .
Then
Lc(uc - i) = f'- f0
a [(I - p ) ( A h k- Ah*) -1auO + +axh ask
d dNhk + pa (ul8--) axj (I
d2u0 dxkdxh
dp dxh
duo axk
-+ -ahk - -
-&
Ad~ axh
hk NS
d2U O -
d ~ ~ d ~ ,
d3u0 d ~ k d ~ ~ d ~ ,
-a p ~ h k ~ s
a -
&
a d~p axj
j
l
d-N h k -d2u0 dxkdxh
W e thus have
ajN lh k ~ p ~-
a~~~ a2~' d3u0 axkdxhdxj
+
11. Homogenization of the system of linear elasticity
152
~1 h --(
1 - cp)(ahk - ~
duo dxk
-- E
h k )
ap axj
-A
\
duo 3N - , axk
h.
d N h k d2u0 F? = &pAjl- -, 3 X I axkdxh 89 d N h k -a2u0 E-A atl dxhdxk axh . dNhk @uO = -&pA" atl d x k d x h d x j dp duo = -'p-, axh dxk
+-&-AjI-d p dxj
Consider the boundary conditions for
hk
d2u0 NSd2kdx8 ' d3u0 dxkdxhdxs '
u' - ii:
a
auO
. . due .. gC(ue- ii) = A'jv, - - AtJvi - (uo + € p N 8 -) dxj dxj 8x8
. due
. . due
..
duo
- uiA'J - axj + +uiA'J axj dxj
= ( 1 - ~Y)~,A'"
. . d N 8 duo - ,cviA'3p - -- E
dxj
~
ax,
duE axj
= ( 1 - c p ) ~ i j v i-+ ( 1 - (p)vi(,$'j
axj
d2u0 dx,dxj
-~
duo
=
.. dp N 8 duo ,cViAS3
A ~ -~
~
(2.42)
i j) d xj
~
N
axs
~
duo ( 1 - p ) ~ 1 .3.v i d xj A
-
.. duo a~Nl j duo - ,cViA13 .. d(p duo cpV,A'3 - - , c ~ , ( p ~- NS - axj axl dxj axj ax,
-
uo due . . duo &viA"pN8 -- ( 1 - p ) ~ " v ;- - ( 1 -cp)A"vi axj dxj
a2 axsaxj
A
.... duo + (1 - p)vi(AV - A") - - p (v,A'j + EV,A'I d xj
-
. d(p duo - EyiA'~cpNs d2u0 ,rviAt3 -N s --dxj dz, dxsdxj '
By virtue o f the boundary conditions in (1.4), (2.40) we have
c p ( v i ~ G+ E U , A ~ I
-)dN'
8x1
=o
,
aNJs
-EU~~A= ~ ~E N~" u ~ A ~ anc ~ .
at1
Therefore taking into account (2.42) we find that
+
$2. Boundary value problem with Neumann conditions
Set w = u"
- 4 + qE,where q'
is a rigid displacement suchlthat
ER .
( w , ~ ] ) ~ I= ( ~O. )for any
Due t o the boundary conditions for uO,ue and the fact that cp = 0 in O\Ol
it follows from (2.41), (2.43) that
Let us estimate the integrals in the right-hand side of this equality. Note,,
89 and 1 - cp vanish in {x that owing t o (2.25) the functions O1,p(x,aO1) 2 cis) and of
E.
ax
IsVcpl 5 c, where c, cl
:
x E
are constants independent
Therefore by Lemma 1.5, Ch. I, we obtain
where c2 is constant and does not depend on
E.
I t follows from (2.42) that
where c3 is a constant independent o f
E.
Taking into account (2.27), (2.28) and setting a = mes Q n w we get
11. Homogenization o f the system o f linear elasticity
154
- -
/
(hahk
anl\s.
duo
-,w)d~+ a xk
/
(viAij
anl ns,
duo
-,w)d~
ax
+
It should be noted here that in the integral over (aR1)\Sc the normal v is exterior t o dR1, whereas in the integral over dRl
n S,
the normal v is exterior
to RE. The last two integrals on the right-hand side o f (2.47) can be estimated by
similarly t o (2.45). Let us introduce the matrices phk([) setting
Then
It follows from (2.47), (2.48) that
$2. Boundary value problem with Neumann conditions
lJll
+ CE"~
I IJ21
llwllHl(ne)
~ ~ U O I I ~ ~ ( ~ )
155
,
where
J / (phk($)uhduo ,W ) ~ +S ax mesQ n w
J2 = -
Q\w
k
anl
($O,
+
W ) ~ S
an
The integral identity for u0 yields that
Therefore by virtue o f Lemma 1.5, Ch. I,
mesQ n u
/ (~O,w)dS
=
an
where IJ3I
5 ~ 1 / 2 ( ~ l ~ 0 1 1Ilwllel(n*) ~ z ( n ) + Ilf011r2cn)llwllxl(n*,) . (2.52)
W e thus obtain from (2.50), (2.51) that
Jz =
/
((mes g\w),Ahk
duo dxh
- v h p k -,w)d~
an1
+J
(Go - @ , w ) d S + J3
+
-
an Set
-yhk = (mes Q \ w ) A ~-~ phk in Lemma 2.9. It is easy to verify that conditions (2.34) are satisfied for -yhk. Indeed due to (2.48) and (2.29) we have
156
J
11. Homogenization o f the system of linear elasticity
-yhk d~ = an-' (mes Q\w)A~*-
ohm
/
or\s*
= an-'(mes Q\w)Ahk
-
/
/
phkds-
phkds =
qnSc
ah*d~ - (mes or n sC)Ahk =
a,"\&
-
(mes or
n sC)Ahk = en-'
(mes Q\w - mes a-' (or\$)
+
t rnes Q n w - rnes €-'(or n sC))Ahk =0 , since
mesQ\wtmesQ~w= 1 , mes E-l (op\S,)
+ mes E-'
(or n S,) = 1
W e conclude from (2.52), (2.53) and Lemma 2.9 that
I t follows from (2.44), (2.45), (2.46), (2.49), (2.54) that
From the well-known results on the smoothness of solutions of elliptic boundary value problems we have
since d R is a smooth surface and
f0
E H 1 ( R ) , 4' E H ~ / ~ ( ~ R ) .
Therefore by virtue of Theorem 4.4, Ch. I, the inequalities (2.55), (2.56) yield (2.26). Theorem 2.4 is proved.
$2. Boundary value problem with Neumann conditions
2.5. Estimates for Energy Integrals and Stress Tensors Slightly modifying the proof o f Theorems 1.3 and 1.4 on the convergence of energy integrals and stress tensors one can establish similar theorems in the case o f the Neumann problem. To this end we should use estimate (2.26) instead of (1.15). Theorem 2.1Q (On the convergence of energy integrals). Suppose that all conditions of Theorem 2.4 are satisfied and E,(uc), Eo(uO) are defined by (1.25), (1.26). Then
where C is a constant independent of
E;
u E , uO are solutions o f problems
(2.22), (2.23) respectively. The proof of this theorem in the main repeats that o f Theorem 1.3. However, slight modifications should be made. In particular we consider the solutions uOand uc such that
J(u',q)dz=~(u0,q)dz=O, V ~ E R . nz
(2.58)
nc
This choice of ue and u0 is possible since solutions of problems (2.22), (2.23) are defined t o within a rigid displacement. In this case one can take
qc = 0 in (2.26), and use the estimates
which are well known from the theory o f elliptic equations (see 111). Similarly t o Theorem 1.4 we establish
II. Homogenization o f the system o f linear elasticity
158
Theorem 2.11 (On the convergence of stress tensors). Suppose that all the conditions o f Theorem 2.4 are satisfied and u E ,uO are orthogonal t o the space o f rigid displacements as in (2.58). Let the stresses
a,P(x),a:(x) be defined by the formulas (1.42), (1.43). Then
1 0:
- (mes Q
auO ax, 11L21fi) 5
n w ) o i - GPq(-) 8
where c i s a constant independent o f E , the matrices GPq are defined by (1.45). Moreover
UP(.)
4
(mes Q n w)u:(x) weakly in L 2 ( R ) as
E +0
2.6. Some Generalizations For the homogenization o f eigenvalues and eigenfunctions related t o the Neumann problem (2.22) for the system o f elasticity in a perforated domain we shall need some results on homogenization of an auxiliary system. Consider the Neumann problem
L c ( u e )- pe(x)ue = f" in Re
,
and also the corresponding homogenized problem
L ( u O )- pO(x)uO= f0 in
,
(mes Q n w)&(uO)= $0 on d a
E
,
where operators L,, are the same as in problems (2.22), (2.23), the functions PC
E
L w ( R c ) ,PO E L m ( R ) are such that
159
52. Boundary value problem with Neumann conditions and constants co, cl, c2, cg do not depend on
E.
In Theorem 2.4 we established the closeness of solutions of problems (2.60), (2.61) when p,
= 0, po = 0.
If we introduce a parameter characterizing the
closeness o f p, t o po it becomes possible t o prove a similar theorem for the problems (2.60), (2.61) under the conditions (2.62). In particular it is o f interest t o consider the case in which p,(x) = 2
p ( ; , x ) , p ( ( , x ) is 1-periodic in ( and satisfies the Lipschitz condition with respect t o x E R uniformly in <, i.e. p((, x ) E 1.6, Ch. I. Let pO(x)
J!,(R"x 0 ) in terms of Lemma
-- ( p ( . , z ) ) ,where ( p ( . , x ) )is defined by (1.23),
Ch. I, and is
equal t o the mean value o f p([,x) with respect t o t . I t follows from Lemma 1.6, Ch. I, that for any vector valued functions
u , v E H'(RE)we have
) Lemma 1.6, Ch. I, where Indeed, set g(<,x)= ( p ( < , x )- p o ( x ) ) x w ( (in
xw(<)is the characteristic function of the domain w with a 1-periodic structure. It is easy t o see that g(<,x)E
L(nnx !=I),
g((,x)d( = 0. Consider the extensions P,u, P,v o f
u,
v t o the domain R
6which were constructed in Theorem 4.2, Ch. I. Then
Note that the set fl\R1 order
E.
belongs t o a 6-neighbourhood o f dR and 6 is of
Therefore applying Lemma 1.6, Ch. I, t o estimate the first term in
II. Homogenization of the system o f linear elasticity
160
the right-hand side o f this inequality, and Lemma 1.5, Ch. I, to estimate the second term, we obtain qc 5 cl&IIpeuII~l(n)IIPevIIH1(n)
This estimate together with (4.17), Ch. I, yields (2.63). x Therefore the functions p(- ,x ) and po(x) are close in the sense of the E inequality (2.63). In a more general situation we shall characterize the closeness of p, and po by the norm
where the supremum is taken over all vector valued functions u , v in H 1 ( R c ) . Relation (2.64) implies that for any u , v E H 1 ( R c )we have
It is easy t o see that estimate (2.63) implies Lemma 2.12.
x Let pc(x) = p ( - , x ) , po E
( p ( . , x ) ) ,p ( t , x ) E ~ ( I R "x
where c is a constant independent of
fi). Then
E.
Theorem 2.13. Suppose that f' E L Z ( R c ) ,f 0 E H 1 ( R ) , $f E L 2 ( a R ) , go E H ~ I ~ ( ~ R ) ,
E C 1 ( Q )and ue, u0 are the solutions of problems (2.60), (2.61) respectively. Then
PO
-1
x duo Ilue - u0 - a p N 8 ( - ) E d x , HIPc)
5
$2. Boundary value problem with Neumann conditions
161
where the constant C does not depend on E , the function cp is defined by the conditions (2.25) and is the same as in Theorem 2.4. The proof of this theorem is almost identical to that of Theorem 2.4. Here we briefly outline its main steps referring t o the proof of Theorem 2.4. An approximate solution of problem (2.60) is sought in the form
x duo ii = uO &(pNd(-) - , E ax,
+
where u0 is the solution of problem (2.61), N" are the same as in Theorem 2.4. Applying the operator C, - p,I to uc - ii we obtain
C,(uC - ii) - p,(u" - fi) =
where
e,e, are defined by the formulas (2.42) and
F;,q ,
For u,(uE - ii) the formula (2.43) remains valid. Setting w = uC-ii we obtain from the integral identity for problem (2.68), (2.43) the following relation which replaces (2.44):
Owing to (2.65), (2.69) we have
162
11. Homogenization of the system of linear elasticity
Formulas (2.45)-(2.50)
remain the same.
In order to obtain (2.53) one should use the integral identity in R\RI for the solution u0 of problem (2.61). Further changes in the proof of Theorem 2.4 are obvious.
$3. Asymptotic expansions for solutions o f boundary value problems 163 $3. Asymptotic Expansions for Solutions of Boundary Value Problems o f Elasticitv in a Perforated Laver
3.1. Setting of the Problem Consider a domain REo f the form
where w is an unbounded domain with a 1-periodic structure satisfying the Condition
B
of $4.1, Ch.
I,
E
> 0 is a small
parameter, and
E-'
is an integer
number. Set
If w In
# Rn,then
RE is a perforated layer.
0' we consider the following boundary value problem
Here A h k ( [ ) are ( n x n ) matrices o f class E(tcl, t c 2 ) ( t c l , tc2
> 0 ) whose
elements a ; / ( ( ) are functions 1-periodic in (. Existence, uniqueness and estimates for solutions o f problem (3.1) under suitable assumptions on
f, Q 1 , a2 are established
in Section 6.3, Ch. I (The-
orem 6.5). In this section it is assumed that are 1-periodic in 2, j = 1,2.
f E C " ( R n ) , iP3
E
C m ( R n - I ) ,f , i P j
II. Homogenization of the system o f linear elasticity
164
Our aim is t o find an asymptotic expansion for the solution u' in powers of the small parameter
E
and t o obtain an estimate for the remainder.
In the case o f a single second order elliptic equation such an expansion was constructed in
[loll.
For any integer k
Here we reproduce the results obtained in [87].
> 0 the solution ue of problem (3.1) can
be represented
in the form
where PA((,&)are n x n matrices such that
and PA, are 1-periodic in (, PA1(()and PA,([,(, ers, the components of the vectors
f ) define
boundary lay-
Y;(x,E)are polynomials in s whose co-
efficients can be expressed in terms of solutions o f boundary value problems for the homogenized system of elasticity with constant coefficients in the layer
{x
: 0
< x, < d ) , the remainder p k ( s , x ) satisfies the inequality
with a constant M k , independent o f
E.
3.2. Formal Construction of the Asymptotic Expansion We seek the solution o f problem (3.1) in the form
In contrast t o Chapter I, here for the sake o f convenience we use the following notation
Dav =
d'v axa1...dx,,
,
a = ( w,..., a ' ) ,
( Q I ) = ~ ,
a, takes the values 1, ..., n; N a ( ( ) are matrices whose elements are 1-periodic in J; v e ( x )= ( v f ,...,v i ) is a vector valued function 1-periodic in 4. Substituting the series (3.2) in (3.1) and taking into account that
$3. Asymptotic expansions for solutions of boundary value problems 165
we obtain the formal equality
+
x 03
E-I
&I
I=O
a N ( I ) dD0v, A ~ ~ (A I) (a)=l 8Ij dxk
+
C
Here we used the following notation
+ Aalj(o a
N"2 ..." , ( I )+ A""(0Na3
,,, ( I )
>
for (a) 2, a
I
a
a
a
(ak'(<) 6T;N] ( I ) ) + BF; (Aka1( < ) N o ( ( ) )t
11. Homogenization of the system of linear elasticity for ( a )= 1,
for (cr) = 0. Substituting the series
( 3 . 2 ) in
the boundary conditions (3.1) we obtain
the equalities
For
x E dRc\(I'o U I'd), E = C ' X ,
we have
w
+ aka'( ~ ) ~ a ~ . . . Da ~a v(et()x )) s C &I-' I=O
C
(a)=l
Ba(C)Daul(x)2 0
where
B,(F) = v k ( ~ * j ( taNa;i;(t) ) for
( a )> 0 and
+ A ~ ~ ~ ( ~ ) N , ~ . . . , ~ (( O3 . 3) )
$3. Asymptotic expansions for solutions of boundary value problems 167
for ( a ) = 0. Let us represent
N,(t)
in the form
where N:(t) are matrix valued functions 1-periodic in J,
N i , Nz define bound-
ary layers near the hyperplanes x, = 0 and x, = d respectively.
: = E , N,' = No2 = 0, where Set N
E is the unit ( n x n)-matrix. Denote
where N,P= 0, if the length o f the index a is negative. Set
The matrices N:(t)
are defined as solutions of the recurrent sequence of
problems
N:(t)
is 1-periodic in
t,
J
N:([)dt
=0
,
Qnw
(a)= 1,2, ... . Existence and uniqueness o f N : can be easily established by induction due t o Theorem 6.1, Ch. I. We define the matrices NA, N : successively with respect t o ( a )= 1 , 2 , ... as the solutions o f the problems
11. Homogenization of the system of linear elasticity
where h i , h i are (n x n)-matrices with constant elements chosen in such a way that the inequalities
5 c;eXp(-K;(-
hold with constants C,", C,",
K,:
d E
- S))
n i independent o f
,
E.
Existence o f the solutions for problems (3.6), (3.7) and existence o f the constant matrices h i , h i can be proved by induction on the basis o f Theorem
8.4, Ch. I. Note that because o f the boundary condition in (3.7) on the hyperplane
d
tn= - , the matrices N,2 and h i depend on E . E
If d is a multiple o f E it follows
from the 1-periodicity in J o f the matrices Ahk(J) that the dependence of
N:([) on
E
is determined by the relation
53. Asymptotic expansions for solutions o f boundary value problems 169 where
N:([) are solutions of the corresponding sequence of
problems of type
( 3 . 7 ) in w ( - m , 0 ) with the boundary conditions
Obviously the matrices
N:, k i
do not depend on
Having thus defined N,P, p = 0,1,2,,
let
US
E.
substitute v, in ( 3 . 1 ) . We get
the formal equalities
It is easy t o verify that here
and the boundary conditions on (dOE)\(ro U
r d )are satisfied
due t o the
boundary conditions in ( 3 . 5 ) , ( 3 . 6 ) , ( 3 . 7 ) for the matrices N:, NA, Note that by virtue o f (3.4) the constant matrices
N:.
h0,,,2 are defined by the
formulas
h:l,2 = (mes Q n w ) - l
J
+
-8% )dt
( ~ " l " ~~ ( t" )l j
Qnw
atj
Comparing these equalities with (1.3) we conclude that hyj =
. aij,
i, j =
1, ...,n, i.e. h:j are the coefficient matrices o f the homogenized elasticity system. Let us seek
v, in the form of a series
Substituting v, given by (3.13) in (3.10) and taking into account (3.12) we obtain the following formal equalities
11. Homogenization of the system o f linear elasticity
Therefore by virtue of (3.10) we find that
Consider the first equality in (3.11). Due t o (3.13) it is obvious that
By virtue of (3.9) we have
Therefore the first equality in (3.11) yields
In the same way we find that
Equating the terms of the same order with respect t o & in (3.14), (3.15), (3.16) we get the following recurrent sequence of problems for V,(x):
$3. Asymptotic expansions for solutions of boundary value problems 171
Here j
(PP=-CC
@P, 'PoP --
hEVmK-l,
p=1,2,
(")=I
I=1
jt2
q0= f
,
~j
=-
C C
(3.18)
h 0 , 2 ) " ~ , + ~, - ~
1=3 (m)=l
j'
= 1,2, ... .
Existenceof
Vj followsfrom Theorem 6.5, Ch. I, when w = IRn and the
coefficients of the elasticity system are independent of
E.
3.3. Justzjication of the Asymptotic Expansion. Estimates for the Remainder In the previous section we constructed a formal asymptotic expansion for the function ue which is the solution of problem (3.1). This asymptotic expansion has the form (3.2) where
1%
= N:
+ NA + N:,
N:, NA, N: are
solutions of problems (3.5), (3.6), (3.7) respectively, v, is given by (3.13),
V,
are solutions o f the problems (3.17). Let us seek an approximate solution o f problem (3.1) in the form
where k
In the next theorem we give an estimate for the remainder term of the asymptotic expansion for the solution u' of problem (3.1). Theorem 3.1. Let uc be the solution o f problem (3.1). Then for each integer k
3 0 we
have
11. Homogenization of the system of linear elasticity
where Mk is a constant independent o f E , u ( ~is)defined by the formula (3.19). Before giving the proof o f this theorem let us establish the necessary estimates for the matrices N:, p = 0,1,2. Lemma 3.2. The solutions N:, p = 0 , 1 , 2 , o f problems (3.5), (3.6), (3.7) satisfy the inequalities
where ME, c j , yj are positive constants independent of
Proof. Let us establish
E
(3.22) for p = 0. By induction with respect t o (cr) =
0,1,2, ... we obtain from Theorem 6.1, Ch. I, that
Changing the variables x =
E<
and taking into account the 1-periodicity
of N i ( J ) and the fact that the domain
+
cells
E(Z
fY
+
contains not more than (d
+ Q n w ) , z E Zn,we get (3.22) for p = 0.
Let us prove (3.22) for p = 1. Summing estimates (3.8) with respect t o s = 1,2, ... we deduce that
where M, is a constant independent of
E.
Passing t o the variables x =
E(
in this inequality we obtain (3.22) for p = 1 since N : ( t ) are 1-periodic in and the domain Z
= (2,O).
he contains exactly
+
domains E ( Z d(0,
[
d)), z E Zn, E
$3. Asymptotic expansions for solutions of boundary value problems 173 Estimate (3.22) for p = 2 is proved in the same way as for p = 1. However, in this case one should use the inequalities (3.9) instead of (3.8). Estimates (3.23), (3.24) follow directly from (3.8), (3.9) and the definition 0 of the norms in H ' / ~ ( F ~ ~) , l / ~ ( f 'Lemma ~ ) . 3.2 is proved. Proof of Theorem 3.1. Let us show that the vector valued function u(k)given by (3.19) is the solution of the problem
L , ( u ( ~ )=) f
a d m( x ,E ) + E ~ + ' ~ ~ (E )X +, E ~ + ' ax,
dk)(?, d ) = i p 2 ( f ) + ~ ~ + l d E~) (o nf , I'd ,
in R'
,
1 1
where
Mo, M I , Mz are constants independent of E . Then estimate (3.21) would follow directly from Theorem 6.5, Ch. I. Consider first the boundary conditions for u @ ) ( x ) We . have
11. Homogenization of the system of linear elasticity
where
and hz are assumed t o be zeros if the length o f the index cr is negative or is larger than k
+ 1. Due t o the conditions (3.18)
Taking into account the smoothness of
we have
V, in the layer {x
: 0
< x, < d )
we conclude from (3.23), (3.28), (3.29) that ~ ( ~ ) ( ? = , d@) 2 ( ? ) + ~ k f 1 2 9 2 ( ? , ~ ) and the second inequality (3.27) is satisfied. =) Q 1 ( ? )+ ~ ~ + ' t 9 ~ ( ?and , & )that In the same way we prove that u ( ~ ) ( ? , o the first inequality (3.27) is satisfied. ) Let us now calculate u , ( u ( ~ )on
Setting
5 = E - ' x , due t o (3.3)
dRE\(rou rd).
we have
$3. Asymptotic expansions for solutions of boundary value problems 175 Here we used the equality
Ba(()= 0 for
E a-'
( ~ R ' \ ( I ' ~u I'd)) which holds
owing to the boundary conditions in (3.5), (3.6), (3.7). Substituting
d k )in (3.1) we obtain
Since N:, N i , N: are solutions of problems (3.5), (3.6), (3.7) respectively we can replace the expression in the square brackets in (3.31) by
Therefore
LO,
Let us transform the expression (3.32) setting = 0 for ( a ) 2 k 3. W e have
+
ft:
= h: for ( a ) 5 Ic
+ 2,
II. Homogenization of the system of linear elasticity
Therefore it follows from (3.17), (3.18), (3.33) that
where
Estimate (3.26) holds due to the inequalities (3.22), Lemma 3.2 and the smoothness of V,. satisfies (3.25). Theorem 3.1 is proved. It is obvious that Remark 3.3. It follows from the estimate (3.21) and the equalities (3.19), (3.20) that
53. Asymptotic expansions for solutions of boundary value problems 177 where J ~ ~ ( X , E ) J5I ~M ~ with ( ~ ~ a, constant M independent o f E . Therefore
where llq1(x,~)llL2(n.)
I MI,
and constant Ml does not depend on
E,
N, = 0
for negative (a).In particular we obtain for k = 0
where C is a constant independent o f
E.
It also follows from (3.37) for k = 0 that
where C1 = const and does not depend on
E.
It is important t o note that having taken into account the boundary layers we obtain in the first approximation an estimate o f order
E
for the remainder
term, whereas without the boundary layers we can only get an estimate of order
as in Theorem 1.2 with
a0= aE,f0 = f" (see estimate (1.15)).
II. Homogenization o f the system o f linear elasticity
178
$4. Asymptotic Expansions for Solutions of the Dirichlet Problem for the Elasticity System in a Perforated Domain Here we consider asymptotic expansions in E for solutions o f the Dirichlet problem for the elasticity system in a perforated domain RE with a periodic structure. The displacement vector is assumed t o vanish on the surface o f the cavities
S,.
Similar asymptotic expansions for solutions o f the Dirichlet problem for the equation AuE = f in a perforated domain RE were obtained in [52], where the estimates for the remainder term were proved in the case f E C r ( R ) . In order t o justify the asymptotic expansion, when f ( x ) is sufFiciently smooth and may be non-vanishing in a neighbourhood of d R , we construct boundary layers which exponentially decay in x with the increase o f the distance from x to dR.
4.1. Setting of the Problem. Auxiliary Results Consider a perforated domain RE = R domain o f
Rnwith
Zn.It is
by the vectors z E
n EW,
where w is an unbounded
a 1-periodic structure, i.e. w is invariant under the shifts assumed here that Q\w contains a surface of
class C 1 and R is a smooth bounded domain. Note that in this section we do not impose any restrictions on the smoothness o f w. In R' we shall study the following Dirichlet problem
where Ah'(<) are (nx n)-matrices o f class E ( n l , n 2 )and their elements a$(<) are 1-periodic in (. The aim o f this section is t o justify the asymptotic expansion
u E ( x )%'
x x &It2
1=0
(a)=/
NN,(&, ()Do f (x),
179
$4. Asymptotic expansions for solutions of the Dirichlet problem for solutions of problem (4.1). Here a,ID" are the same as in $3,
No([)are
N, = NL + N:, where the elements of N t are functions x defined in w and 1-periodic in (, the elementsaof N ~ ( E -),decay exponentially matrices of the form
in Re with the increase of the distance from on
E
x t o d R , N:, N: do not depend
f. Let us now prove some auxiliary propositions t o be used below for the jus-
tification o f the asymptotic expansion (4.2). Lemma 4.1. For any vector valued function w E
HA(R')the following inequalities are valid
where M is a constant independent of E.
W f . The inequality (4.4) follows directly from the First Korn inequality (2.2), Ch. I, in R, applied t o the function 6 E HA(R)such that 6 = w in Re, 6= 0 in R\Rc. Let us prove (4.3).
Obviously we can assume that w is defined in
and vanishes in Rn\Re. Denote by E(Z
+ Q)n R #
0
Tc the
and consider the function
Rn
Znsuch that W([)= w(EJ).Taking into
set o f all z E
account the properties o f dw and the fact that W = 0 on dw, we can apply the Friedrichs inequality o f Lemma 1.1, Ch. I, t o
W(t)in
(z
+ Q) n w . We
thus get (4.5) (~+Q)nw (~+Q)nw Summing up these inequalities with respect t o z E
TE
variables x = E[we obtain (4.3). Lemma 4.1 is proved. Lemma 4.2. Let
U(x)E H1(Rc) be a weak solsution
of the problem
and passing t o the
II. Homogenization o f the system o f linear elasticity
180 where
fj
E
L 2 ( R c ) ,j = 0 ,...,n , iP E H 1 ( R c ) . Then
where the constants C, C1 do not depend on
E.
Proof. It follows from the integral identity of type (3.5),
Ch. I, for w = U -
that
Due t o (3.13), Ch. I, we have
This inequality combined with (4.3), (4.4), (4.8) implies
where the constants
K2,
do not depend on
E.
Therefore estimates (4.7) are valid, since w = U-iP. Lemma 4.2 is proved.
The next theorem shows in particular that the solutions o f problem (4.6) have the form o f a boundary layer in the vicinity o f
d R , provided that
=0
$4. Asymptotic expansions for solutions o f the Dirichlet problem on
181
(an.) n R and f ' ( x ) , i = 0, ..., n , rapidly decay in Re with the growth of
the distance from x t o d o . Consider a scalar function ~ ( xE) C 1 ( n )such that
T
= 0 in a neighbour-
2 0 in R , (Vr1 5 M = const. It is assumed that E is so small that there is a subdomain R1 c R whose closure consists o f the cubes EQ + E Z with z belonging t o a set T, C Zn, and dR1lies in the neighbourhood of d R where T = 0. hood of 8 0 ,
T
a'
Theorem 4.3. Let U ( x ) be a weak solution of problem (4.6) with @ ( x )= 0 on (aRE)\dR (i.e. Q, E H1(RE,S,). Then
where K, 6 are positive constants independent of e.
Proof. Set
v = (ep7 -
where p = const
/ (A" na
> 0 is a
dU dU ax, -)ax,
-- ,
Since
T
l)U in the integral identity (3.15), Ch. I, for U ( x ) , parameter t o be chosen later. We have
exp pr dx = -
dU dr / (Ahk, U )p exp(pr)dx axk dzh
n
= 0 outside R', we find by virtue o f (3.13), Ch. I, that
-
II. Homogenization of the system of linear elasticity
182
+ C4 /
le(U)12dx
.
(4.10)
R
D u e t o t h e Korn inequality for vector valued functions w E H 1 ( Q n w , a w n
Q ) (i.e. w = 0 on ( d w ) n Q) we have
This inequality follows from Theorem 2.7, C h . I, if we extend w as zero t o
Q\w and note t h a t Q\w contains a surface o f class Passing t o the variables
w)
+
EZ
c Oc n O',
t
C1.
= E-'x in (4.11) we obtain for any w, = E(Q
z E T e ,the following inequalities
n
183
54. Asymptotic expansions for solutions of the Dirichlet problem Setting p = a ( 2
a&)-', where = const E ( 0 , l ) will be chosen later, a
we get from (4.12)
for any w, C REno'. Therefore
By (4.11) we obtain
It follows that
a
Therefore since p = -we find from (4.13) that
2 d G ~
<
C,(a
+ a')
/
+
l ~ U l ' e x p ( p ~ ) d xCSI
ncnnl
J
Ivu~'
exp(pr)dx
I ~ ( u e) Ix 'p ( p ~ ) d x.
ncnnt
Thus for all a E (0,min(l,
ncnnl
J
1 =)) we have
5 Cia
J
.
le(u)/'exp(p~)dx
Rcnnl
I t follows from (4.10), (4.13), (4.14) that
(4.14)
II. Homogenization o f the system o f linear elasticity
184
t
/ le(U)12dx.
(4.15)
n
Choosing u sufficiently small and independent o f E and taking into account u that p = -we obtain from (4.15), (4.14), (4.13) the estimate (4.9).
2 G &
Theorem 4.3 is proved. For the justification o f the asymptotic expansion (4.1) we shall also use the following result. Consider the boundary value problem for the system o f elasticity
w = 0 on aw , w is 1-periodic in J
,
where Ahk(E)are matrices o f class E(rcl, n 2 ) ,w = ( w l , ..., w,)',
3j E L 2 ( Q f l
t ,j = 0,1, ..., n. A weak solution of problem (4.16) is defined as a vector valued function w E$ ( w ) = r/i/;(w)n H 1 ( Qflu,Q n 8u)which satisfies the integral identity (6.2), Ch. I, for any v E$ ( w ) .
w), 3 3 are 1-periodic in
Theorem 4.4. There exists a weak solution of problem (4.16) which is unique and satisfies the estimate n
I l w l ! ~ l ( ~ n5w )
IIFjll~2(~m) j=O
185
54. Asymptotic expansions for solutions o f the Dirichlet problem
The proof of this theorem is based on Theorem 1.3, Ch. I, and is quite similar t o that o f Theorems 6.1, 3.5, Ch. I.
4.2. Justification of the Asymptotic Expansion Let us substitute the series
in the equations (4.1). Formal calculations similar t o those o f 53.2 yield (4.18) I=O
(,)=I
where
Let us seek N , ( [ ) in the form N , = N,O([)
+ N A ( [ ) , where N : ( [ ) are 0
matrices whose elements are 1-periodic in [ functions belonging t o W (w), and
x
the elements of N : ( - ) decay exponentially with the increase o f the distance from x t o
dR.
E
We introduce the notation
T,OEI, T t ~ 0 ,
1
11. Homogenization o f the system o f linear elasticity
186
I is the unit matrix. Define the matrices N,O(J) as weak solutions of the problems
where
The matrices Nt(J) are defined as weak solutions o f the problems
Existence of N:, NA can be easily proved by induction with respect t o 1 on the basis o f Theorems 4.3, 4.4, Ch. I. Lemma 4.5. The matrices
x N : ( - ) satisfy the following inequalities E
where C, are constants independent o f
E.
x
Proof. Relations (4.19), (4.20), (4.21) show that N , ( - ) E
x
= N:(-) E
are solutions of the following boundary value problems:
Let us use induction with respect t o 1. For 1 = 0 i t follows from (4.23) due to (4.7) with
= 0 that
+ N ~ ( E-)x, E
187
§4. Asymptotic expansions for solutions o f the Dirichlet problem
where Co is a constant independent o f
E.
Let 1 = 1. By virtue of (4.7), (4.24) we get
These inequalities and (4.26) imply that for k <_ 1 we have
Suppose now that the inequalities (4.27) hold for k
5 1 - 1.
Let us show
that they also hold for k = 1.
It follows from (4.25), (4.7) that
5 1 - 1 we obtain (4.27) for k = 1. matrices N:(() are 1-periodic in J . Therefore
Therefore due t o (4.27) for k The elements o f the
esti-
mates (4.22) for j = 0 are obvious. For j = 1 estimates (4.22) follow from (4.22) for j = 0 and the inqualities (4.27). Lemma 4.5 is proved. Lemma 4.6. subdomain
R0 such that
2
N:(a, -) are o f boundary layer type, i.e. € noc R the following inequalities are valid
The elements o f matrices
where C,, y are positive constants independent of
k f . Consider a domain R' such that
0' c R,
for any
E. Q0
c R'
and the distance
11. Homogenization of the system of linear elasticity
188 between of
E,
-
and
52' and dR' is larger than
0'
K
+
consists of the cubes E ( Q z), z
The parameter
E
is assumed so small that
Let us construct a scalar function
T
> 0, where
K
T(X)
0 outside the --neighbourhood of 2
R'
K
is a constant independent
E T , for some subset T
C
Zn.
with the above properties exists.
such that T E C 1 ( n ) T, r 1 in
RO,
RO,J V T5J C K - ' , C = const.
x Using the induction with respect t o s = 0 , 1 , 2 ,... , let us prove that N : ( - ) E
for a = ( a 1.,. . , a , )satisfy the inequalities
where C,
6 are positive constants independent of
E.
Let us first show that (4.29) holds for the matrix N,' which is a solution of the problem
x
Since N,'(-) = 0 on E
dRE\bQ, we
can apply the estimate (4.9) of Theorem
4.3 t o N,'. We get
<
K
J
IV.NiJ2dx.
n* This inequality together with (4.22) implies (4.29) for N,'. Fix a positive integer s and suppose that (4.29) is valid for all NA with a = ( a , ,..., a , ) ,
1
< s - 1. Let us show that (4.29)
holds for
NA
with 1 = s, where NA
is a solution of the problem
x
Taking into account the fact that N : ( - ) = 0 on BRE\BR and using the e estimate (4.9) o f Theorem 4.3 applied t o N;,,,,,,, we obtain
189
$4. Asymptotic expansions for solutions o f the Dirichlet problem
67
+ c4E2 J n*nnf
IN:a...a* '1 ~ x P E( - ) ~ +x E
- ~
J IN:^...^^ I'
~ X P ( - ) ~ X E
ncnni
6r
I
.
Estimating the first integral in the right-hand side o f this inequality by (4.22), and applying the assumption o f induction t o the other integrals, we get (4.29) for NL The estimates (4.28) follow from (4.29), since
T
r 1 in RO. Lemma 4.6 is
proved. Theorem 4.7. Let u E ( x )be a weak solution o f problem (4.1) with f E CS+'(i=l).Set
u:(x) =
2
v:(x) = where N, = Nz
Na(;)Da f ( x ) ,
of+'
1=0
2f=o ol+'
+ NA, N:,
(a)=[ ..
(a)={
N,o(:)D" f ( x ) ,
1 J
NA are weak solutions of problems (4.20), (4.21)
respectively. Then
l l " z ( ~ ) - u C ( x ) I I ~ l ( n5*CO&~+' ) I l f llca+2(n), where R0 is a subdomain of R such that
(4.31)
no c R, the constants Co, C1 do
not depend on E ; Cl may depend on RO.
Proof. for (a)
Let us apply the operator LC t o u: - uc. Assuming N, = 0 in (4.17)
2
s we obtain in the same way as (4.18) that
11. Homogenization of the system of linear elasticity
Note that, because of (4.22), the matrices
x
d x N , ( - ) , -N , ( - ) E
atj
L 2 ( R e ) norms
of the elements of the
are bounded by a constant independent of
E.
E
Therefore applying Lemma 4.2 with @ = 0, f j = 0, j = 1,...,n , to u: - uc we get the estimate (4.31). T o prove (4.32) i t suffices to observe that
N , = N:
the inequalities (4.28). Theorem 4.7 is proved.
+ N:
and
Ni
satisfy
191
55. Some generalizations for the case o f perforated domains
$5. Asymptotic Expansions for Solutions of the Dirichlet Problem for the Biharmonic Equation. Some Generalizations for the Case o f Perforated Domains with a Non-Periodic Structure
5.1. Setting of the Problem. Auxiliary Propositions The methods suggested in $4.1 and $4.2 can also be used t o justify asymptotic expansions for solutions o f the Dirichlet problem for higher order elliptic equations.
In this section we consider a special case which is particularly
important for mechanics, namely, the Dirichlet problem for the biharmonic equation:
and obtain a complete asymptotic expansion for solutions of this problem. Here RE is a perforated domain of type I with a periodic structure described in 54.1, f ( x ) is a sufficiently smooth function in R ; v is the outward normal. We seek the asymptotic expansion for the solution o f problem (5.1) in the form
u: = where
N,(E,[)V" f ( x )
E'+' l=O
,
[ = E-'x
,
(5.2)
(+I
D",a are the same as in $3.2.
We shall prove that solutions of (5.1) admit asymptotic expansions of type (5.2) after establishing some preliminary results. Lemma 5.1. For any v E H i ( R E )the following inequality is satisfied
where E 2 ( u )=
a2v a2v -)dx;dxj (-dx;dxj
Proof. Obviously it is sufficient
' I 2 , MI is a constant independent of
E.
t o prove (5.3) for v E C,"(RE). Set v = 0 in
Rn\Rc and denote by T' the set o f all z E iZn such that ~ ( +zQ) n R # 8. Consider the function W ( [ )= v ( e [ ) . Since W = 0 in Rn\w, the Friedrichs inequality for each o f the sets w, = z Q yields
+
II. Homogenization o f the system o f linear elasticity
Summing these inequalities with respect t o z E TE and passing t o the variables
x
Let O
= E<, we obtain (5.3). Lemma 5.1 is proved.
E H 2 ( R E )fj , E L 2 ( f l E )j, = 0,1,2, ...,n. U ( x ) is a weak solution o f the problem
We say that
if W = U
-@
for any v E
belongs to
H i ( f l e ) and satisfies
the integral identity
H,2(RE).
Denote by
H ~ ( W )the
E C 2 ( 3 ) , v = 0 in a neighbourhood o f dw and v ( J )is 1-periodic in J . Here w is an unbounded domain with a 1-periodic
the functions
v ( t ) such
completion with respect t o the norm IIvIIKlcsnw, of
that v
structure, the same as in s4.1. We say that w is a weak solution o f the problem
where
Fj E L 2 ( un Q),F j ( J ) are 1-periodic in J , j
and satisfies the integral identity
= 0,
...,n , if w E H ; ( W )
$5. Some generalizations for the case o f perforated domains
193
for any v E H;(W). The existence and uniqueness of solutions of problems
(5.4),(5.6) follow
from Theorem 1.3, Ch. I . Lemma
5.2.
A weak solution U ( x ) of problem (5.4) sdtisfies the following inequalities
where
K1, I<2, I<3 are constants independent
of
E.
Proof. Set v = W = U - @ in the integral identity (5.5) for v = W . W e get
II. Homogenization o f the system o f linear elasticity
194
It follows that
Since
W = U - O,this inequality implies (5.8).
Due t o (5.3) we have
llVW11~2(n*) I M I EllEz(W)II~~(nc) . From these inequalities and (5.11) we obtain (5.9), (5.10), since W = U - cP. Lemma 5.2 is proved.
7 ( x )be a function o f class C 2 ( n )such that 7 = 0 in a neighbourhood of a R , 7 > 0 in R. Consider a subdomain R' defined just before Theorem 4.3 and assume that 7 = 0 outside 0'. Let
Theorem 5.3.
a@
U ( x ) be a weak solution of problem (5.4), cP = - = 0 on aRe\aR (i.e. av @ E H Z ( R caRc\aR)). , Then Let
55. Some generalizations for the case of perforated domains
195
KO > 0 , 6 > 0 are constants independent o f E . ( N o t e t h a t I(o and S can depend on 52' and 1 1 ~ ( x ) 1 1 ~ ~ ( ~ ) . )
where
Proof.
For any v(x)
E HZ(Rc)the function U ( x )satisfies the integral
Set v = (epT - 1)U, where p
Since
T
-
(5.13) that
> 0 is a
identity
parameter t o be chosen later. W e have
0 outside of R', we obtain by virtue of t h e Holder inequality and
II. Homogenization of the system o f linear elasticity
196
In the same way as in the proof of Theorem 4.3 t o obtain (4.14), we find that
where p = u / I ( a , I( is a constant independent of to be chosen later.
E,
u E (0,l) is a constant
Since U ( x ) can be approximated in the norm of H 2 ( R " )by functions vanishing in a neighbourhood of BRe\aR it follows that inequality similar t o (5.15) holds for the first derivatives o f U ( x ) , i.e.
/
IVU12e"dx
< K2a2
nennl
1
I E ~ ( U ) ~ ~ ~ ' ~ ~. X
ncnnl
Estimates (5.15), (5.16) yield
where K 2 , I(3 are constants independent of From (5.14), (5.16), (5.17) we obtain
E.
(5.16)
55. Some generalizations for the case o f perforated domains
197
(5.18)
n* where p = u I K E . If we choose
a sufficiently small but independent of c, we
get from (5.18) the following inequality
where
M I , M2, M3 are constants independent of c.
Estimate (5.12) follows from (5.19), (5.16), (5.17). Theorem 5.3 is proved.
5.2. Justification of the Asymptotic Expansion for Solutions of the Dirichlet Problem for the Biharmonic Equation
C"t4(fi)in (5.1). Let us seek an asymptotic expansion N O ( € , [ )= N : ( [ ) + N:(e,E), x N:(() are 1-periodic in (, N;(E, -) are functions o f boundary layer type in E RE,which decay exponentially with the increase of the distance from x t o d R . Suppose that f E
for the solution o f (5.1) in the form (5.2) where
It is easy t o verify that
Therefore
11. Homogenization of the
198
system of linear elasticity
From (5.2), (5.20) we obtain
where
6,,
is the Kronecker symbol.
Let us define the functions
Na(e,t)as
weak solutions of the following
boundary value problems
a
A ~ ,,,, N =-~-A~N,,-~~,,,,AN4 ata1
O
-
N~ ata 2at,, a1
in
& - ~ a e7
$5. Some generalizations for the case o f perforated domains
199
On the basis of Theorem 1.3, Ch. I, we can easily prove by induction that
N o r ( [ )exist. ) N: Let us show that N , ( E , ~ = periodic in
+ NA, where
N : ( [ ) are functions 1-
t and belonging t o H ~ ( w )N:(E, ; f ) are of boundary layer type in
Re.Set
Define the functions N:(t) as solutions o f the following boundary value problems
Obviously Theorem 1.3, Ch. I, guarantees the existence of N,O([). In the domain Dirichlet problems
&-'RE
define the functions N: as weak solutions o f the
II. Homogenization o f the system o f linear elasticity
200
Obviously
N , = N,O
Lemma 5.4. The functions
+Nt.
x x N:(-), N ~ ( E-), satisfy the inequalities E
where the constants
E
M, do
not depend on
E;
j =0,l.
This lemma is proved by induction in the same way as Lemma 4.5. Lemma 5.5.
x N ~ ( E-), are of boundary layer type, i.e. for any subdomain e such that !=lo c R the following inequalities hold The functions
where on
Q0
C,, y are positive constants independent o f E , (C, and y may depend
RO).
Proof. The estimate (5.26)
is obtained in the same way as (4.28) in Lemma
4.5. Let us indicate the main steps of the proof.
R' c R which consists of the cubes ~ (+zQ) for ~ ( x E) C 2 ( O ) possess the same properties as in the
Consider a subdomain some z E Zn,and let proof of Lemma 4.5. The function
x . N t 1 . , , , , ( ~ -) , IS a weak solution o f the problem E
55. Some generalizations for the case of perforated domains
Due t o (5.12)
aN:
-= 0 on dRE\dR, we can apply Theorem 5.3 av for U = N:l,,.,m we get
Since N o =
201
to
U = N:.
where IC2 is a constant independent of E . From these inequalities and (5.25) we obtain by induction with respect t o
m = 0,1,2, ... that
where the constant I~,,,,,,, does not depend on T
= 1 on RO,we obtain the estimates (5.26).
E.
Taking into account that
Lemma 5.5 is proved.
Theorem 5.6 (On the asymptotic expansion o f solutions of problem (5.1)). Let u E ( x )be a weak solution o f problem (5.1) and let ti:(.)
=
2 I=O
mlt4
x (,)=I
N,(E, f)Dof ( x ) , E
f E Csf4(fi),
202
11. Homogenization of the system of linear elasticity
where N , ( E , [ ) = N:
+ NA, N:,
Ni
are weak solutions of problems (5.23),
(5.24). Then
C1, C2are constant independent o f E , R0 is a subdomain o f R such that fiOc R, the constant C2may depend on RO.
where
Proof.
It is easy t o see that by virtue o f (5.21), (5.22) u:
solution of the problem
Due t o the estimates (5.8) we get
m=s+l a s ,...,o m = ]
-u '
is a weak
$5. Some generalizations for the case o f perforated domains 3+4
+E
C m=s+l
~ + ~ E ~
Since Na = Nz
I
n
C as,...,
o,=l
+ N:,
203
IINa5...~,ll~~(n*) I l f Ilcs+4(ii). we obtain by virtue o f (5.25) that
This estimate and (5.9), (5.10) imply (5.27). Inequalities (5.28) follow from 0
(5.27), (5.26). Theorem 5.6 is proved.
5.3. Perfarated Domains with a Non-Periodic Structure Analysing the proof o f Theorems 4.3, 5.3, we can easily see that estimates similar t o (4.9), (5.12) can also be obtained in the case o f some non-periodic structures. d*
Suppose that a subdomain
R' c R
is such that
fi' c R and fi' =
U B,', s=l
B: are bounded domains of Rn such that B; n B; = 0 for i # j . Suppose also that ,:'I s = 1, ...,d,, are closed sets r: c @ and for each v E C1(&) such that v = 0 in a neighbourhood o f r:, the Friedrichs inequality
where
holds with a constant C * independent o f c and s. Let
T(X)
be a function in C2(fi) such that
) ) ~ ) ) ~5 z (M~ *) , where
T
= 0 in R\S2',
M* is a constant independent of
E,
s.
Theorem 5.7. Let U ( x ) be a weak solution of the boundary value problem
T
2 0 in R,
II. Homogenization of the system of linear elasticity
204
acp
iP E H Z ( R c ) ,cP = - = 0 on R' n d o c , fj E L2(R'), j = 0 , ...,n. av Then for U the estimate (5.12) is valid with constants KO > 0, 6 > 0 depending only on C* and M*. acp Suppose that f j = 0, j = 0 , ...,n in R e n R 1 ,@ = - = 0 on R 1 n d R eand dv the domain R0 is such that c R', p(dRO,d o ' ) 2 K > 0 with x independent of E . Then the solution U ( x ) satisfies the inquality
where
no
where
C > 0 is a constant depending only on C * , RO.
The estimate o f type (5.12) in this case is proved by the same argument as Theorem 5.3. The estimate (5.30) follows from (5.12) if we take that
T(X)
~ ( xsuch )
= 1 in RO, T ( X ) = 0 outside the ~/2-neighbourhood o f RO, the
C 2 ( n )norm of ~ ( xis) bounded by a constant independent o f E. Consider now the system of elasticity. Suppose that the sets
rz,s = 1, ...,d,, are such that for each v E C 1 ( & ) ,
v = 0 in a neighbourhood o f
l?:
the following inequality is valid
where C; is a constant independent o f
R',
1 1 ~ 1 ) ~ 1 ( <~ ) M;, where M:
E.
let
~ ( xE) C 1 ( O ) ,~ ( x=) 0 outside
is a constant independent o f
E.
Theorem 5.8. Let
U ( x ) be a weak solution o f the boundary value problem for the elasticity
system
L2(R'), j = 0,..., n , @ E H 1 ( R ' ) , @ = 0 on R' n do' and ) t o the class E(nl, n 2 ) with n l , n2> 0 independent matrices A h k ( x , & belong
where f j E
$5. Some generalizations for the case of perforated domains
205
U ( x ) the estimate (4.9) holds with constants I( > 0, S > 0 depending only on C: in (5.31), M:, K,, KZ. = 0 on 0' n 80' and Suppose that f j G 0, j = 0, ...,n , in RE n R', the domain R0 c R' is such that p(8R0,dR') 2 K > 0, where 6 is a constant independent of E . Then the solution U ( x ) satisfies the inequality of
E.
Then for
where C is a constant depending only on C;,
M;,
K,,
n z , RO.
The estimate (4.9) in this case is proved in the same way as the corresponding estimates in Theorem 4.3. The inquality (5.32) follows from (4.9),
if we take T ( X ) such that of
RO,I
T
= 1 on
RO,T = 0 outside the ~/2-neighbourhood
( T ( I ~ ~ ( ~ is) bounded uniformly in E .
11. Homogenization o f the system o f linear elasticity
206
$6. Homogenization of the System of Elasticity with Almost-Periodic CoefFicients In this section we consider homogenization o f solutions o f the Dirichlet problem for the system of elasticity with rapidly oscillating almost-periodic coefficients.
6.1. Spaces of Almost-Periodic Functions Denote by TriglRn the space o f real valued trigonometric polynomials. Thus Trig Rn consists o f all functions which can be represented in the form of finite sums
ctexp {i(y,E)) ,
U(Y)= C
y,[ E R n , (y,() = y;&, ct =
= const
The completion of TrigRn in the norm sup
.
(6.1)
Iu(y)l is called the Bohr
R"
space of almost-periodic functions and is denoted by A P ( R n ) (see [50],[51]). The space of all finite sums having the form (6.1) and such that
Q
= 0 is
0
denoted by Trig Rn. Let
1I, E LLc(EP). We say that M ($1 is the mean value of +, if $(e-'z) + M ($1 weakly in L ~ ( G )as
for any bounded domain G C
e
-+ 0
Rn.
I t is well known that for any function g E L:,,(Rn),
which is T-periodic in
y, the mean value exists and is equal t o
[O,TIn={y : O s y j S T , j = l , ...,n ) . Thus each function belonging t o TrigRn possesses a finite mean value, and therefore we can introduce in T r i g R n the scalar product defined by the formula
$6. Homogenization o f the system o f elasticity
207
The completion of T r i g R with respect t o the norm corresponding t o the scalar product (6.2) is denoted by
B 2 ( R n )and is called the Besicovitch space
of almost-periodic functions. We keep the symbol M ( $ 9 ) for the scalar product of the elements II,and
g in B 2 ( R n ) . As before we say that a matrix (or vector) valued function belongs t o one of the spaces Trig Rn, B 2 ( R n ) ,A P ( R n ) , if its components belong t o the corresponding space. In this case the mean value is a matrix (or vector) whose components are the mean values o f the components o f the given function. We shall also use the notation (1.8), (1.9), Ch. I, for matrix (or vector) valued functions.
e(u) denotes the symmetric matrix with elements e l j ( u ) = auj (k+ -), where u is a vector valued function u ( y ) = ( u l ,...,u,). 2 ay, 8~1 As usual
1
Lemma 6.1. Suppose that f , g E Trig Rn,and
Moreover for any functions
u = ( u l ,...,u,) E Trig IRn. Then
Flh E Trig Rn such that Flh = Fhl, 1, h = 1 , ...,n ,
there is a vector valued function w E Trig Rn such that
Proof. Note that M {ei("t))= 0 for [ # 0 . Let f
=
C E
fEe'(~'P)
,
=
C g,e'("d 7
11. Homogenization of the system of linear elasticity
208
Then by virtue of (6.6) we have
...,
Let us prove inequality (6.4). Let u = (ul, u,), u j =
4 e ' ( ~ ~ Then ). E
due t o (6.6) we find that
Eta
This implies (6.4). Let us show now the existence of the solution of equations (6.5). Suppose that
W e seek w in the form w =
C wCe"ylC).Then E
§6. Homogenization o f the system o f elasticity
Obviously for each
# 0 the coefficients w:
must satisfy the system
For each [ # 0 system (6.7) has a unique solution, since the corresponding homogeneous system has only the trivial solution. Indeed, let
I, h = 1, ..., n. Then multiplying the equations (6.7) by with respect t o I from 1t o n we obtain
# 0, c y
= 0,
and summing up
Therefore w: = 0. Let us replace
by -( in (6.7) and write the complex conjugate equation.
One clearly has w: = wkt, since ckh = z'$.
Lemma 6.1 is proved.
Consider the Hilbert space of ( n x n)-matrices whose elements belong t o
B 2 ( F )and denote by W the closure in this space of the set
Elements o f W will be denoted by e, Z, etc. The norm o f an element e E W is given by
M {eueu)'12 = M {(e, e))'I2 . It should be noted that not every element e E W can be represented as Nevertheless for every e E W there is a sequence e = e(u) with u E B2(Rn). o f vector valued functions {u6) with components in T r i g R n and such that
M {(e - e(u6)I2) + 0 as 6 -,0. 6.2. System of Elasticity with Almost-Periodic Coeflcients. Almost-Solutions Consider the system o f linear elasticity
11. Homogenization o f the system of linear elasticity
= const. > 0, whose elements belong t o A P ( R n ) , u = ( u l ,...,u,), fj = ( f i j , ..., f n j ) are column
where A ~ ~are( matrices ~ ) of class E ( K ~ c 2, ) , ~
1 K Z,
vectors, fjl = fij E A P ( R n ) . In the general case o f almost-periodic coefficients in A P ( R n ) no proof for the existence o f a solution u E B 2 ( B n )o f system (6.8) has yet been found. However we can construct the so-called almost-solutions u6 o f (6.8) with components in Trig Rn. This fact was established in [149]. Following [I491 we shall outline here a method for the construction o f such almost-solutions. Due t o the conditions
(3.2), Ch. I, one
can rewrite system (6.8) in the
form:
, In the rest o f this paragraph we shall denote by qh the column ( q I h..., qnh)* o f the matrix q with elements qih. Then system (6.9) becomes
where ek(u) = (elk(u),...,e,,k(u))*.
If the coefficients akk(y)and the functions f l j ( y ) are 1-periodic in y, then the definition o f a weak 1-periodic solution u ( y ) o f system (6.8) can be reduced to the integral identity
for any v E
w ; ( R n )where , f
is a matrix with elements fib and
Let the coefficients a:/ be almost-periodic functions of class A P ( R n ) . Then in analogy with (6.10), (6.11) we consider the system
21 1
§6. Homogenization of the system of elasticity and define a weak solution o f (6.12) as the element E E W, 2 =
{Eij),
which
satisfies the integral identity
for any e E W.
It follows from Lemma 3.1, Ch. I, that the bilinear form M ( ( M 2 , e ) ) is continuous on W x W, i.e.
fot any 2,e
E W , since for a ( y ) E A P ( R n ) , f E B 2 ( R n ) we have a f E
B 2 ( R n )and Ilaf (
IB~R~)5
SUP
R"
lal
llf I I B ~ R ~ ) .
Moreover the condition (3.8), Ch. I, yields the inequality
for any e E W. By virtue o f (6.14), (6.15) the bilinear form
M { ( M 2 , e ) ) satisfies all
conditions of Theorem 1.3, Ch. I, with H = W. Therefore, the solvability of problem (6.12) in W follows directly from Theorem 1.3, Ch. I. Let us show that we can find vector valued functions
u:) E Trig Rn which approximate solutions o f the system (6.9)
= ( U f , ..., in the sense
o f distributions. To this end we need the following Lemma 6.2. Let f j , A h k E A P ( R n ) and let E E W be a weak solution o f system (6.12). Then there exist sequences of vector valued functions U s E Trig Rn and matrices gs E A P ( R n ) with columns g: = ( 9 4 , ...,g$), gfj = g:,, j , 1 = 1, ...,n , such that
lim M {1g612)4 0
6-0
,
lirn M (12 - e(u*)12)+ 0
6-0
as 6 -+ 0, 6
> 0,
and the integral identity
(6.17)
II. Homogenization o f the system of linear elas t i c i t y
holds for any $(y) = (&,
Proof. By U sE
...,4,)
E C,O"(Rn).
the definition o f the space W we see that there is a sequence
T r i g R n which satisfies the condition (6.17). Therefore due t o (6.13),
(6.14) we have
for any e E W, where y(6) -+ 0 as 6 + 0. Set
Since the elements @fh of matrices @6 belong t o AP(Rn), we can represent Q6
in the form
where Q 6 ,F 6 , G6 are symmetric matrices with elements ath,
G ,Gfh, F6 E
T r i g R n , G6 E AP(Rn), and
lim M ( 1 ~ ~=10~. ) 6-0
(6.21)
Since
it follows from (6.19), (6.21) that
for any e E W, where+y1(6)
-t
0 as 6 + 0.
According t o Lemma 6.1 there is a vector valued function w6 E T r i g Rn such that
213
$6. Homogenization o f the system o f elasticity
Multiplying each o f these equations by wf and summing with respect t o 1 from 1t o n, we find by virtue o f (6.3) and Lemma 6.1 that
Therefore
It follows from (6.20), (6.17), (6.21) that
M IF^^^)
are bounded by a
constant independent o f 6. Therefore due t o (6.24), (6.25) we obtain
Obviously by virtue o f (6.20) we have
+
where g6 = e ( w 6 ) G6; and the equations (6.27) hold in the sense of distributions. The convergence (6.16) is due t o (6.26), (6.21), and the integral identity (6.18) follows from (6.27) and the conditions (3.2), Ch. I, for ahk. Lemma 6.2 is proved. The vector valued functions
U6are called almost-solutions o f system (6.9)
with almost-periodic coefficients. Let us now establish some other properties o f the almost-solutions U s , which are essential for the study of G-convergence o f elasticity operators with almost-periodic coefficients. Lemma 6.3. Suppose that fj, Ahk E
AP(Rn), 2: is a weak solution o f system (6.9),
2: E
W,
and U6 (6 + 0 ) is a sequence o f almost-solutions o f system (6.9). Then for any sequence E
-+
0 there exists a subsequence € 6
(6)+
.c6 (u6
C 6 ) -+
-+
0 as 6
0 weakly in H 1 ( O ) ,
+ 0 , such that
(6.28)
II. Homogenization of the system of linear elasticity
214
where cs is a constant vector,
rap(:) + M { A P ~-Ef,} ~ weakly in L 2 ( R ) ,
p = 1, ..., n
,
(6.29)
a
-r 6 h
axh
x (G) +0
in the norm of H-I (R)
,
(6.30)
as 6 -+ 0 , where
0 c Rnis a bounded Lipschitz domain.
Proof. Taking into account
the inequality (6.4) o f Lemma 6.1, the fact that
U s E Trig Rn, and the convergence (6.17) we obtain
where
I<
is a constant independent of 6.
-
Denote by G 6 ~ " ( xthe ) matrices whose elements are
G~:(x)
E
a
x
ax,
E
-U f ( - ) E Trig Rn.
Note that the matrices G61c are not necessarily symmetrical. By the definition of mean value we have
Similarly
lim
EO '
/ (g6(Z)I2dx ( m e s a )M {1g61'} , n
E
=
where g6 are the matrices from Lemma 6.2.
It is obvious that
56. Homogenization of the system of elasticity
215
M e ( u 6 ) ( 5+ ) M { M e ( u 6 ) } weakly in L 2 ( R ) as e + 0 . (6.35) Moreover
G ~ , ' ( X+ ) 0 weakly in L 2 ( R ) as e -+ 0 dU6 x
,
(6.36)
0
since -(-) €Trig Rn and ei(: R, -i0 weakly in L 2 ( R )as E -+ 0 for J # 0. ~ Y IE Let V = {q1,v2,...} be a countable dense set in the Hilbert space of all matrices with elements in L2(R). For each 6 by virtue of (6.33)-(6.36)
we can find
€6
such that
m = 1 , 2,...; qm E V . It follows from (6.32), (6.39) that the norms llG6"611Lz(n)are bounded by a constant independent of 6, and inequalities (6.40) imply that for any qm E V we have
form
Therefore
G6"6(x)+ 0 weakly in L 2 ( R ) a s 6 + 0 Set
where the constants c6 are chosen such that
11. Homogenization of the system of linear elasticity
Then due to the Poincari inequality we have
where c is a constant independent of 6. Since the right-hand side o f (6.44) is bounded in 6, it follows from (6.42). (6.44) that E ~ . V ~ ' ( ? + ) V weakly
€6,
in H1(SZ) and strongly in L 2 ( R )for a subsequence 6' + 0. Here we used the weak compactness of a ball in a Hilbert space and the compactness of the imbedding H 1 ( R )c L2(R). By virtue of (6.43), (6.42) V 0. Thus the convergence (6.28) is established. Since the elements a k of matrices are bounded, it follows from (6.17), (6.37) that the norms e(u6) are bounded by a constant inde-
=
JIM
(E)I l u ( n )
pendent of 6, and
lim M { M e ( u 6 ) )= M { M 2) .
6-0
Therefore we conclude from (6.41) that
M e ( u 6 ) ( f )3 M { M 2) weakly in LZ(SZ) as 6 --+ 0
.
E6
To complete the proof of (6.29) it is sufficient t o observe that given by (6.31). Let us prove (6.30). For any $ ( I ) = ($,, ..., $,) (6.18) we obtain
Therefore
rs(-)x
are
€6
E
C,"(R) due t o (6.31),
$6. Homogenization o f the system o f elasticity
217
This inequality together with (6.38), (6.16) implies (6.30). Lemma 6.3 is proved.
6.3. Strong G-Convergence of Elasticity Operators with Rapidly Oscillating Almost-Periodic Coeficients In a bounded Lipschitz domain R consider the Dirichlet problem for the system of elasticity
where const
f E H-'(R), matrices Ahk(y) belong t o the class E(nl, n2), n l , nz = > 0, and their elements akk(y) are almost-periodic functions of class
AP(Rn). If matrices Ahk((y are 1-periodic in y, then according t o $1, Ch. II, the homogenized elasticity system corresponding t o the strong G-limit of the sequence
{C,)
has the following coefficients
,: is the s-th column of the matrix Nq,ejk(N:) = where N: = (N:a, ...,N) 1 dNZ* dN!a and the columns N : are 1-periodic solutions o f the system 2 a y j d ~ k
(-
+ -),
Setting
A q :
= (a;:,
(6.47) in vector form
..., a),:
A q :
= (A,:;
...,iK), we
can rewrite (6.46),
11. Homogenization o f the system o f linear elasticity
218
A P ( R n ) . I t was shown above that for fixed q, s we can find weak solutions e"" E W (P" is a matrix with elements dg:) o f the Now let
belong t o
system
which is similar t o (6.12) with
Set
and denote by
ah¶the matrices with the elements 6::
Theorem 6.4. Suppose that
APq(y) are matrices of class E(n1, n 2 ) , 61,n2 = const > 0, and are almost-periodic functions belonging t o A P ( R n ) .
their elements
Then the sequence of operators
is strongly G-convergent t o the elasticity operator
k
whose coefficients are
given by (6.50).
Proof. Let
us show that there is a sequence 6 -+ 0 and matrices
1, ..., n . such that matrices A"(:), I,'as 6 + 0, where
dhksatisfy the Condition
N:, q
N of $9, Ch.
dhkare matrices whose elements are defined by (6.50).
virtue o f Theorem 9.2, Ch. I, this means that
L,,
9
=
By
as 6 + 0. Due t o
the uniqueness of the strong G-limit (see Theorem 9.3, Ch. I) it follows that
L,
Sk
as
E --+
0.
Fix q, s and consider the almost-solutions (6.48) constructed in Lemma 6.2. Set
U& = ( U h S ..., , U:ns) of system
$6. Homogenization of the system of elasticity
219
where c:, are constant vectors satisfying the condition (6.28) with U s = UsP,.
N:(x)the matrices whose columns have the form (6.51). Let us the matrices N i , APQ - , h'4 satisfy the Condition N as 6 -+ 0.
Denote by verify that
(2)
Indeed, the Condition N 1 follows from (6.51) and (6.28).
Consider the
Conditions N2, N3. Due t o (6.29)-(6.31)
we have
weakly in LZ(R),
in the norm o f HW1(R), as 6 -t 0. These relations show that Conditions N2 and N3 are satisfied, since due t o
(6.50) the expression in the right-hand side o f (6.52) is equal t o 6.4 is proved.
,@.
Theorem
11. Homogenization o f the system o f linear elasticity
220
57. Homogenization of Stratified Structures
7.1. F o n u l a s for the Coeficients of the Homogenized Equations. Estimates of Solutions Consider a sequence
{LC)o f differential
operators o f the linear elasticity
system
, )with constant n l , n2 > 0 independent o f E , x (see belonging t o class E ( I c ~n 2 53, Ch. I). Here
E
is a small parameter,
E
E ( 0 , l ) ;the elements o f matrices
AF(t, y ) are bounded (uniformly in E) measurablefunctions of t E R 1 , y E Rn with bounded (uniformly in E) first derivatives in yl, ...,y,; p ( x ) is a scalar function in C 2 ( o )such that 0
5 p ( x ) 5 1 , ( V p l 2 const > 0; R is a bounded
smooth domain. Let us also consider the following system o f linear elasticity
whose coefficient matrices belong t o E ( k l , 22) and k l , k2 are positive constants which may be different from n1, n2; the elements of the matrices
a ' j ( t , y ) are bounded measurable functions o f t E R1,y E Rn, possessing bounded first derivatives in yl,
...,y,.
In this section we consider the following Dirichlet problems
Problems of type (7.3) serve in particular t o describe stationary states of elastic bodies having a strongly non-homogeneous stratified structure formed by thin layers along level surfaces o f a function cp(x) (see [go]). Here we obtain estimates for the difference between the displacements uc and u , the corresponding stress tensors and energies. We establish explicit
221
§7. Homogenization of stratified structures
dependence of the constants in these estimates on the coefficients of system (7.3). We also obtain the necessary and sufficient conditions for the strong G-convergence of the sequence
{L,)t o
explicit formulas for the coefficients of
the operator
k
as
E +
0, and give
k.
The corresponding spectral problems are studied in $2, Ch. Ill. Let the matrices
where
N;(t, y), M$(t, y ) be defined by the formulas
( y 1 ( y )..., , y n ( y ) )=
(3, ..., *) = V y , B-I ayn
is the inverse matrix
ayl
of
B. It will be proved in Lemma 7.5 that the matrix [cpl(y)cpk(y)A,kl(~, y)]-'
exists and that its elements are bounded functions (uniformly in c). To characterize the closeness between solutions of problems (7.3), (7.4) we introduce a parameter 6, setting
6, =
max
{ I M G ( P ( X ) ~ X ~) IN, ; ( P ( x ) ~, x ) \
XER
l , i , j = 1,..., n
For a given matrix
B with elements bra we set IBI = (bk'bk')1/2.
Theorem 7.1. Let u",u be thesolutions of problems (7.3), (7.4) respectively, and u E Then the following estimates hold
H2(R).
11. Homogenization o f the system o f linear elasticity
222
.auc
depend on
.
A:] -, jt axj
where yf
-..au
At3 -, axj
,
2
= 1, ..., n , the constants q,cl do not
E
Proof. Define v c ( x ) as the solution of the problem
Then it is easy to calculate that
Therefore
The right-hand side of this equation is understood as a n element of Let us show using the definition of
S,, N,", MG
that
H-'(0).
$7. Homogenization of stratified structures where IP;,(X,E)I 5 ~ 2 Icx,(x,~)I 6 ~ ~ 5 stants c2, c3, c4 do not depend on E .
c3&,
223
Icx;,(x,~)lI ~ 4 6 , )and the con-
Indeed, we obviously have
Multiplying these equations by
cpk
and summing them up with respect t o
1- aM;
k. we obtain (7.11) due t o the inequality < c6,. ay1 Setting k = i in (7.14) we find by virtue of (7.5) that
This equality implies (7.12). According t o the formulas (7.6), (7.5) we have
A .
= A'"
- A>
v c p , c p [j ~$ yO ~ ~ ~ A , -~ A' ]r -) +' ( ~ ~ ~
( Y ; , ( x ,E
)
=
Let us estimate the H-'(a)-norm of the right-hand side o f (7.10). For any column vector $ =
=
E
- J ( -aMi; n
+J n
at
au dx,
C,"(R), due to (7.13), (7.11) we obtain
a$
-, -)dx
ax;
du dlC, (ais(x,&)-7 -)dx ax, axi
+ =
II. Homogenization o f the systern
o f linear elasticity
Therefore, taking into account (7.12) and the definition of
where c5 is a constant independent of
6,
we find that
E.
Let us estimate the second term in the right-hand side of (7.10) in the norm of
H-'(0).
Using the definition of
6, we
get
I t thus follows from (7.10), (7.14), (7.15) that
5 7.
225
Homogenization of stratified structures
where C, is a constant independent of E and u. Therefore by virtue o f Theorem 3.3, Ch. I, and Remark 3.4, Ch. I, we obtain from (7.10), (7.16) the following inequality
where c8 is a constant independent of E. We now estimate the norm I l ~ , l l ~ l (Set ~).
+,
where $, = 1 in the 6,-neighbourhood of dR, = 0 outside the 26,neighbourhood of d R , $, E CW(n),0 5 $, 5 1, 6, IV$,I I const. It follows from Theorem 3.1, Ch. I, that
Let us estimate IldcIIHl(n).We have
and therefore
where wl is the 26,-neighbourhood of d R . By virtue of Lemma 1.3, Ch. I,
IlVuIIZzcw1) L cldc IIull&Zcn,.Hence
Estimates (7.17), (7.18) imply (7.7). Let us now prove (7.8). It follows from (7.7) that
where IIqf Il~z(n)I ~ 1 4 6 , "1~( ~ 1 ( ~ 2.( n )
Due to (7.13) we get
II. Homogenization o f the system o f linear elasticity
and thus the estimate (7.8) is valid. Theorem 7.1 is proved. Corollary 7.2. Suppose that the coefficients o f system (7.4) are smooth in
fi and f
E
L2(R),
E H3I2(dR). Then under the conditions o f Theorem 7.1 we have
where
Q,
cl are constants independent of
E.
Estimates (7.20), (7.21) follow from (7.7), (7.8) due t o the inequality
which is known from the theory o f elliptic boundary value problems in smooth domains (see [I]). Now we shall obtain an effective estimate for the energy concentrated in a part G of the stratified body
R.
Let G be a smooth subdomain o f
R . We define the energies corresponding
to uc and u by the formulas
Theorem 7.3. Let uc, u be the solutions o f problems (7.3), (7.4) respectively, u E Then
H2(0).
§ 7. Homogenization o f stratified structures where c l ( G ) is a constant independent of
E.
Proof. For the sake o f simplicity we prove this theorem assuming the elements of the matrices
A?
t o be smooth functions. It is easy t o show using smooth
approximations for the coefFicients, that the result is valid if the coefFicients are not smooth.
It follows from (7.8) that
Taking into account
-
(7.19), (7.11) we find
J [(dMi", ax.
G
Vk
* az,)
, V V , ~ax. '
au
+
d2u
G
Ivv12 d x ,
,N c -)dx
dx,dxj
-
II. Homogenization of the system of linear elasticity
where lqZl I c36;l2 11~11&2(~). By virtue of (7.12) we have
=-
J ,.( L a axk ( 1 ~ ~ ax, 1 2
G
-
-
pk au a au J (a. - - ( N C -))dx ax, ' axk axj +
G
1
~
dML +
3
du -ldX axj
~
pk
1
2
3
du
aG
Therefore it follows from (7.25) that
From (7.19) we obtain
du ,NC -)ukdS
axj
.
§ 7. Homogenization o f stratified structures
<
where ( p iI ~ 6 , " ' ( ( u ( ( & = ( ~ ) . Since by the imbedding theorem we have IIVulJLa(ac)5 c ~ ~ u I I ~ z ( for ~) any u E H 2 ( R ) (see also Proposition 3 of Theorem 1.2, Ch. I), it follows from (7.24)-(7.27)
that the estimate (7.23) is valid. Theorem 7.3 is proved.
Corollary 7.4. I f the coefficients of system (7.3) are smooth, it follows from (7.22). (7.23) that
Note that the matrix [cpkcpl~,kl]-' was used in (7.5) t o define N,', MG. Let us show that this matrix exists and its elements are bounded functions (uniformly in
E).
Lemma 7.5. Let A'j(x), i, j = 1, ...,n , be matrices o f class E(nl, n2), where nl, n2 are positive constants independent o f x. Let cp E C1(Q), ( V y ( 2 const.
> 0,
Vcp = (91, ..., cpn). Then there exist two constants n3, n4, depending only on n l , n2 and cp, such that for any
E
Rn
!%d.Set T i h = ( ~ i t h4-9 h E i
in (3.3), Ch. I. Then
II. Homogenization o f the system o f linear elasticity
Set K ( x )
= cp,(x)cp,(x)APq(x).Then by (3.3),
where the constants cl,
Ch. I, for any [ E Rnwe get
MI depend only on n l ,
K Z , (P.
I t follows that I{-'
exists. Setting [ = K-lC we obtain
•
These inequalities imply (7.28). Lemma 7.5 is proved.
7.2. Necessary and Suficient Conditions for Strong G-Convergence of Operators Describing Stratified Media In the case of stratified structures the general results on strong G-convergence together with formulas (7.5) and Theorem 7.1 make it possible t o formulate necessary and sufficient conditions for the strong G-convergence o f
2 in terms o f convergence o f certain combinations of the coefficients o f L,,and t o obtain for the coefficients o f E explicit
the sequence {C,) to the operator
expressions involving only weak limits o f the above mentioned combinations of the coefficients o f
C,.
We shall need some auxiliary results about compactness in functional spaces. Denote by COIPthe space of bounded measurable functions g(t, y), ( t ,y ) E
[O, 11 x 0, equipped with the norm
t varies over a set of full measure.
By C1@we denote the space o f functions g(t, y) such that g(t, y ) .
Co8P,j = 1,..., n.
9 E ayj
§ 7. Homogenization of stratified structures
231
Lemma 7.6. Consider a family o f functions & ( t , Y ) whose norms in CotP are uniformly bounded in
@E
C0,P
E
E ( 0 , l ) . Then there exists a subsequence E'
0 and a function
-+
0
such that
&(t, y) for any y
-+
E
-+
@ ( t ,y ) weakly in ~ ~ ( 0 , as l ) E'
a.
Proof. Let V
be a dense countable set in L 2 ( 0 , 1 ) . For a fixed v E V consider
the tamily o f functions o f y:
Due t o the assumptions o f Lemma 7.6 this family is uniformly bounded and equicontinuous with respect t o
E.
Therefore by the Arzeli lemma there is a subsequence E'
f t t ( t , y ) v ( t ) d t -+ Q,(y) uniformly in y
-+
0 such that
,
(7.29)
0
where Q,(Y) is a function of y
E
n. Since V is a countable set, one can use
the diagonal process t o construct a subsequence E'
-t
0 such that (7.29) holds
for any v E V. Now let w be an arbitrary function in L 2 ( 0 , 1) and v j -+ w in L 2 ( 0 , 1 ) as
j
-+
m, vj
E V.
Let us show that there exists Q,(y) such that
Qu,(y) -+ Q,(y)
uniformly in
a
as j
-+
oo
.
Indeed, it is easy t o see that
Choosing
EO
sufficiently small in order that for
E'
< EO
we have
232
11. Homogenization of the system of linear elasticity
we get
+
IQv,(y) - * v k ( ~ ) I 5 cllvj - vk11~2(0,1) 612 for any j , k ; y E
a. It follows that { Q , , ( y ) )
and therefore there is a function Qw E
is a Cauchy sequence in
C 0 ( O )such
Q v 1 ( y ) + Q W ( y ) uniformly in y E
c0(i?)
that
as j
-t
co
.
Choosing a sufficiently large j in the inequality
we find that
uniformly in y
E
a.
Obviously Q w ( y ) is a bounded linear functional on w y E
E L 2 ( 0 , 1 ) for any
a. Therefore
where @ ( t ,y ) E L 2 ( 0 , 1 ) for any y E
a.
Thus
for any w ( t ) E L 2 ( 0 , 1 ) . The function @ ( t ,y ) satisfies the inequalities -C
(y' - y"(P 5 @ ( t ,y') - @ ( t ,y") 5 c lyl - ytl(*
,
(7.30)
5 7. Homogenization of stratified structures owing t o the fact that iff, then m
5 f 5 M for
-+
233
f weakly in L2(0,1) as E -+ 0 and m 5
fc
<M,
almost all t E ( 0 , l ) . Therefore correcting, if necessary,
cP on the set o f measure zero we get iP E COl@ due t o (7.30). Lemma 7.6 is proved. Corollary 7.7. Let {&(t, y ) ) ,
E
E ( 0 , I ) , be a family of functions, whose norms in C'VP are
bounded uniformly in
E.
Then there exists a subsequence E' -+ 0 such that
weakly in L2(0,1) for any y E
0, j
= I , ..., n , where
~ E C1@.
Proof. It follows from lemma 7.6 that there is a subsequence E'
weakly in L2(0,1) for all y E Obviously for any g E
-+
0 such that
0 where $,cpiE C0?O
C,"(R) we have
a'(t' ' ) in the sense of distributions. Since Therefore p j ( t , y ) = -
8~j
COIPthe last equality holds in the classical sense for almost all
@, rl E
t.
Lemma 7.8. Suppose that the functions to
E
y E
&(t,y) are bounded in COIPuniformly with
E (0,1), and that $,(t,y)
0. Then
--t
0 weakly in L2(0,1) as
E -+
respect
0 for every
11. Homogenization of the system of linear elasticity
in the norm of CO([O, 11 x
a*
a) as
E --t
.
0
Moreover, if 2 , j = 1, ...,n , are also bounded in
1 ayj 1
@,(cp(x),x ) -+ 0 weakly in H 1 ( R ) as
E,
E
then
-+
0
for any cp(x) E C 1 ( 0 ) .
Proof. The family { @ , ( t , y ) ) , E E ( 0 , 1 ) , is equicontinuous and uniformly bounded in [O,1]x a . Therefore due t o the Arzela lemma there exists a function
$ ( t , y ) such that @,, E' -+ 0. Since
/ A,(.,
-+
$ in the norm of CO([O, I] x
a) for a subsequence
t
@. =
y)dr
-+
$ ( t , y ) for all t , y E [ O , l ] x
0, and $.(t, y)
-+
0
0
0 for any fixed y E a , it follows that 11, = 0. -+ 0 weakly in H 1 ( R ) as E -+ 0 . Indeed, we have already proved that @,(cp(x),x) -+ 0 in the norm o f L m ( R ) as E + 0 .
weakly in L2(0,1) as
E
4
Let us prove that @,(cp(x),x )
Moreover the derivatives
a
-@,(cp(x),z)are
axi
bounded uniformly in
due t o the compactness of a ball o f L 2 ( R )there is a subsequence E'
a that - @,t(cp(x),x )
ax,
-+
E.
-+
Thus
0 such
~ ( xweakly ) in L Z ( R ) ,and therefore x = 0. Lemma
7.8 is proved.
We introduce the following notation for i, s = 1, ...,n:
$7. Homogenization of stratified structures
Let us now apply the general results, established in
$9, Ch. I, on strong
G-convergence t o obtain the necessary and sufficient conditions for the strong G-convergence o f operators describing stratified media, in terms of weak convergence of the combinations (7.31) o f the coefficients o f system (7.1). Theorem 7.9. Suppose that the elements of the matrices A y ( t , y ) , norms in
C'fP uniformly bounded in
G-convergent t o the operator
E.
i, j = 1, ...,n ,
Then the sequence
have
{L,)is strongly
as E --+ 0 if and only if the following conditions
are satisfied
weakly in L2(0, 1) as
E
-+ 0 for any y E
0.
Proof. Assume first that the conditions (7.32) in this case 6, +
0 as
E
are satisfied. Let us show that
+ 0, where 6, is defined by formula (7.6). Indeed,
one can easily check that
"A Therefore
-B: + (B;)*(B,O)-'B;,
= -bq ((B')*(@)-ljjs .
II. Homogenization of the system of linear elasticity
236
Denote the integrands in the above formulas for N,'(t,y), M,",(t,y) by n:(t, y), rn;$(t,y) respectively. By virtue of (7.32) we have (7.34)
n:(t,y),rnk(t,y) -+ 0 weakly in L2(0,1) as E
-+
0 for any y E fi.
According t o Corollary 7.7 it follows that
weakly in L2(0,1) as as E
-+
0 for any y E
E
-+
0 for any y E
a.
a.
Lemma 7.8 and (7.34), (7.35) imply that the matrices N,'(t, y), M;",(t,y ) ,
a a -N,E(t,y ) , -M;E,(t,y ) converge t o zero in the norm o f CO([O, 11 x a) as 8~j
E
-+
8~j 0. Therefore due t o (7.6) we have
Moreover, it follows from Lemma 7.8 that
N,'(cp(x), x)
+
0
weakly in H 1 ( R ) as
ML(cp(x),x ) E -+
+
0
(7.37)
0.
Taking into account (7.11), (7.36), (7.37) we find that
a
- M i " , ( ~ ( x ) ,-+ x ) 0 weakly in
at
L 2 ( R ) as
E +0
,
(7.38)
a
Pk -Mi",(cp(x),x)-+ 0 weakly in L 2 ( R ) . since IVPI2
ask
Let us prove the strong G-convergence of
L, t o c as E
-+
0.
Set f = L ( u ) E H V 1 ( R ) ,iP = 0, u E C,"(R) in Theorem 7.1. Then estimates (7.7), (7.8) are valid. By virtue o f (7.36), (7.37), (7.38) we have u' -+
u weakly in H i ( R ) , -yf
-+
+' weakly in
L2(R)
57. Homogenization o f stratified structures
237
Now let us show that the set { E ( v ) , v E C F ( R ) ) is dense in H - ' ( 0 ) . Then the convergence
L,
3 E
will follow from Remark 9.1, Ch. I. According
t o Remark 3.1, Ch. I, every g E H - ' ( R ) can be represented as g = k ( v ) ,
v E H,'(R), and for any f = L ( w ) E H - l ( R ) , w E
(?,"(a),we have
This means that
Therefore choosing w E C,"(R) close t o v in H i ( R ) we get a functional
f
= E ( w ) close t o g in
H-'(a).
Let us now prove that the conditions (7.32) are necessary for the strong G-convergence of
L, t o E.
Suppose that
L,
3 k as
E
+ 0.
Due to our assumptions about the coefficients of system (7.1) and Lemma
7.5, the elements o f matrices B l ( t , y ) , s = 0,1, ...,n , B f j ( t , y ) , i, j = 1, ...,n , belong t o C'vP and have norms in C 1 @uniformly bounded in
E.
Therefore
by virtue o f Corollary 7.3 there exist matrices B O ( t y, ) , B S ( t ,y ) , B i j ( t ,y ) ,
s, i, j = 1, ...,n , with elements in C 1 @and such that for a sequence E'
+
0
we have
weakly in L2(0,1) for any y
E
a.
Set
Define the matrices f i ~ ( ty,) , ~ i E j ( yt ,) by the formulas (7.5) with k j ( r , y ) replaced by 2 j ( r , y ) and define 8c by (7.6) with
N,E, MiEj replaced by N,E, M;.
The same argument that we used a t the beginning o f the proof of this theorem shows that
II. Homogenization o f the system o f linear elasticity
-t
&I
0 in H 1 ( R ) ,
N ~ ( ~ ( X s) ) , + 0 weakly
a
-~
at
as
E' t
: ( ~ ( 2 2 )) --t ,
O weakly in L 2 ( R )
0.
Let ii E
C r ( R ) . Denote by uc solutions of the following problems
Set
+y:=Ac
,k duC -, dxk
'ik
?'=A
dii
-
Similar t o the proof of Theorem 7.1 we obtain the inequalities
-
u
- ii - N
II"t
- 7;
t
-a1dGx ,
<
lliill,plnl
H1(Q) -
aMfal aii - at h/ l a c o l
,
I)iill~2(n) .
< - clJ;l2
Therefore by virtue o f (7.41) we have
u"' + ii weakly in H 1 ( R ) , y;, -+
9'
weakly in
L 2 ( R ) (7.42)
as E' + 0. Denote by u0 the solution of the problem
L, t o i? and due t o (7.42) a3 we have u0 = 6, Ahk - = Ahk - almost everywhere in R. Since G is axk dxk an arbitrary vector valued function from C r ( R ) , it follows that Ahk = almost everywhere in R . Thus we have shown that from any subsequence E" -t 0 we can extract B B', another subsequence E' + 0 such that relations (7.39) hold for " -.. . . B'3 = B'J,s = 0 , ...,n , z , = ~ 1, ...,n , where B \ B'J are expressed in terms By the definition o f the strong G-convergence of
33
ahk
h
.
.
of the coefficients of the G-limit operator
A
A . .
by the formulas (7.31).
Since
5 7.
Homogenization o f stratified structures
239
{E") is an arbitrary subsequence, it follows that (7.32) is valid for
E -+
0.
Theorem 7.9 is proved. In the proof of Theorem 7.9 we have actually established Theorem 7.10. Let the elements o f the matrices A y ( t , y) be such that their norms in C 1 @are uniformly bounded in
E.
Suppose that there exist matrices ~ " ( y), t , ~ ' j ( ty ,) ,
s = 0 , ...,n , i, j = 1, ...,n , such that (7.32) holds for the coefficients of system (7.1).
Then the sequence o f operators
C,
corresponding t o the co-
, is strongly G-convergent t o operator?!, whose efficient matrices A y ( c p ( x )x) coefficients a ' j ( t , y ) have the form
a's = ( b ) * @ o ) - l B s
- &S
,
z , s = l , ..., n .
(7.43)
Let us consider some examples o f strongly G-convergent sequences
{C,)
which satisfy the conditions (7.32). Theorem 7.11. Suppose that the elements of the matrices A:]($) o f class E ( K , ,K
~ have )
the
form aY1(E-'zl), where a?,([) E A P ( R 1 ) are almost-periodic functions of
E
E
R1.Then the sequence C, strongly G-converges t o the operator k whose
matrices of coefficients are given by the formulas
where ( A i j ) is by definition the matrix with elements
Moreover estimates (7.7), (7.8) hold and 6, -+ 0 as
E + 0.
Proof. In the case under consideration we have A:j(t, y ) = Aij(&-It),p ( x ) = XI.
Set
II. Homogenization of the system of linear elasticity ( s )=
( ~ l(3))-' l
(A1]- A ' ~ ( s ) ,)
+
'3 z i j ( s ) = ~ i l ( s ) ( ~ l l ( (Alj ~ ) )- ~ l j ( s ) ) A..(
The elements o f matrices any almost-periodic f , g, f
I.;., Zij
- A'3
6..
.
are almost-periodic functions, since for
2 const > 0 , the functions fg
almost-periodic.
and
1
-
f
are also
It is easy t o see that
(5)=
0 and the elements o f N j , M,Fj are uniformly bounded and equicontinuous. Therefore 6, -+ 0 in Theorem 7.1, since N j , M$ converge t o zero as E -+ 0 at any point x1 E ( 0 , l ) . The strong G-convergence o f L, t o 2 follows from the conditions (7.32),
Obviously ( Z i j ) =
t
which hold due t o the fact that f (-) any almost-periodic
f. Theorem
E
-t
( f ) weakly in L2(0,1 ) as
0 for
7.11 is proved.
Let us consider some examples where the coefficients of We introduce a class
E -+
c depend on 2.
A, consisting of functions f ( t , y) such that for some
~ f ( Y ) 9t f ( t , Y )we have
af
dcf ( y ) , g f , cj(y),agf , 1 = 1, ...,n, are also dYl dYl assumed t o be Holder continuous in y E R uniformly in t E [O,l],and such
The functions f ( t , y ) , -,
8~1
that
where the constants co, a do not depend on t, o E ( O , l ] . Set
5 7.
Homogenization o f stratified structures Obviously for any f E A, we have ( f
(a,
y)) =cj(y).
A few examples of functions that belong t o A, are listed below. 1. Functions f ( t , Y ) E C'lP that are 1-periodic in t belong t o A, with cr = 1 . 2. Consider a function f ( t ) of the form f ( t ) = M
+ cp(t), where M = const.,
Ip(t)l 5 C ( 1 + Itl)-N, N > 0 . We can easily check that f E dl, if N f € & f o r a n y U E ( O , l ) , ifN= 1 ; f € A N , i f 0 < N < 1. 3. The sum
$1
+ $2,
where
$1
E
A,,
+2
E
.Aaz, belongs t o A,
> 1; with
cr3 = min(cr1, a z ) , c r ~ , u zE ( 0 , 11. Lemma 7.12. Let f ( t , y ) E
A, for some a E ( O , l ] , and let ( f ( . , y ) ) = 0 for all y E a. Then
where cl is a constant independent o f Moreover, for any y E
52 fixed,
E,
y, T
we have
weakly in L 2 ( 0 , 1 ) (as functions of 7 ) .
Proof.
Let us prove (7.46) for a = 0 .
c j ( y ) = 0 in (7.44), and tain E-'
jf
i o
Since ( f (., y ) ) = 0 , therefore
f ( s , y ) d s = g ( t , y ) . Setting s =
E-'T
we ob-
f ( ~ - ' r , ~ ) d= r g ( t , y ) . Therefore setting T = ~ t by virtue of
0
(7.45) we get
Thus (7.46) is valid for a = 0. For a = 1 the estimate (7.46) is proved in the same way, since we can differentiate (7.44) with respect t o yl, and d c j ( y ) / d y I = 0 . The convergence (7.47) follows directly from (7.46). Indeed, due to (7.46) we have
II. Homogenization of the system of linear elasticity
where 0
and X[a,b] is the characteristic function o f the interval
[a, 61.
Approximating v E L2(0,1 ) by linear combinations o f characteristic functions and taking into account that f , d f / d y rare bounded, we get
,y)v(s)ds+ 0 as E
J %(: 0
--t
0. Lemma 7.12 is proved.
For a given matrix B(t,y ) with elements B;j(t,y ) let
( B ( . ,y ) ) be the ma-
, trix with elements ( B i j ( -y)). Theorem 7.13. Let the elements of the matrices A:j have the form
and define for
i, s = 1, ...,n , A" = {a;kj') the following
matrices
Suppose that the elements o f BO(r,y ) , P ( r ,y ) , BiS(r,y ) belong t o
A,
for some u E ( 0 , l . Then the sequence of operators t,corresponding t o Q ") x) is strongly G-convergent t o the operator k whose the matrices
1
coefficient matrices are
243
$7. Homogenization o f stratified structures Moreover, the number CEO,
6, used in Theorem 7 . 1 satisfies the inequality 6, 5
where the constant c does not depend on
Proof. According t o
weakly in LZ(O,1 ) as
E.
Lemma 7.12 we have
E
-+ 0 for any y E
a.
Therefore due t o Theorem 7.9 one can take
B S ( t ,y) = ( B s ( .y, ) )
,
&j(t, y ) = ( B i i ( . ,y ) )
,
Thus by virtue of (7.43) the coefficients of the G-limit operator
are given
by (7.49). Let us now show that 6,
5 e".It is easy t o see that
Therefore, from (7.48) we see that the matrices Njc(t, y ) , MiC,(t,y ) defined by
( 7 . 5 ) can be written in the form
7
7
Denoting the integrands in (7.50) by n j ( - ,y ) , m i j ( - , y ) , respectively, we E E see that the elements o f n j ( t ,y ) , m i j ( t , y ) belong t o A, and ( n j ( . ,y ) ) =
( m i j ( . , y ) )= 0 . Thus by the definition of 6, and (7.46) we get 6, proved.
5 E". Theorem 7.13 is
II. Homogenization of the system of linear elasticity Corollary 7.14. If cp(x)= x1 in Theorem given by the formulas
7.13, then the coefficients of the G-limit system are
58. Estimates for the rate o f G-convergence $8. Estimates for the Rate o f G-Convergence o f Hieher Order E l l i ~ t i cO ~ e r a t o r s
8.1. G-Convergence of Higher Order Elliptic Operators (the n-dimensional case) In a smooth bounded domain R
c Rnconsider
a differential operator of
the form
where aap(x) are bounded measurable functions in R , a,/? E a1
Z;,(a(=
+ ... + a,, U(X)is a scalar function in H,"(R).
We say that a differential operator L : H,"(R) + H-"(R) o f the form (8.1) belongs t o the class E(Xo, XI, Xz), if its coefficients satisfy the following conditions
Ao, X I , A2 are positive constants independent o f U. It follows from the last inequality (see (1341, [9])that for any t E Rnand any x E R we have
for any u E C F ( R ) , where
> 0, ta=
where
Q
= const.
is elliptic.
Now following [I481 we give the definition for the strong G-convergence of a sequence of higher order elliptic operators.
{Lk)of class E(X0, XI, Xz) is strongly k o f class ~ ( iXI,~I,),, if for any X > j, (j, =
We say that a sequence of operators G-convergent t o the operator
11. Homogenization o f the system o f linear elasticity
246 const.
> 0 ) and any f E H-"(R)
the sequence o f solutions o f the Dirichlet
problems
converges in H,"(R) weakly as k
m t o the solution u of the problem
t
and moreover, if the sequence of functions
converges in L 2 ( R ) weakly as k
t
oo t o the functions
Here { a k p ( x ) )and {Zl,p(x)) are the matrices of coefficients of operators
C k and
respectively.
Note that the difference between the strong G-convergence and G-convergence consists in the requirement of the weak convergence of the weak gradients r , ( u k , . C k )t o r & , L )
in L 2 ( n ) as
k
-t
oo.
It is shown in [I481 that the strong G-convergence o f C k t o
E
as k
is equivalent t o the following conditions, the so-called Condition N: There exists a sequence of functions { N , k ( x ) ) such that N1
~ , Ek H m ( R ) , ~ , +kO
N2
k
weakly in H m ( R ) ,
lyl
<m
+
;
a & ~ ' ~ p k a t p -+ hop weakly i n L 2 ( R ) , I~l=m
N3
Do($p 1a1=m
- hop)
+
0 in the n o r m of H - " ( 0 )
,
t
oo
$8. Estimates for the rate of G-convergence
247
A similar condition for the system o f linear elasticity was formulated in $9, Ch. I. If we impose some additional restrictions on the functions N,k we arrive at a stronger condition (the so-called Condition N1) which not only implies the weak convergence o f uk t o u in H,"(R) as k
-+
co,but enables us t o estimate
the difference between uk and u. We say that a sequence of operators { C k ) E E(Xo,X1,X 2 ) with the rnatrices o f coefficients { a $ ( x ) ) , l a [ ,IPI
5 m , satisfies
the Condition N' in 0 ,
if there exists an operator E E E ( ~ O , K ~i2) , with the matrix of coefficients {Zlap(x))and a family of functions N,k E H m ( R ) ,lyl m , such that
<
DaN,k E L m ( R ) for la1 5 m , Iyl 5 m , DON; + O
N'1
in thenormof L M ( R ) , la1
< m , )yl 5 m
in the normof H-'*"(a),
lal,IPI 5 m ;
in the norm of H-mlm(R) ,
5m
as
;
k + co
(for the definition o f H-"*"(R) see $9.2, Ch. I). For the sake o f simplicity we assume that the coefficients 21,p are infinitely smooth . Let us introduce the parameters which characterize the rate of convergence in the Conditions N'l, N12, N13. Set
II. Homogenization o f the system o f linear elasticity ,f?f) =
Ilidtp-
max
,
idapll~-~,m(~)
lol<m IPlSm
a=
max IP16m
1
oa(id:,- idar) lol=m
I
H-m$03(R)
Theorem 8.1. Let the Condition N' hold for the operators C k ,
ji such that for p
> ji
and s
k . Then there is a real constant
5 m - 1 the solutions o f the
Dirichlet problems
satisfy the inequalities
lluk
- u l l ~ s ( n5 )
+ + BY + n)I I ~
IlvkllHs(Q)
+ K[B!" I I 1 1~ ~ 1 c o+) ( u k
where I( is a constant independent of
,
IIL~(Q)]
k,
f,
and
vk
(8.11)
is the solution o f the
Dirichlet problem
Proof. For any operator C E E ( X o , XI, Xz) there is a real constant jl depending only on Xo, XI, Xz, and such that if p > fi, then the solution w of the Dirichlet problem
58. Estimates for the rate of G-convergence satisfies the inequality
where c is a constant depending only on Xo, X I , X z (see p
>
Lk
>
[9]). Let us choose
0 such that the solutions of the Dirichlet problems for operators
+ p , k + p satisfy (8.13)
with a constant c the same for all Ic.
We shall use the following Leibnitz formulas (see [127]):
where ( that
Pj
a
P
)=(
a1
)
< aj for each j
(
an
P,
Q!
)=
= 1, ...,n.
P!(a-P)!
, a! = a l ! ... a n ! ,P I a means
Set
where u is the solution of problem (8.9) and N,k are the functions entering the Condition N'. By virtue o f (8.14) we find (L(u),)
lallm IPlSm
/
n
a$ D@U:
V av d x
=
II. Homogenization o f the system o f linear elasticity
250
Denote by
where
CI
JO the last integral. Then because o f (8.4) we have
is a constant independent o f
Transposing the indices y and
P
k. in the integral next t o the last one in
(8.16), we obtain
where
J1,
Jz,
J3
stand for the respective integrals on the left-hand side of the
last equality. From (8.4) we have
I t follows from (8.6) and Lemma 9.1, Ch. I, that
58. Estimates for the rate o f G-convergence
Let us estimate J I . Using (8.15) we find
Applying Lemma 9.1, Ch. I, and (8.7), (8.5) to estimate the first two integrals in the right-hand side of (8.21) we get
1 J ~ I 5 Cl [ 7 k l l u l l ~ m ( R IIvIIH"'(f2) ) + + /?!I) IIvIIHmm)
~lullHzm*l(R)]
.
Thus ((.lk
for any v E
+ P ) u : , v ) = ((i.+ P ) U . V ) + e ( u ,v )
HF(fl),
where
W e obviously have u: - v k - u k E
HF(fl)
and
II. Homogenization of the system of linear elasticity
Hence, setting v = u i
- vk - uk
and using (8.23), (8.13), we obtain the
inequality
which implies (8.10), since for the solution u of problem (8.9) the following a
priori estimate is valid
(see
[55]).
Due t o (8.4) we have for s
5 m - 1 the following
inequality
Theorem 8.1 is proved. One o f the simplest examples, when the Condition N' holds, is provided by the sequence o f operators
L k
with coefficients a k p ( x ) such that a k p ( x ) +
Lm(R) a s k + m. In this case we can take N,k G 0. Obviously the Conditions N'l, N12, N'3
Zl,p(z) in the norm of
are satisfied and
where c is a constant independent of k,
$8. Estimates for the rate of G-convergence
According t o Theorem 8.1 we have
Actually one can prove a stronger inequality in the case under consideration, namely:
To obtain (8.25) we note that in the proof of Theorem 8.1 the norm
11 f
llHlcn,
estimates
IIuII~~,+I(~)
in (8.22). The norm
I I u I I ~ z ~ + I ( ~ ) is needed
t o estimate the first integral in the right-hand side o f (8.21). It is clear that under our assumptions this integral can be estimated by
Let us now consider a less trivial example, when the Condition N' issatisfied. Assume that the coefficients a k p ( x ) of operators
Lk
depend only on x l , i.e.
a!&(z) = akp(xl). Let the coefficients Aap(zl) o f operator
k
be such that for all la1
5
m,
1/31 5 m , a = (m,O, ...,0 ) we have
&
1 atp Asp Aaa -'^, +7, a,, a,, a,, a:, a,, abuatp ~a,A,p a& - --$ ZlaP - - weakly in ~ ' ( 0 , l )as k a:, a00 1
- 7 , k
a:,
where 1 is such that
R c {x
: 0
(8.26) -+ m
< XI < I).
Define the functions N j ( x l ) as solutions o f the equations
such that
,
11. Homogenization o f the system o f linear elasticity
It follows that the Condition N13 is satisfied and /If)t 0 as k t m, -yk = 0 in (8.5), (8.7). By virtue of (8.26) the right-hand side of (8.28) tends t o zero weakly in d" L2(0,I ) as k t oo. Therefore -N ; ( x ~ )t 0 in the norm o f CO([O, 11) as dx: k + m , s = O , l , ...,m - 1 . Owing t o (8.28), (8.4) one has
and
m . Thus the Condition N 1 l is also valid. Let us consider the Condition N12. We obtain due t o (8.28) that ak t
0 as k
t
-
-~*B.P
Gap - ( a b 2 P
aka
4,) .
By virtue of (8.26) we have iikp(xl)- i a p ( x l )+ 0 weakly in L2(0,1). Therefore
/ (i*,(s) - ~ , ~ ( s ) ) d s 0 in the 21
@k,(z,) =
--+
norm of CO([O, 11)
0
d dx1
Since hk0(x1) - ;,p(xi) = -@ k p ( x l )we , can assume in (8.6) that
P?'
=c
max IaIsIPl<m
58. Estimates for the rate o f G-convergence
p;)
and
-+
0 as k + oo,c = const.
Now in order t o obtain an effective estimate for the closeness of uk(x) t o
u ( x ) it is sufficient t o estimate I I ~ ~ l l ~ r nWe ( n ) .have
where cj are constants independent of
k.
Here we used the definition o f a k , the a prion' estimate (8.24), the inequality
IIuIIL*(an)5 c I I u I I H # - ~ ( ~ ~ , lIuII~*+t(an) ,
>0
t >0
[9]),and the fact that Npk possess derivatives up t o the order m, which are bounded uniformly in k. (see
Define the parameter bk, by bk = rnax { a k , ,&)). Then, according t o Theorem 8.1, we obtain the inqualities
lluk - uIIHm-l(n) 5 c6:I2 I l f llHl(n) 7
8.2. G-Convergence of Ordinary Differential Operators The results of the previous section are obviously valid for ordinary differential operators.
However in the latter case we can obtain more accurate
estimates. Here we prove some theorems in this direction.
II. Homogenization of the system of linear elasticity
256 Let
R =(0,l)
and let
.Lk, J?
be ordinary differential operators of the form
Theorem 8.2. Let u k , u be the solutions of the following Dirichlet problems
(Lk where C k ,
J?
+ P ) U ~= f ,
( k + CL)U = f ,
max
max rE[O,l]
and
,
(8.34)
f, k,
[(*-L)dil+ mm
~€[O,ll
+
1)
are ordinary differential operators (8.32), (8.33). Then
where the constant c does not depend on
Ak =
u k , E~ H,?(O,
aim
11
[(':f:q
N,k are the solutions of the equations
satisfying the boundary conditions
(aim';q
hPq)- -- a
0
akm
)d
(8.37')
58. Estimates for the r a t e o f G-convergence
257
Proof. It follows from the above result for higher order elliptic equations, whose coefficients depend only on XI,that in order t o prove estimates (8.35), (8.36) we have only t o estimate the functions vk which are solutions o f problems (8.12), namely
For the functions
Moreover, if p
N,k we have
5 2m - 1, it follows from
Sobolev's lemma that
where c is a constant independent of u . Therefore due t o (8.24) we get
Set m
'Pk =
C
p=o
dpu
N kdxp
'
It follows from (8.41), (8.42) that
b!Pk)~, I ~ $ 5I czAk Ilf ~ILz(o,I) .
P mapping any pair o f i = 0,1, ..., m - 1, into a smooth function cp(x)
One can construct a continuous extension operator numerical sets {aj0)), {a!')),
(8.43)
defined on [O,1] and such that
II. Homogenization of the system of linear elasticity
Obviously Q ( X ) can be defined by the formula
where e p ) ( x ) , e l 1 ) ( z ) are smooth functions which satisfy the conditions
Therefore vk is the solution of the Dirichlet problem
Lk(vk)
+
P V ~=
0 on ( 0 , l )
,
vk
- cpk
where ~k are the functions defined by (8.44) with $1
EHr(0,l)
,
= a!?, a!') =
By
vritue o f (8.43), (8.44) we get
Set wk = vk
- (pk.
Then wk is the solution o f the Dirichlet problem
C k ( w k ) =Ck(Qk)
Wk
E
1)
.
Using the inequalities (8.13), (8.45), we find that
I I C k ( ~ l k ) l l ~ - m ( o , l5) cs I I ~ l k l l ~ m ( 0 , L l ) G A l l~f IIL~(o,I)
.
11 f llLa(o,l) and We finally obtain C S A. ~I l f l l ~ 2 ( o , r .)
Hence l l w k l l ~ m5 c7Ak
I
11vkll~m
It is clear that
~k
= ,Bf) = 0 ; ,Bf'),cyk
< c A k , c = const, and therefore
estimates (8.10), (8.11) imply (8.35). Theorem 8.2 is proved. Remark 8.3. Suppose that the coefficients a;, o f the operators L k have the form a k , ( x ) =
a , , ( k x ) where a , , ( ( )
are 1-periodic bounded functions. It then follows from
(8.26) that the coefficients o f the G-limit operator
2 are given by the formulas
98. Estimates for the rate of G-convergence
/
1
where p,p
9 rn - 1. (f)=
f ( t ) d < . We also have Ak 5
0
C -
k
.
Moreover, the
estimates (8.35) become
where c is a constant independent of k and
d d If C k = - ( a k ( x )-), . . dx dx
-
L
reduced t o
f.
d d = - ( h ( x ) -), dx dx
where C is a constant independent o f
then the estimate (8.31) is
k.
Let us consider the latter case in more detail, so as t o obtain an explicit expression for the constant C.
It is easy t o see that
and
Therefore
M
- - max. <
*o
.6[0,1]
h(t)
11. Homogenization o f the system o f linear elasticity
260 where
so < a k ( x ) 5 M for any k = 1,2, ... and the function vk is such that
Due t o (8.24) there exists a constant
The dependence of
R
such that
on the coefficients of the operator
1 -1
du below. It is easy t o see that ax IIuk
R
- ~IlL2(0,15 )
1
<so 11 f l l L 2 ( o , l ) .
L2(0,1) -
2 will
be specified
Thus we have
M R + 1
Ak I l f llL2(o,l) + I I v k l l ~ 2 ( o , l ) 60
7
where
In order t o estimate the norm
J J V ~ J J ~we Z ( apply ~ , ~ ) the
maximum principle,
which yields
Ivkl
=€[o,ll
5 max { l v k ( O ) l , l v k ( l ) l ) .
Therefore
since obviously
Thus
R > 0 in terms of the ~oefficientsof the d u diL du G-limit operator L.Squaring both sides o f the equation i - = f - - Now let us estimate the constant
and integrating it over [ O , l ] , we obtain
dx2
dx dx
$8. Estimates for the rate of G-convergence
I t follows t h a t
where
/ , Idii
P > 0 is any constant such t h a t max
5 P.
Thus
z€[O,lI
2112
and therefore we can take
P2
R = - (1 + -) 60 6:
112
W e finally obtain t h e inequality
where the constants M,R,60 can b e easily calculated for t h e given coefficients o f t h e operators
Ck,2.
This Page Intentionally Left Blank
CHAPTER Ill SPECTRAL PROBLEMS
$1. Some Theorems from Functional Analysis. Spectral Problems for Abstract Operators Here we formulate and prove some results in the spectral theory o f linear operators, which are useful for applications considered below. Moreover in
$1we prove theorems on the convergence of eigenvalues and eigenvectors of a sequence of abstract self-adjoint operators depening on a parameter defined on different Hilbert spaces which also depend on non-self-adjoint operators are considered in
E.
E
and
Such questions for
$9.
These theorems provide means for the investigation o f spectral problems in the homogenization theory; they can also be applied t o study asymptotic behaviour o f spectra o f some other singularly perturbed operators considered in this chapter.
1.1. Approximation of Eigenvalues and Eigenvectors of Self-Adjoint Operators Following [I321 we give here a proof of an important lemma which has wide applications for the approximation of eigenvalues and eigenvectors of self-adjoint operators. Let H be a separable Hilbert space with a real-valued scalar product (u,v)H; and let
A be a continuous linear operator d
:
H
-+
H. By
lldll we denote the norm sup IIAUIIH , where the supremum is taken over all llullH ti E H , u # 0; and llullH as usual stands for (u,u)z2. A : H -+ H is denoted It is well known that L ( H ) is a Banach space with the norm by L ( H ) . The space of all continuous linear operators
I l d l l ~ c= ~ )11All.
111. Spectral problems
264 Lemma 1.1. Let A :
H
H be a continuous linear compact self-adjoint operator in a H . Suppose that there exist a real p > 0 and a vector u E H ,
-t
Hilbert space
such that llullH = 1 and
( I A u - pullH
5a,
a = const
Then there is an eigenvalue p, of operator
Moreover, for any d
>o .
(1.1)
A such that
> a there exists a vector 'll such that
and ii is a linear combination o f eigenvectors of operator eigenvalues of
A from the segment [ p - d, p
Proof. Consider in H tors o f
A
:
A corresponding t o
+ 4.
an orthonormal basis { ' P ~ )which , consists of eigenvec-
Avk = p k ( ~ k k, = 1,2, ... .
Such a basis exists according t o the
Hilbert-Schmidt theorem (see [40]). Then
The assumptions of Lemma 1.1imply that
Let pi be the eigenvalue o f A such that Ip
- pil = rnin k
have
and therefore Ip; - pI
5 a , since
Ip - pkl. We then
§1. Some theorems f r o m functional analysis
Let us prove the second statement of Lemma 1.1. Set
Then
where
Without loss o f generality we can assume that p
#
pj for any j . Therefore
ck = (pk - ~ ) - ' f f k .
Set uo =
x
c,pl, where the sum is taken over all indices 1 such that
1
pi E [p - d , p
+ dl. We have
where the sum is taken over all k such that pk
6 [p - d , p + dl.
Let us show
that ii = IIuolljjluo is the vector we seek. Indeed, since Ilu
- u o l l ~= l l v l l ~I a d - ' , l l u o l l ~I 1, l l u o l l ~2 l l u l l ~-
l l v l l ~W , e have
Lemma 1.1is proved.
III. Spectral problems
266
1.2. Estimates for the Difference between Eigenvalues and Eigenvectors of Two Operators Defined in Diflerent Spaces In this section we prove some important theorems on the behaviour of eigenvalues and eigenvectors o f a sequence o f abstract operators defined in different Hilbert spaces under certain restrictions imposed on this sequence (Conditions C1-C4).
The Hilbert spaces can be chosen in such a way that
homogenization problems as well as many other singular perturbation problems for differential operators can be associated with such sequences o f operators satisfying Conditions C1-C4, which enable us t o study the corresponding spectral properties. Let
'FIE,X0 be separable Hilbert spaces with
real valued scalar products
respectively, and let
be continuous linear operators,
I m A C V C 7-10, where V is a subspace o f
7-10,
In the rest o f Chapter 111 we consider spaces 'Ido,'H,, V and operators
A, subject
do,
t o the following Conditions C1-C4.
a. There exist linear continuous operators
Re : 'Flo + 7-1,
and a constant y
>0
such that
f0 E V. (If 'Flo = V
for any
= L2(R), 'FI, = L2(R" and R' is a perforated domain of
type I or II (see 54, Ch. I), then we can take as such that
Ref
=f
In* for any f
RE the
restriction operator,
E L2(R). It is shown below that in this case
51. Some theorems from functional analysis
y = mes Q n w ) . C2. Operators JZ,
:
1-I, + 'He,Jb
self-adjoint; their norms
Il&ll, c c. H I)
'Ho + 'Flo are positive, compact and
:
are bounded by a constant independent of
&.
c.3. For any
f EV
C4. The family of operators
{A,)
is uniformly compact in the following sense.
' , such that sup From each sequence f' E H
11 fcllnc
< CX),
one can extract
6
a subsequence f" such that for some
w0 E V
Remark 1.2. Condition
C1 implies that if the sequences f", gc
and the elements
fO,
are
such that
11 f'- R, follw,
+0
, IJg' - R 6 g o ( l ~ , 0 as +
&
--+
0
7
then
(f',gC)n. 7 ( f 0 ,s").HO >
(1.8)
+
Indeed, due t o (1.4), (1.7) we have
(f', gC)n.- ( R e fO, R,gO)n.= (f'- Ref0,gC)ne+
+ ( g c - REgO,REfO)nI5 I l f '- REfolln,llgelln. + 0. + 119" - RcgOlln,IIR6f011n. 0 +
&
+
III. Spectral problems
268
It is easy t o see that (1.4) implies the convergence ( R e f 0 Reg0)%. , -+ y ( f o , since ( u , v )= 4-'(llu v(I2- IIu - v1I2).
+
Remark 1.3. Condition C3 implies that iff'
[ I f' Refollnl
-+
E 'FIE, f 0 E V and
0 as
+
0
(1.9)
then
llAef' since
IIA,f'
- R c A o f O l l ~-+ . 0 as E
- RcAofOllx,I
IIAe(f'
and the norms of the operators
+
0
,
- R,f0)l17i,
(1.10)
+ IIAeR,f0 - R e d o f O I I ~ ~
are bounded by a constant independent
Consider the spectral problems for the operators
&.
A,, do:
d , u , k = p , k u ~ , k = 1 , 2 ,..., u , k € ' H c , p f 2 p: 2
k 1 pc"' ,
( U ; , U ~ ) N ,= 4 r n
p;
>0 ,
,
= 1 for 1 = m , 61, is the Kronecker symbol: tilm = 0 for 1 # m, the eigenvalues p,k and ph, k = 1,2, ... , form decreasing sequences and each where
eigenvalue is counted as many times as its multiplicity. Our aim is t o estimate the difference between eigenvalues and eigenvectors of problem (1.11) and those o f problem (1.12) for small
E.
Theorem 1.4.
' ,, 'HO, V and operators A,, Let the spaces H Then there is a sequence
{@) such that P,k
A, R, satisfy -+
0 as
E -+
Conditions C1-C4.
0, 0
< P,k <
and
$1. Some theorems from functional analysis
269
where p,k, pk are the k-th eigenvalues o f problems (1.11), (1.12) respectively,
N ( & , d o ) = { u E ?lo, &u = pku) is the eigenspace of operator docorresponding t o the eigenvalue p;. In order t o prove this theorem let us first describe some properties o f operators
A, A.
Lemma 1.5. Let u, E V and let {u,k), {p,k)
be sequences o f eigenvectors and eigenvalues
of problems (1.11) such that
for a fixed
k.
Then u, and p, are respectively an eigenvector and eigenvalue
o f d o , i.e. d O u , = p,u.,
U,
Proof. Setting f' = u,k, f 0 that
It is easy t o see that
# 0. = u, in (1.9) and using (1.14), (1.10) we find
111. Spectral problems
270
Due t o the conditions (1.14) the first two terms in the right-hand side o f this inequality converge t o zero as e
-+
0, and the third term converges t o zero by
virtue o f (1.15). Thus
Hence, by (1.4) we deduce that Au, = p,u..
llR,~,11~.
-$
Due t o (1.14)
0 as e -+ 0. Therefore according t o (1.4)
which means that u,
# 0.
71/2
I J u , ~ \-J ~ ,
IIu.llXo = 1,
Lemma 1.5 is proved.
Lemma 1.6. Suppose that Conditions C1-C4 are satisfied. Then
where pt,
are the k-th eigenvalues o f problems (1.11), (1.12) respectively.
Proof. Let us first establish the inequalities
where
Q,
c ( j ) are constants independent o f E , q,does not depend on j .
The upper bound for p i follows from the fact that the norms of operators
A, are bounded uniformly in e. Fix an integer j and il;,...,
1 = 1, ...,j of
> 0.
let
> ... > pi+'
>
ilitlthe corresponding
+ 1. The fib
exist since pk
be eigenvalues of operator
eigenvectors such that
# 0, k
dohas a finite dimension. Setting f = 6; for each k = 1, ...,j
((iiL(17.1,
= 1,
= 1,2, ... , and each eigenspace
+ 1 in Condition C3, we obtain from
(1.5) that
IIA,R,G~
- R,Aofi;lln,
-+
0 as
E
-+
0
Therefore
JJA,R,E,~- p , k ~ , ~ , k l J-+~ o, Then by Lemma 1.1 for d = A,, there exists a sequence p,"(k"):
; t ~E -4
0.
H = 'HE,p = fit,
u =
II~~ckll
$1. Some theorems from functional analysis
where
are eigenvalues of problem (1.11) such that
pr(klc)
for all E smaller than some 6j. Therefore the inequality (1.16) is valid, since 2 Pr(j9c) and p;(J+l") + hit1 as E + 0.
pi
Taking into consideration the conditioan (1.16), the fact that Acu: = p:u:, and using the diagonal process, we conclude from Condition C4 that there are vectors u; E V and numbers p i such that
for a subsequence E' + 0, j = 1,2, ... .
It follows from (1.17), (1.18) that
According t o Lemma 1.5 u; is an eigenvector corresponding t o the eigenvalue .
.
i.e. Aoui = p3,u:, u;
Setting
fc
= ui,
(1.8), (1.19) we get
as
E'
f0
# 0; =
1
.
u3,, g'
= u,k,
P:
=
U,S1
in (1.7) and using
P*
+ 0.
Let us show that the vectors U j = ( p i ) - 1 y 1 / 2 ~j i , = 1,2, ... , form an orthonormal basis in 'Ho. Assume that this is not the case. Then there is a vector U E V such that for some
Set
f0
= U in Condition C3. Then
we have
III. Spectral problems It follows that
where UC,= I I R c t U I I ~ ~ , R e ,since U,
Let us apply Lemma 1.1with
By Lemma 1.1the relations (1.22) imply that there is a sequence o f eigenvalues
0. Therefore due t o (1.18) among the p i , j = 1 , 2 , ... , there is a ppk such that prk = pgk. Set
pet o f the operators
d=
1
A t
inf
which converges t o pgk as
E'
-t
IPk - p i [
j P:#P;
in Lemma 1.1and suppose that the multiplicity o f p r k is equal t o
...
=
prk+~-l
. Then the segment [pk - d , p $
+ dl can contain only such
that coincide with p r k , and therefore by (1.18)
eigenvalues of the operator
we see that for e' sufFiciently small the segment [pgk - d,pk only the eigenvalues p y k , ...,pc,
mk+l-l
u F k ,...,uFk+I-l
.
x 1-1
E'
sufficiently small
,
C : ~ U ; ~ + ~IIiiclllNzI , =
1 , such that
Choosing a subsequence E" -+ 0 such that cf,, + cl as due t o (1.19), that [\iictj- R c ~ ~ i i * l l+ ~ z0, , as where
+ d] can contain
o f A,, corresponding to the eigenvectors
By virtue o f Lemma 1.1and (1.22), for
there is a vector ii,, =
6, pFk =
E"
-+
0,
E"
-t
0, we obtain
§ 1. Some theorems from functional analysis Consequently by virtue o f (1.23) we have
Thus
U is a linear combination o f the vectors urktj, j = 0 , 1 , ..., 1 - 1 , which
is in contradiction with (1.21). Therefore the vectors U j , j = 1,2, ... , form an orthonormal basis in 'Flo. Obviously we can assume that pi = and p!
corresponds t o
2 p2 2
and ui = U3 in (1.12), since
Ui
... . Lemma 1.6 is proved.
Proof o f Theorem 1.4. Fix k and consider the sequence ~,ku,k= dCu,k.Since by Lemma 1.6 j ~ , k+ p,k as E -+ 0 , it follows from the proof o f Lemma 1.6 (see (1.19)) that there exists a sequence E' -+ 0 and a vector ut E N ( & ,
d o )c V ,
such that
Observing that the operators
A,
are self-adjoint we have
~,k(u,k, RCut)n.= (A,u,k,R C u f ) x = , (u,k,d c ~ E u f ) .n . Therefore
+
0 = ~,k(uR ; , C ~ ~ )-HP,: ( u , ~R,c u f ) ~ . ~ : ( u , k~, c u f )-~ . -
.
(U,~,AR~U:)~*
Hence
(P," - P!)(u;,Rcu$)x.= (u!, dCRcu$- ~ l o k ~ ~ .u f ) ~ (1.25) ,
It follows from (1.24) and (1.8) that
, Setting (u,k,,R . , U : ) ~ ,= from (1.25), (1.24) that
pk +
actt
where a,,
--t
0 as
E' --+
0, we deduce
111. Spectral problems
0 and @, =
Therefore estimate (1.13) holds for the subsequence E' -+ Let us prove that it also holds for
E -t
laell.
0.
E (0,l)denote by a, the infimum o f P,k 1 0 such that the and (1.13) is satisestimate (1.13) is valid. It is easy t o see that 0 5 a, < For each fixed
fied with
E
P,k = a,.
Let us show that a,
Then there is a subsequence
E"
-t
-t
0 as E
+ 0. Suppose the contrary.
0 such that a,,,> c > 0. According t o
what has been proved above, there is a subsequence E' o f the sequence E" such that the estimate
a,, 5
P,k,,,which
+ 0. By the definition of a,) we have
(1.13) holds with
is inconsistent with the inequality a,~,> c
> 0.
Theorem
1.4
is proved. Estimates for the difference between eigenvectors of problems (1.11), (1.12) are established by Theorem Let k
1.7.
2 0, n 2 1 be integers such
pi >
that
=
o f problem
i.e. the multiplicity o f the eigenvalue
... = pt+m> pi+m+', (1.12) is equal t o m,
IIW~~~,
p: = m. Then for any w E N ( ~ ; + ' , A ~ ) , = 1, there is a linear o f problem (1.11) such that
combination iic o f eigenvectors u,k+', ...,
where
M k is a constant
independent o f
E.
Proof. Set
H = 'He,A = A,, u = II~,wll;;:~,w, p = - d,&' and choose d > 0 so small that the segment no eigenvalues o f
7 llwllgo = 7 as
E
d,, other
than
=
...
=
&".
in Lemma
directly from Lemma
1.1. Theorem 1.7 is proved.
+ dl contains
Since IIR,wII$.
+ 0, the existence o f .iie and the estimate
1.1, -t
(1.26) follow
$2. Homogenization of eigenvalues and eigenfunctions $2. Homogenization of Eigenvalues and Eigenfunctions of Boundary Value Problems for Strongly Non-Homogeneous Elastic Bodies
2.1. The Dirichlet Problem for a Strongly G-Convergent Sequence of Operators Let L C ,k be the elasticity operators in a domain R, considered in $9, Ch.
I, and let LC be strongly G-convergent t o
as
E +
0 (L,
3
k).
Consider the following eigenvalue problems for operators L, and f?:
L,(ut) = -X,kp,(x)u~ in R
O < X f <XZ<...<X,k<
J
P ~ ( x ) ( u u:)dx :, = 61,
,
ut = 0 on d R
,
...,
(2.1)
7
n
where bl, is the Kronecker symbol, the eigenvalues of problems (2.1), (2.2) form increasing sequences and each eigenvalue is repeated as many times as its multiplicity. We impose the following restrictions on the scalar functions and
PE(x)
PO(X):
O < c o < p o ( ~ ) < ~o < ~C ; zIpC(x)Ic3; p,
+ po
weakly in L Z ( R ) as
where c2,cg are constants independent of
E
-+ 0
,
E.
Theorem 2.1.
If L,
k
as
E
+ 0, and conditions (2.3) are satisfied, then
Moreover, suppose that the eigenvalue
Xo = Xit1 has multiplicity m , i.e.
111. Spectral problems
u ( x ) is the corresponding eigenfunction o f problem (2.2), Ilull.r,2(n) = 1. Then there is a sequence o f functions {u,) such that ii, -+ u in L 2 ( R )as E + and
0, ii, is a linear combination of eigenfunctions of problem (2.1) corresponding
..., A:+".
t o the eigenvalues A;+',
To prove this theorem we shall reduce it t o Theorems 1.4 and 1.7 for abstract operators, making a suitable choice o f spaces 'FI,, ?lo, V and operators
dc, do. Denote by 'If, the Hilbert space consisting of all vector valued functions with components in
By
u
L2(R).The scalar product in 3-1, is defined by the formula
7Io = V we denote the space o f vector-valued functions L 2 ( R ) ,where the scalar product is given by
with compo-
nents in
Lemma 2.2. Let the conditions (2.3) be satisfied. Then
provided that v c + v O , us -t u0 in the norm o f
Proof. It follows from (2.3)
for any cp continuous in bounded uniformly in
E
G.
L 2 ( R )as E
4
0.
that
Taking into consideration the fact that p, are
and that functions continuous in
fi form
a dense set
L 1 ( R ) ,we easily verify that (2.5) holds for any cp E L 1 ( R ) . Let v E -t v O , uc + u0 in L 2 ( R )as E -+ 0. Then in
$2. Homogenization o f eigenvalues and eigenfunctions
0, we obtain (2.4). Indeed, the last two integrals in the right-hand side of above equality converge t o zero as E --+ 0, since u E ,v" converge in the norm o f L2(R)and p, is bounded uniformly in Passing here t o the limit as
E.
E
4
The difference of the first two integrals in the right-hand side converges t o
zero due to (2.5) for cp =
(uO,vO). Lemma 2.2 is proved.
It follows from Lemma 2.2 that Condition C1 is satisfied if we take as RE R,u = U . In this case y = 1. Let us define operators A,, setting A E f E= u E ,where uEis a solution of
the identical operator the problem
It follows from Theorem 3.3, Ch. I, that the norms llA,ll are bounded by a E . Operators A, are compact, owing t o the compact H1(R)c L2(R)and Theorem 3.1, Ch. I. The integral identity for = Akhyield a solution o f problem (2.6) and the equality (Ahk)*
constant independent o f imbedding
L2(R),where we = A,gE. Therefore A, is a positive self-adjoint operator in 3-1,. We take as & : 'Flo 4 No the operator which maps f 0 E 'Ho into the solution uOo f the
for any f',gC E
problem
278 i.e.
111. Spectral problems
JZo f0
= uO. By the same argument that was used for the operators
A, we
?to
Thus
can show that
is a positive compact and self-adjoint operator in
Condition C2 o f $1is satisfied. Consider now the Condition C3. Let f0 E
V and define w c as a solution
of the problem
Since u0 = &fO
is a solution o f problem (2.7), the G-convergence of
LE t o
E as E -+ 0 implies that w E- uO -+ O strongly in L 2 ( R ) . The function u E = A ,f0 is a solution of problem (2.6) with
(2.8) = fO. Therefore
v E = uc- w C is a solution of the problem
It follows from the integral identity for vc that
The norms o f vc in
H 1 ( R ) are bounded by a constant independent o f c , since
v E = uE- w E .Therefore vc' -t v0 weakly in H 1 ( 0 )and strongly in L 2 ( R )for a subsequence E'
-+
0; v0 E H 1 ( R ) .
It follows from Lemma 2.2 that the integral in the right-hand side of (2.10) converges t o zero as E' + 0, and therefore vc' -+ 0 strongly in H 1 ( R ) . Since each sequence {vc) contains such a subsequence vE' -+ 0, it follows that 'v + 0 in H 1 ( R )as& -+ 0. Thereforedue t o (2.8) we have IJuO-uEllWc-+ 0. This means that Condition C3 is satisfied. Condition C4 is also valid owing t o the compact imbedding
L 2 ( R ) and the inequality Ildcf ' l H ~ ( n )
5
H1(R) c
11
c f'll~*(~), where c is a constant
independent o f c . The last equality follows from Theorem 3.3, Ch. I.
It is easy t o see that in the case under consideration the eigenvalues of problems (2.1), (1.11) and (2.2), (1.12) are related by the formulas
§ 2 . Homogenization of eigenvalues and eigenfunctions
279
Thus we have shown that all conditions of Theorems 1.4, 1.7 are satisfied, and therefore Theorem 2.1 follows directly from (2.11) and Theorems 1.4, 1.7, since I(A,u -
+ 0 as E -+
0 due t o the Condition C3.
Note that Theorem 2.1 implies in particular the convergence o f the eigenvalues and eigenfunctions o f the elasticity operators with almost periodic coefficients, considered in $6, Ch. II. In the case o f periodic coefficients it is possible t o give estimates for the difference between eigenvalues and eigenfunctions of problem (2.1) and those of problem (2.2). Such estimates in a more general situation o f perforated domains are obtained in Section 2.3.
2.2. The Neumann Problem for Elasticity Operators with Rapidly Oscillating Periodic Coeficients in a Perforated Domain In this section we study spectral properties of operators associated with problems (2.22), (2.23), Ch. II. Here RE is a perforated domain of type 11,
C, is an elasticity operator with rapidly oscillating periodic coefficients, L, is given by (1.1), Ch. II, E is the corresponding homogenized operator whose coefficients are defined by the formulas (1.3). Ch. II. In order t o simplify the derivation o f estimates for the difference of eigenvalues o f problems (2.22), (2.23), Ch. II, i t is convenient t o deal with suitably "shifted" operators. To this end we consider the following eigenvalue problems for operators of type (2.60), (2.61), Ch. II:
III. Spectral problems
where
61, is the Kronecker symbol, the eigenvalues form increasing sequences
and each eigenvalue is counted as many times as its multiplicity.
It was shown in Section 2.2, Ch. II, that the operators
C,, 2
are "close"
to each other in the sense that the solutions of problems (2.22), (2.23), Ch.
II, satisfy the inequalities (2.26), Ch. II. In contrast t o Section 2.1, here we impose some additional restrictions on the scalar functions p,, po, namely
where c2, cg are constants independent o f (2.64), Ch. II, and
( I l p o - p,(l(
E,
the norm
111 . 1 1 1
is defined in
characterizes the closeness of the functions po
and p,. Applying here the method suggested in Section 2.1 for G-convergent operators, in order t o compare the spectral properties o f problems (2.12) and (2.13) we reduce these problems t o the form which allows us t o use Theorems 1.4, 1.7 for abstract operators in Hilbert spaces depending on a parameter. The main result of the present section is Theorem 2.3. Let conditions (2.14) be satisfied. Then for the k - t h eigenvalues o f problems (2.12), (2.13) the estimate
52. Homogenization of eigenvdues and eigenfunctions holds with a constant ck independent of
281
E.
Moreover, if the multiplicity of the eigenvalue Xo = Xf;tl is equal t o m , i.e.
and u o ( x )is the corresponding eigenfunction of problem (2.13), then there is a sequence
I I u ~ ~ ~ =~ 1,z ~ ~ )
{u,) such that
where Mi is a constant independent of
E
and uo, uE is a linear combination of
eigenfunctions of problem (2.12) corresponding t o A;+',
...,
Before giving a proof t o this theorem we establish some auxiliary results. Let us introduce in L 2 ( R c )the following scalar product
and denote the obtained space by 3-1,.
The space L 2 ( R ) equipped with the
scalar product
( u OvO)n, , =
/ po(x)(u0,vO)dx
(2.18)
R
is denoted by 3-10. Set V =
3-10
and take as RE in Condition C 1 of $1the
restriction operator
L ~ ( R3) f
-+
il,. E ~ ~ ( .0 ' )
In order t o show that 7 f o ,3-1,, V,
(2.19)
R, satisfy Condition C 1 we shall need
Lemma 2.4. Let Re be a perforated domain of Type II, and let conditions (2.14) be satisfied. Then for any uO,v0 E L 2 ( R )we have
III. Spectral problems
Proof. Set f ( ( , x ) =
xw(E),II) = gC= pouO,cp = cpc
= v0 in Corollary 1.7, Ch. I, where xu(<)is the characteristic function of the domain w. Then by
virtue o f (1.21), (1.22), Ch. I, we have
as
E
+ 0. Formulas (4.2), (4.3), Ch. I, show that
Therefore since the measure of the set R\Rl (2.21) implies that
0'
= (I(l\Rl) U Rl
is of order
E,
n EW.
the convergence
as c -t 0. Taking into account estimate (2.65), Ch. II, and the fact that p,, PO are bounded and lllpc - polll 0 , we get for any u , v E H 1 ( R ) -)
Obviously (2.23) is also valid for u = uO,v = v0 since we can approximate u O , H1(a). The convergence (2.20) follows from (2.22) and
v0 by functions in
(2.23) with u = u O ,v = vO. Lemma 2.4 is proved. Relations (2.19) and (2.20) show that for the above defined spaces 'He,' H o p V and operators Rcf = f Condition C1 is satisfied with 7 = mes Q fl w . Let us introduce operators A, : 'Hc + IHc setting A, f' = u', where u' is a solution of the problem
In,
The existence of an upper bound for the norms 1]A,)1independent of
E
follows from Theorem 5.4, Ch. I, and the compactness of A, follows from the compactness o f the imbedding H1(R") C L2(R'). Let us show that A, is a positive self-adjoint operator in ' H e . Indeed, using the integral identity for solutions o f problem (2.24) and setting w c = A,gc, we find
$2. Homogenization o f eigenvalues and eigenfunctions
=
J ( w Cpcf.)dx , = (.%9', fe)n. nc
for any p , g c E 'He, since ( A h k ) *= Akh. It follows that operators A, : 'HE-+
'MEare positive and self-adjoint. Denote by
the operator mapping f0 E 'Ho into the solution u0 of the
problem
Obviously
is a positive compact self-adjoint operator in 'Ho.
Thus we have checked Condition C2 of
$1. Consider
now Condition C3.
Let us show that for any f0 E 'Ho we have
where .uE = A, f O , u0 = & f O . Let
Since the norms
f" E H 1 ( R ) . Then
IIAcll are bounded
where c is a constant independent of
A,
and
E.
uniformly in
E,
it follows that
According t o our choice o f operators
& the functions we = d C f ,w0 = dof are solutions of the problems
111. Spectral problems
284 Estimate (2.67) of Theorem 2.13, Ch. II, implies that
By the definition of the norm
I/
Therefore from (2.28)-(2.31)
we conclude that
Choosing
J
.
1IH1.
in Section 2.2, Ch. II, we have
t o be such as t o make the first term in the right-hand side
of this inequality less than 613, and choosing €6 such that for e second term be less than 613, we obtain the inequality (luE- u for
E
5 ~ 6 c,= const.
5
56
the
5 CS
O ~ ~ ~ Z ( ~ . )
Hence the convergence (2.26), which is equivalent t o
Therefore Condition C3 is also valid. Let us establish Condition C4. Consider the extension operator rem 4.2, Ch. I, and suppose that of
E.
11 f'll.~.
PCof Theo-
5 C , where c is a constant independent
Then due t o Theorem 5.4, Ch. I, we get
Therefore
IIPCdcFllHlcn)
5 c3 and
the constant c3 does not depend on
E.
H 1 ( R ) in L 2 ( R ) there is a vector valued function w0 E H 1 ( R ) such that ((P,,&,f"- w O [ ~ ~ Z ( -+~ ) 0 for a By virtue of the compact imbedding of
$2. Homogenization o f eigenvalues a n d eigenfunctions
IIA,tf"'
subsequence E' --+ 0. This means that
285
- R,,W~~~,~(~,I) + 0, and
therefore Condition C4 is satisfied. Let us now consider the eigenvalue problem (1.11) and (1.12) for the operators A,,
do defined
above. It is easy t o see that
According t o the proof o f Lemma 1.6 we have
where the constants we get for any f E
Q,
c(k) do not depend on e. Taking into account (2.14)
H1(R)
where uc,u0 are solutions o f the problems (2.24), (2.25) with
fc
= f, f 0 = f
respectively. Using the estimate (2.67) of Theorem 2.3, Ch. II, we obtain
where c2 is a constant independent of e. To derive (2.34) we also used the estimate (2.31) for
f
= f. Thus Theorem 2.3 follows from Theorems 1.4, 1.7
and estimates (2.33), (2.34). Corollary 2.5.
x
Suppose that p.(x) = p ( ; , x), p(t7 x) is l-periodic in the Lipschitz condition in x E
C
E Finand satisfies
uniformly in [. Then according to Lemma
2.12, Ch. II, we have lllp, - polll 5
E.
where c i is a constant independent of
In this case estimate (2.15) implies
E.
III. Spectral problems
286 Remark 2.6.
Estimate (1.13) allows us t o obtain a more accurate expression for the constant
ck in (2.15). Indeed, according t o (1.13) the constant
ck
in (2.15) can be
replaced by
where
p:
4
0 as
E
-+ 0 and c is a constant independent of
k, E .
Note also
that in the proof of Lemma 1.6 it was established in particular that A,k where
-yk
is a constant independent o f
5 yk,
E.
2.3. The Mixed Boundary Value Problem for the System of Elasticity in a Perforated Domain Here we consider free vibrations o f elastic bodies with a periodic structure. The boundary o f the body is free of external forces at the surface o f the cavities and fixed at the outer part. The corresponding boundary value problem of elasticity was studied in $1,Ch. II. It should be noted that in the case under consideration
Re
is a perforated domain of type I, the elasticity operators
LC
are the same as in (1.1), Ch. II, and have rapidly oscillating coefficients, l? is the corresponding homogenized operator whose coefficients are given by the formulas (1.3), Ch. II. For these operators we consider the following eigenvalue problems
$2. Homogenization o f eigenvalues and eigenfunctions
where 6,, is the Kronecker symbol, the eigenvalues form increasing sequences and each one is counted as many times as its multiplicity. In Section 2.1 we already considered a particular case o f problems (2.37), (2.38) when
Rc = R, i.e.
the domain
RE is
not a perforated one. Under the
assumptions (2.3) on p,, po we proved the convergence of the eigenvalues
If RE is a perforated domain and the elasticity coefficients are €-periodic, it
of problem (2.37) t o the corresponding eigenvalues of problem (2.38).
is possible t o obtain more accurate results compared with those of Section 2.1. In this case the key role is played by the closeness o f operators
2,
L, and
which is expressed in terms o f estimate (1.15), Ch. II, for solutions of
the corresponding boundary value problems. In similarity with the case of the Neuman problem considered in Section 2.2, t o estimate the difference of the respective eigenvalues o f problems (2.37), (2.38) we must have some
pE(x) and p,(x). It will be shown x in particular that if p,(x) = p(-, x) and p([, z) E Lm(IRnx 0 ) is 1-periodic E in E and satisfies the Lipschitz condition in x E 0 uniformly in ( then the knowledge o f the closeness of the functions
eigenvalues of problem (2.37) converge t o those of problem (2.38) with
PO(X) = (mes Q n w)-'
/
P(E, x ) 4
1
(P(.,x)) .
Qnw
In this section the closeness of pc and po is characterized by the norm
Illplllo, which is defined by (2.64), Ch. II, where the supremum is taken over all
U ,v
E
H 1 ( R c ,re). H 1 ( R c ,r C ) we obviously
For all u,v E
have
III. Spectral problems
288
u.
x Suppose that p(E, x ) E L(R", f l ) (see (1.13), Ch. I),pc(x) = p(- ,x ) , pO(x)= E
( P ( . , 2 ) ) . Then
where c is a constant independent of a. The proof of this lemma is based on an estimate of type (2.63), Ch. II, for vector valued functions u , v E H1(R',I',). This estimate can be obtained in much the same way (with obvious simplifications) as the estimate (2.63), Ch. II, for u , v E H 1 ( f l e ) . We assume here that the functions p,, po in (2.37), (2.38) satisfy the following conditions
where the constants c2, cg do not depend on E . The closeness o f the eigenvalues and eigenfunctions o f problems (2.37), (2.38) is established by Theorem 2.8. Suppose that conditions (2.41) are satisfied. Then for the k-th eigenvalues A,: A; of problems (2.37), (2.38) the estimate
holds with a constant ck independent of E . Suppose that X o = A"; is an eigenvalue of problem (2.37) of multiplicity m, i.e.
xt, < xt;tl = ... = xt,+rn
< xt,+rn+'
(A: = 0)
,
and u o ( x ) is the corresponding eigenfunction such that 11uollL2(n)= 1. Then there is a sequence ii, such that
$2. Homogenization of eigenvalues and eigenfunctions
where Mk is a constant independent o f
E , UO;
289
21, is a linear combination of
eigenfunctionsof problem (2.37) corresponding t o the eigenvalues A;+',
...,A:+".
The proof of this theorem can be easily reduced t o the verification of Conditions C1-C4 and application of Theorems 1.4, 1.7, in the same way as in the proof o f Theorem 2.3. In the case under consideration we take as 'He ('HO) the space L2(R') ( L Z ( R )with ) the scalar product (2.17) ((2.18)) respectively, and set
V = 'Ho. The operator Re is defined as the restriction t o Rc of vector L2(R). Operators A, : 'Hz -t 'He, & : No -t No are
valued functions in defined as follows
where
u', u0 are solutions of the boundary value problems
The verification o f Condition C 1 is based on Lemma 2.9. Suppose that R q s a perforated domain of type I and that the conditions (2.41) are satisfied. Then for any
uO,v0 E L 2 ( R )the convergence (2.20) takes place.
This lemma is proved similarly t o Lemma 2.4; the convergence (2.22) follows from (2.21) since
Rc = R n E W .
Conditions C1-C4 are checked similarly t o Section 2.2 with the following modifications: problems (2.24), (2.25) should be replaced by (2.45), (2.46), and instead o f Theorem 2.5, Ch. II, one should use Theorem 1.2, Ch. II; in the proof o f C4 one should consider the extension 4.2, Ch. I, and use the compact imbedding
Pcu constructed in Theorem
III. Spectral problems
2
As in Corollary 2.5, if p, = p(- ,x)and po = ( p ( . , x)), p ( J , x) E L ( R ~x E R),then estimate (2.35) is also valid. The inequality (1.13) allows us t o obtain a more accurate expression for the constant ck in (2.42). Thus we can take as ck the constant defined by (2.36), where X,k, XfS are the k-th eigenvalues of problems (2.37), (2.38) respectively.
2.4. Free Vibrations of Strongly Non-Homogeneous Stratijed Bodies Consider the problems (2.1) and (2.2), where L,,k are elasticity operators, studied in 57, Ch. II. It was shown in $7, Ch. II, that L, % 2 as E + 0, provided that conditions (7.32), Ch. II, are satisfied. Therefore under the conditions (7.32), the genreal Theorem 2.1 is valid. In order to obtain estimates for the closeness of eigenvalues of problems (2.1), (2.2) for stratified bodies, we shall assume that the coefficients of the G-limit operator k are smooth in 0 and that in addition to the conditions (2.3) on p,, po we have
where lllplllo is defined by (2.64), Ch. II, with
Re= R, u , v
E
Ht(R).
Theorem 2.10. Let LC,k be operators of the form (7.1), (7.2), Ch. II, and let C,, k satisfy the conditions (7.32), Ch. II. Suppose also that the coefficients of 2 are smooth functions in 0. Then the eigenvalues of problems (2.1), (2.2) satisfy the inequality
where c k is a constant independent of E , 6, is defined by (7.6), Ch. II. Moreover, if Xo = A;+' is an eigenvalue of problem (2.2) of multiplicity m, 1.e.
52. Homogenization of eigenvalues and eigenfunctions
291
and u o ( x ) is the corresponding eigenfunction, IluollL~(n) = 1, then there is a sequence
{ u , ) such that
where c is a constant independent of
E,
uo, and u, is a linear combina-
tion o f eigenfunctions o f problem (2.1), corresponding t o the eigenvalues
XL+l , ..., This theorem is proved by the same argument as Theorems 2.3, 2.8. However instead o f the inequalities (2.67), (1.15), Ch. II, one should use the following estimate
c*
[ I I-~ f llH-'(n) ' + a:'2
1 1 f llL2(n)]
which holds for solutions o f the problems
where
LC(ue= )
in Rc
,
uEE HJ (R)
k(uO)=f
in R
,
u0€H;(R),
f',f
,
E LZ(R). Let us prove the inequality (2.50).
Denote by iic a solution of the problem
Then estimate (7.7), Ch. II, with i9
= 0 holds for u c - t i c . The function uE-iic
is a solution of the problem
Therefore according t o the inequality (3.25), Ch. I, we have
Hence the inequality (2.50) is valid. Let us consider some examples of functions p c , po satisfying conditions (2.47).
111. Spectral problems
292
Let p,(x) = p ( W , x ) where cp(x) isdefined in Section 7.1, Ch. II, p(f, y ) E
belongs t o the class A, a E (O,1), po(x) = ( p ( . , x ) ) (see Section 7.2, Ch. 11). Our aim is t o show that
~ I ~ P ~ - P O I ~ ~ O - < C E ~ ,c=const.
(2.51)
0bviousl y p ( M ,2)- p o ( x ) = E
- - $0;
where g:,
fi
a
(Pi
-9 3 x 1 - -f A x ) ,
lVcp12 a x ; IVcpI2 denote the respective integrals in the right-hand side of the first
equality. Since (p(., y ) - po(y)) = 0, Lemma 7.2, Ch. II yields
where the constant c is independent of
1 [ a ( 1~igf
=
R
d f
E.
For any u , v E H ; ( f l ) we have
22.-f:] u v d x .
- Vcp2
Due t o the inequality (2.52) and the fact that cp E C 2 ( n ) , ( V y I > c = const > 0, it follows that the right-hand side of (2.53) can be estimated by C ~ E ,~ ~ u ~ ~ H ~ 1) (v0)) ) ~with I ( ~ c1 ) = const independent of E . Thus (2.51) is established. Therefore, if the coefficients of the operators L,,?, satisfy the conditions of Theorem 7.13. Ch. Il, and p.(x) = p(- cp(x) , x ) , p ( t , y ) E d o , then the E estimates (2.48), (2.49) can be written in the form
52. Homogenization of eigenvalues and eigenfunctions
c:,
M
= const
2 0.
In analogy with Remark 2.6, we can get a more precise expression for the constants ck in (2.48) by using (1.13) and (2.32). In the case o f stratified structures we can also replace ck in (2.48) by c', given by (2.36).
111. Spectral problems
294
$3. On the Behaviour of Eigenvalues and Eigenfunctions of the Dirichlet Problem for Second Order Elliptic Equations in Perforated Domains
3.1. Setting of the Problem. Fornal Constructions Here we consider free vibrations of a perforated membrane fixed at the points o f its boundary. In $4, Ch. II, we constructed complete asymptotic expansions for solutions of the Dirichlet problem for the elasticity system in a perforated domain. Using the same method we can construct asymptotic expansions for solutions of the Dirichlet problem for a second order elliptic equation. In the latter case the maximum principle and the well-known properties of the first eigenfunction make it possible t o study the spectral properties of the corresponding operators. Consider a family of second order elliptic operators
= a
x au x ~ ~ ( u) (aij(-) - b(;)u ax; E ax, where
E
-)
,
E ( 0 , l ) ; a i j ( t ) ,b ( t ) are smooth functions o f
5
E
Rn,1-periodic
in
[, and such that
It is assumed that Re is a perforated domain o f type I (see $4, Ch. I),
Re = R n EW and the domains R and w have smooth boundaries. In this section we study the asymptotic behaviour (as values o f the following problem
E + 0)
of the eigen-
295
$3. On the behaviour of eigenvalues and eigenfunctions where P(() is a smooth function of
const
> 0; each eigenvalue
< E Rn,l-periodic in [, p(()
>_
GI
=
is counted as many times as its multiplicity.
The question o f the behaviour o f A,k as
E
-+ 0 was considered before in
+
[130], [131], [54]. It is proved in these papers that A,k = E - ~ A ~At, where
A. > 0 is a constant independent o f k, E , and AS + At as E + 0, At is an eigenvalue o f the Dirichlet problem in R for a second order elliptic operator with constant coefficients. Here we not only prove the convergence o f At t o At, but also obtain the estimate IA,k - A;[ 5 C ~ Eck, = const, and study the behaviour o f the eigenfunctions o f problem (3.3) as E -+ 0. Let a ( [ ) be the eigenfunction corresponding t o the first eigenvalue A. of the following boundary value problem in the unbounded domain w with a l-periodic structure:
= 0 o n aw
,
a ( [ ) is l-periodic in [ ,
I
The boundary condition in (3.4) is understood in the sense that
(3.4)
a([)
$ ( w ) (see $1,Ch.
I). I t is well known (see [ l l ] , [64]) that a ( [ ) is a smooth function in w such # 0 in a neighbourhood of dw. that a ( [ ) # 0 in w and IVF@I belongs to the space
Let us formally represent the k-th eigenfunction o f problem (3.3) in the form
It is easy t o verify that v,k(x) must satisfy the relations
and
296
111. Spectral problems Thus we obtain an eigenvalue problem for a second order elliptic equation
degenerate on the subset
S, o f the boundary o f Cl".
Making a suitable choice of functional spaces for solutions o f the corresponding degenerate boundary value problem, we shall reduce (3.6) t o an eigenvalue problem for a positive compact self-adjoint operator in a Hilbert space, and show that problem (3.6) has a discrete spectrum consisting of eigenvalues
where each X i is repeated as many times as its multiplicity. If v,k is an eigenfunction o f problem (3.6) corresponding to X,k, then
@(:)v3 belongs t o HA(Cle)
and is an eigenfunction o f problem (3.3) corresponding t o the eigenvalue
Applying the homogenization methods, developed in Chapter II and in $51,
2 , Ch. Ill, t o the degenerate operators
we obtain the estimates
where Xk is the k-th eigenvalue of the Dirichlet problem for a second order elliptic equation with constant coefFicients. These coefficients are expressed through the coefficients of operators (3.7) by means of the homogenization procedure described in Chapter II.
3.2. Weighted Sobolev Spaces. Weak Solutions of a Second Order Equation with a Non-Negative Characteristic Form For our further consideration we shall need the following spaces o f periodic functions:
§3. On the behaviour of eigenvalues a n d eigenfunctions P1(w) is the completion o f
e r ( w ) in the norm
PO(w) is the completion o f
&?(w)
+(w) is the completion of
Q={(
in the norm
e?(w)
: O
in the norm
...,n ) .
It is easy t o see that if IIullPcw) = 0 and u is smooth, then u = 0 in w. Indeed, suppose the contrary, i.e. that u 0 at a point so. Then we can assume that u > a. > 0 in a neighbourhood wo o f xo, and therefore
+
/
Iv@12d( = 0. Multiplying the equation (3.4) by $(()@(<). where $([) E
wo
C,OO(wo), $([) 2 0 in w and integrating over Q n w , we find that @ which is impossible, since @ > 0 in Q n w. Let us also consider the spaces
$ (w), W;(W),
= 0 in wo,
introduced in $1,Ch. I.
Lemma 3.1. The following imbeddings
(w) C
Q1(w)
w; (w) C P1(w) ,
c Q(w)
(3.10)
are continuous. Moreover, the imbedding (3.9) is compact, and for any v E
P1(w) we have @(()v(()
Proof. Let u E W;(W).
E$'
(w).
Consider a function
if ~ ( taw) , > 26, cps(() = 0, if p ( t , d w ) c = const. Then
cp6
E & F ( w ) such that cp6(() = 1,
< 6, 0 I cps
I 1, IVrcp61 < c6-l,
111. Spectral problems
This implies (3.8), since
1@1 5 c16 in the 26-neighbourhood o f dw, and there-
fore the right-hand side o f the above inequality tends t o zero as 6 --+ 0. Let us show now that for any u E
6 ' r ( w ) the following
inequalities are
satisfied
Multiplying the equation (3.4) by cPu2,u E e ? ( w ) , we get
Therefore 112
Qnw
Hence (3.11) is valid. The inequality (3.12) follows from (3.11), since
53. On the behaviour o f eigenvalues and eigenfunctions
299
It is easy t o see that for any u E Q 1 ( w )the inequalities (3.11), (3.12) are also satisfied, and moreover e u E$ ( w ) . The continuity of the imbedding (3.9) is obvious. Let us prove its compactness. Consider a sequence {urn}o f elements o f Q 1 ( w )such that sup IlumllP1(,, m
<
<
It follows from (3.12) that Il@um(lH~(Qnw) < cl, where cl is a constant independent of m. Due t o the compactness o f the imbedding H 1 ( Q n w ) c L 2 ( Q n w ) there is a subsequence m' -+ co such that @urn'--+ w E H 1 ( Q n W ) in the norm o f L 2 ( Q n w ) . It follows that @urn'is a Cauchy sequence in L 2 ( Q n w ) and therefore urn' is a Cauchy sequence in c O ( w ) .Hence urn' + uO E Q O ( w ) .Lemma 3.1 is proved. c
oo.
Lemma 3.2 (the Poincark inequality). For any u E V 1 ( w )such that
/
Q2ud[ = 0 ,
(3.13)
Qh
the inequality
5c
IIUIIZ~~,,
J
l@12
(3.14)
lvcui2dt
Qnw holds with a constant c independent o f u .
Proof. Suppose the contrary. Then there is a sequence u N E c 1 ( w )such that
/ I@12 Qnw
l v e u N l 2 d t5
, 1
,
N
llu
IIQI(~)
=1.
(3.15)
Due t o the compactness of the imbedding of v l ( w ) in Q O ( w )we can assume that the sequence { u N ) is such that
/ Qnw
Ie12IuN - uNtt12dt o -+
as N
-+ m
.
III. Spectral problems
300
-+ 0 and therefore there is a function u E P1(w) such
Thus lluN that lluN -ullol(,)
0 as N
oo. Taking into account (3.15) we conclude that lVEul = 0 almost everywhere in Q n w. Since iP vanishes only on Bw, it follows that u = const in Q n w and IIullplcW,= 1. This contradicts (3.13). Lemma 3.2 is proved. -t
-t
Let Rc be a perforated domain o f type I considered in $4, Ch. I. We introduce the spaces V,'(R6), VO(Rc), V(Rc) as completions of C,""(Re) in the respective norms:
Lemma 3.3. For any u E C F ( R e ) the following inequalities are satisfied
1
IV~@(:)(~ 1ul2dx 5 Q
/
+I
V ~ U ,I ~ ) ~ ~
If u E V,'(Rc), then (9(f)u E HA(Rc). The imbedding V,'(Rc) compact and H1(Re,I',)
Proof.
(3.19)
nr
ne
c VO(R')
is
c V,'(Rc).
The function a(;) satisfies the equation
x
x
E
E
and the boundary condition a ( - ) = 0 on d ~ w Multiplying . (3.21) by a ( - ) u 2 where u E C F ( R e ) , and integrating over Re, we find
30 1
'$3. On the behaviour of eigenvalues and eigenfunctions
J
=-
a@ a@ 1 ~ 1 ~ -d x2 " ax; axj J
a . .-
n
Re
d@ du a;j - @u -dx t ax, a x j
Therefore
It follows that
Since
Iv,@(-)~~
x
= E-'
E
x I v ~ @ ( - ) ~this ~ , inequality implies (3.19). Inequality E
(3.20) follows from (3.22), since
For a fixed
E,
of * ( R E ) as j -+
by virtue o f (3.20) the convergence u3
4 u in the norm x oo,uj E C?(Rc), implies that { @ ( - ) u i ) is a Cauchy
E
5
sequence in Hi(Ctc) and consequently @(-)uj -+ w in E
Hi(R')
as j + m.
The convergence o f uj t o u in V,'(RC) implies the convergence of @uj t o @u
x
in L2(R'). Therefore u @ = w . This means that @ ( - ) u E H,'(R') for any E
E v,(ns). The compactness o f the imbedding V,'(Re)
c V O ( R ' )for
a fixed E can
be proved similarly t o the compactness o f the imbedding Q1(w) c p O ( w )in Lemma 3.1. The imbedding H 1 ( O C , r , )c way as the imbedding W;(U)
V d ( N )is established in the same
c p l ( w ) . Lemma 3.3
Lemma 3.4. Let the sequence u" E c ( R E )be such that
is proved.
III. Spectral problems
302
Then there is a subsequence E' + 0 and a function u0 E H i ( R ) such that
l(uO- ~ " ' l l ~ + ~ (0 ~as~E' ' + ) 0.
Proof. Note that the domain Re has the form R n EW,where w
is a smooth
unbounded domain with a 1-periodic structure, w satisfies the Conditions 81-83 o f 54, Ch. I. Due t o the Conditions B1-B3 for any 6 sufficiently small) there is a smooth unbounded domain w6
c
< b0 (60
is
w with a 1-
periodic structure, which also satisfies the Conditions B1-B3 and such that
0 < c16 < p(x,dw) < c26 for x E dws, cl, c2 are constants independent of 6. Set R; = R n E W ~rt, = d R n d o ; , Sf = 80; n 0. It is easy t o see that Ra c R' is also a perforated domain o f type 1, r,6 c I',. Since a(<)> 0 in w, it follows from (3.23) that
where cs is a constant independent of E ; 6 E (O,bo),u' E H1(R;,I't). For a fixed 6 E (0,60) using Theorem 4.3, Ch. I, let us construct extensions
P:uc E ~ : ( f i )of the functions u E t o a domain fi containing 0. According sup I I P : U ~ I I ~ ' 5 ( ~cs )1
t o Theorem 4.3, Ch. I, it follows from (3.24) that
€
<
m. Using the compactness o f the imbedding
~ i ( fci )~ ~
( fleti )us choose
a subsequence E' + 0 such that P:U"
-+ u & ( x ) weakly in ~ , ' ( f i ) a n d strongly in ~ ~ ( f . i ) (3.25)
By Theorem 4.3, Ch. I, we have u6 E HA(R). Thus
In complete analogy with the above considerations, for any can extract a subsequence E" UJ, E
+
HA(R) such that
Let us show that us, = us. Indeed,
0 of the sequence
E
( 0 , 6 ) we
E' and find a function
$3. On the behaviour of eigenvalues and eigenfunctions
The right-hand side of this inequality tends t o zero as E"
t
0. Setting f
= 1,
$f = $ = (P' = (P = us, -us in Corollary 1.7, Ch. I, we see that the left-hand '/~ Therefore us, -us = 0. side of (3.26) tends t o (mes ~ n w ~ ) llu6, Thus we have shown that there is a function uo = us E HA(fl) such that for any 6 E (O,bO)
It follows from (3.19) that sup
J
lu'12 I v { Q ( ~)12dx 5 c2 < m ,
n* since the norms IIuCllvdcne) are bounded by a constant independent o f E . Therefore, due t o the fact that IV{iP([)I
sup
J
# 0 on dw, we have
luc12dr
nc\n; where c3 is a constant independent o f
E;
6 E (0,6,), i f 60 is sufficiently small.
It is easy t o see that
>0 < 012 for all E'. Taking into account (3.27), let us If' < 0 1 2 for all E' < EO. Hence the convergence t o zero
Due t o (3.28) and the boundary condition a ( ( ) = 0 on dw, for each o there is a 6 such that 1;' choose EO such that
of the left-hand side o f (3.29) as E'
-+
0. Lemma 3.4 is proved.
304
III. Spectral problems
As a consequence from Lemma 3.4 we get a Friedrichs type inequality for functions in
Vd(RE).
Lemma 3.5. For any
u E V,'(Rc) the following inequality of Friedrichs type
holds with a constant c independent o f
Proof.Suppose the contrary.
/ I@(;)I2 IV.
U
I
E.
Then there is a sequence
E -+
0 such that
dx
(3.31)
n
where a, E'
0 as
-+
E
+ 0. According t o Lemma 3.4 there is a subsequence
--+ 0 and a function uo E H i ( R ) such that
Similarly t o the proof o f Lemma 3.4 consider the subdomain a fixed
fi
REg o f RE for
6 E (O,hO), and let P!uc be the extension o f ue from REg t o a domain x 0. Since @(-) 2 cg = const > 0 for x E @ and cs does not
containing
depend on
E
E,
it follows from Theorem 4.3, Ch. I, and inequalities (3.31) that
where c, cl are constants independent of
E.
-
On the basis o f (3.25) we can
uo(2) weakly in ~ , ' ( f i )as E' -+ 0. Therefore, because 0. It of (3.34) and (3.31) we have uo = const E H,'(R), which implies uo assume that
P ~ U " -+
follows from (3.33) that (3.31), (3.32) we have
I I u ~ ~ ~ ~ ~ 0~ as( ~E' ~ I )0. On the other hand, from I I u ~ ~ ~ ~ ~ ~ -+( ~ 1. Ias, E' 0. This contradiction shows -+
-+
-+
that inequality (3.30) is indeed valid. Lemma 3.5 is proved.
305
$3. On the behaviour of eigenvalues and eigenfunctions Let us also introduce the space
V1(R") as the completion o f C"(ae) in
the norm (3.16). Consider the following boundary value problem for a second order equation with a non-negative characteristic form, which is elliptic inside degenerate on Sc c
Re and is
do':
f j E VO(Re),j = 0, ...,n ; 11, E V1(R"), operator M, is given by (3.7). A weak solution of problem (3.35) is defined as a function u E V1(R')
where
u - 11, E V,'(Rc) and the following integral identity holds for any w E G(RE):
such that
Theorem 3.6. There is a unique weak solution u
E V1(R') of problem (3.35). This solution
satisfies the inequality
IIullvlcn*, 5 c
[2 i=O
I
llfillv~co,+ Ildllvlcn*,
where c is a constant independent o f
E,
f',11,.
9
(3.36)
The proof of this theorem is similar t o that of Theorem 3.8, Ch. I, and is based on Theorem 1.3, Ch. I with H = V1(R") and on the Friedrichs inequality (3.30). In what follows we shall need a maximum principle for weak solutions of problem (3.35). Lemma 3.7 (The Maximum Principle). Let
u(x) be a weak solution of the problem
III. Spectral problems
V 1 ( R c )n C O ( @ ) , everywhere in Re.
where II, E
5 M = const.
Then
lu(x)I 5 M almost
Proof. Consider the domains fl; = R ~ E wconstructed & in the proof o f Lemma 3.4. Denote by
Since
v6 a solution o f the problem
a ( ( ) vanishes only a t the points o f dw, this equation is elliptic in
a',, and therefore according t o the maximum principle It follows from the integral identity for solutions o f (3.38) that
= -
av + b ~ ' $ v ]dx , / [m2aijaa$x , axj -
Q; for any
v E H:(R:), where w6 = v6 - II,E HG ( 0 : ) . v = w6 and extend it as zero t o Rc\R:. Then from (3.40) and
Let us take
the Friedrichs inequality (3.30) we find that
w6 = 0 in RE\%, it follow from (3.41) for 6 -r 0 that the sequence {wh) satisfies the condition sup I I ~ ~ l l <~ m. ~ ~ Due ~ e t )o the compactness o f the imbedding V:(Rc) C
where c is a constant independent of 6.
Since
6
V O ( R e )and the weak compactness of a ball in a Hilbert space, there is a sequence 6 4 0 and a function wo E V,'(RE) such that wg
w0 weakly in V,'(Rc) and strongly
-i
in
V O ( R c ).
(3.42)
$3. On the behaviour o f eigenvalues and eigenfunctions
307
For a fixed v E Cp(Rc) the integral identity (3.40) holds for all sufficiently
if
small 6, since 0; C
< 6.
Passing in (3.40) t o the limit as 6 + 0 and
taking into account the uniqueness o f a solution o f problem (3.37), we find that w o t $ = U, where u is a solution o f (3.37). It is easy t o see, by virtue o f (3.42), that we have
and therefore ((ufollows that lu(x)l 5
+ 0 for any open set G such that G c 0". It M , since 1v61 5 M. Lemma 3.7 is proved.
Let us consider the problem
A weak solution of this problem is defined as a function N E P1(w)satisfying the integral identity
for any $ E cl(w). Let
in Theorem 1.3, Ch. I, and take the left-hand side o f (3.44) as the bilinear form a(cp,+), and the right-hand side as l(y5). Then, using estimate (3.14) in complete analogy t o Theorem 6.1, Ch. I, one easily establishes Theorem 3.8. Suppose that
308
III. Spectral problems
Then there is a unique (to within an additive constant) solution o f problem (3.43). This solution satisfies the inequality
where q is a constant, c is a constant independent of N ,
Fi, i = 0,1, ...,n
3.3. Homogenization of a Second Order Elliptic Equation Degenerate on the Boundary Let us now define the coefficients of the homogenized equation corresponding t o the problem (3.35).
Denote by N q ( < ) ,q = 1, ..., n , solutions o f the
problems
Set
where (3.49)
> c 1q(2,c = const > 0.
Let us show that iip,qpqq
By virtue of (3.47) one can easily check that ii,, can be rewritten in the forrn
Thus hPq= hqp. Due t o (3.50), setting w, = ( N ,
+ t p ) q p ,we obtain
53. On the behaviour of eigenvalues and eigenfunctions
309
+
If for some 77 # 0 we have apqqpq,= 0, then w, z ( N , Ep)77, = const for almost all ( E Q n w . Since N , ( ( ) are periodic, it follows that 77 = 0. Set h = d a2(()b(()dt. ~Jnw
Thus we have defined the following second order elliptic operator with constant coefficients
The next theorem establishes the closeness of a solution o f the boundary value problem for Mc and a solution o f the boundary value problem for the homogenized operator M. Set
6 =d
/
02(()p(()d(
(6 = d
due to ( 3 . 4 ) )
Qnw
Theorem 3.9. Let u c , uO be solutions o f the boundary value problems
and
f0
E
~ ' ( a ) ,E V O ( R c ) .Then fC
where c, cl are constants independent o f e.
Proof. Set
where v is a solution o f the problem
III. Spectral problems
4 belongs t o Vd(RC).
Therefore the function Applying the operator
M, to uc - 4 , in a similar way to (1.16),Ch. II, we
obtain the following equalities, which are understood in the sense of distributions:
a
a
auO + auO
Mc(uc - 4 ) = MC(uc)- - (ahkm2- ( U O + EN. -)) axh axk ax*
auO= Mc(uc)- a + m2bu0+ &bm2Njaxj x a EC +E 8x1, 6x6 axk )8
aN, duo
h
(m2ijhk
-)axk +
d2u0
auO= M,(uc) - m2i$f(u0)+ Q2(b- &)uO+ Q2bu0+ &ba2Njaxj - ihk
duo dm2 dxk axh
8 axh
- -+ - [ ( a 2 & h k - @'ahk- Em2ahj
a
aNk duo
auO + & b ~ j@ ~= Me(uc)axh axj duo dQ2 am2 duo - a2i$f(u0) + a2(b- i))uO- ihk-+ ahk - - axk axh axh axk a - - [m2a,,*+ m -E
a2u0 8xk6x,
- (ahk@'~.-)
ax h
a axj puO auO = - €@'ahkN, + &bNjm2axkaxhax, axj = pG2(f0- fr) + (6 - p)@2 + m2(b- i))uO+ dNk a(a2aihNk) + - a2ahk- m2ahj--
+ (m28hk -
2
a~~ axj
ahk - &@ ah' -- E - (ajh@
a2u0
fO
(02iihk
atj
ati
+
$3. On the behaviour o f eigenvalues and eigenfunctions
Define the functions N , , ( t ) , B ( ( ) , R ( J )as 1-periodic in
311
J
solutions of the
following boundary value problems
a
-(
at i
a
a - B ) = ( b ) - ) a 2 in w
at1
,
B E Q 1 ( w ),
These problems are solvable, since the relations (3.48), (3.49), (3.52) allow to apply Theorem 3.8. W e thus have
a
a~
M.(ue - O) = pQ2(f0 - f')+ a - (m2ai1-) ax; at1
fO
+
It follows that u' - fi satisfies the equation
M,(ue - 6 ) = a m 2 P where
a ( Q ~ F+~pm2(f0 +a ) - f'), ax,
(3.59)
111. Spectral problems Due t o the periodicity o f R, B , Nhk, N8 we have
From this estimate and Theorem 3.6 with 11,
-
0 we deduce that
where cl is a constant independent of e. Since 6 has the form (3.57), it is easy t o see that the inequalities (3.55), (3.56) will have been proved if we establish the estimates
where c2, cg are constants independent of
E.
The proof o f the estimate (3.61) is based on the maximum principle for solutions o f problem (3.58), established in Lemma 3.7. 2
the functions N j ( - ) are continuous in independent o f e.
E
@
Let us show that
and are bounded by a constant
~i (w). Since N, is a solution
It follows from Lemma 3.1 that @ ( O N , ( ( ) of problem (3.47), we have
It follows that the function QN, = w E$ (w) satisfies the equation
313
$3. On the behaviour of eigenvalues and eigenfunctions
since
# 0 in w .
Due t o the well-known results on the smoothness of solutions
of elliptic boundary value problems and our assumptions about the smoothness of w and akj, the function w ( ( ) is smooth in 5. Moreover w = 0 on aw. Since
a(() = 0 on d w , but its gradient does not vanish the function N , r w/@is continuous in Thus by Lemma 3.7 we have Ivl
in a neighbourhood o f dw,
w.
5 C ~ IluOllclcn) E and therefore the inequal-
ity (3.61) is satisfied.
) a truncating function defined Let us prove estimate (3.62). Let q ~ , ( x be x auO
immediately after the formula (1.23), Ch. II. Set Q , = c p , ~ N , ( - ) -. Then E
@, E V 1 ( R ' ) . It is easy t o see that
ax,
Therefore, taking into consideration the smoothness o f u0 and the periodicity of N , , we obtain the inequality
where c5 is a constant independent of E . Using the integral identity for u - Qc we get the estimate (3.62). Theorem 3.9 is proved.
3.4. Homogenization of Eigenvalues and Eigenfunctions of the Dirichlet Problem in a Perforated Domain Consider now the question o f the closeness of the eigenvalues and eigenfunctions of the following problems
III. Spectral problems
+
~ ( v , k ) X;,v,k
= 0 in R
o<X;<X:I...<X;I
I
B V ~ V = ~ ~ 61, X
,
v,k E HA ( 0 ) ,
...,
,
n where
1
Theorem 3.10. Let X,k and X b be the k-th eigenvalues of problems (3.63) and (3.64) respectively. Then
where ck is a constant independent o f
E.
Suppose that the multiplicity o f Xo = A;+'
is equal t o m, i.e. Xf,
< Xf;tl
=
... = x L + ~< ~ f , + ~ +XE' , = 0, and v o ( x ) is an eigenfunction of problem (3.64) corresponding t o Xo, IIvollLzcn, = 1. Then for every
E
E ( 0 , l ) there is a
function ve such that
where MI is a constant independent of E , vo; v e is a linear combination o f eigenfunctions o f problem (3.63) corresponding t o the eigenvalues A:+',
...,A:+".
Proof. Let us apply the abstract results obtained in Section 1.2. Denote by 'He the space V O ( R e )equipped with the scalar product
By 'Ho we denote the space L 2 ( R ) with the scalar product
( u , v ) ~ ,= ,
/ n
6 uv dz .
315
$3. On the behaviour of eigenvalues and eigenfunctions
V = 3-10. We Re, u E L2(R).
Set
define Re as the restriction operator: Rcu = uc, uc = u on
Let us check the conditions (1.4). According t o Lemma 1.6, Ch. I, and inequalities (3.49), (3.52) we have
/ ~ u ~ l ~ ~ ( q ) @ ~ ( : )/d xI ~ ~ / ~ d x =
-t
nc
Thus Condition
n
C1 o f Section 1.2 is satisfied.
Let us introduce the operators A,
:
3-1,
-t
'H,, do : 3-10
-+
3-10,
setting A, f' = uc,
&fO = uO,where uc and u0 are solutions of problems (3.53) and (3.54) respectively. Using the corresponding integral identities we see that these operators are positive and self-adjoint. The compactness of A, and d,,follows from the compactness of the imbeddings Vd(Rc) C VO(R') and H,'(R) c L2(R) respectively. Due t o (3.36) the operators A, have norms uniformly bounded in E . Therefore Condition C2 of Section 1.2 is also satisfied. The validity of Condition C3 is guaranteed by the estimate (3.55) o f Theorem 3.9 and by the density o f C1(!?) in L2(R). Let us check the Condition C4. If sup 11 f'llNt < 00, then according t o c
(3.36) we have sup IIA. f ' l l v l c n * , < 00 and therefore due t o Lemma 3.4 we c
can find a subsequence
E'
and a function w0
E H,'(R) c L2(R) such as t o
satisfy (1.6). Due t o the smoothness of eigenfunctions of problem (3.64) the estimate
(3.55) yields for any vk
Since the eigenvalues of problems (3.63), (3.64) and (1.11), (1.12) are related by (2.11), the assertions o f Theorem 3.10 follow directly from Theorems
1.4, 1.7. Using the above results we can easily compare eigenvalues and eigenfunctions of problems (3.3), (3.63), (3.64). Thus we have actually proved
111. Spectral problems
316 Theorem 3.11.
At, A$, Xk
Let
be the k-th eigenvalues of problems (3.3), (3.63), (3.64) re-
spectively. Then
where
A.
depend on
is the first eigenvalue of problem (3.4), the constant ck does not E.
Suppose that the multiplicity o f the eigenvalue
Xo
=
A;+'
of problem (3.64)
is equal t o m, i.e.
and
vo(x)is an eigenfunction corresponding t o Xo.
Then for each
where
E
there is a function Ue such that
M,' is a constant independent of
E,
vo; U E is a linear combination of
eigenfunctions o f problem (3.3) corresponding t o the eigenvalues A :',
..., A:+"
$4. Third boundary value problem for second order elliptic equations 3 17 $4. Third Boundary Value Problem for Second Order Elliptic Eauations in Domains with R a ~ i d l vOscillatinn Boundarv
4.1. Estimates for Solutions Let R be a simply connected bounded domain in
R2whose boundary 8 0 is
smooth and is described by the natural parameter s , which takes values from
0 t o 1 and is equal t o the curve length counted from a fixed point on 8 0 . In a neighbourhood of d R we introduce the coordinates ( s , t ) , where t is the distance from a given point t o d R along the normal t o d R containing this point. Consider the domain Rc
c IR2 containing R
and bounded by the curve
E = l l m , m > 0 is integer, $([) is a smooth 1-periodic function of R1,$(<)2 0. Thus for small e the domain RE has a rapidly oscillating
where
6
E
boundary. Let L ( u )
a (a", .( 5 ) a au,)be a second order =-
coefficients a i j ( x ) are smooth functions in
a 13. , a,. - 3% KO
7
= const
a i j ( ~ ) ~Li K~Oj171'
>0 ,
E
elliptic operator whose
R2such that 7
R2 .
By a ( u ) on d R or on dRc we denote the conormal derivative a ( u ) =
au a;j - vj, where u = ( y , u 2 )is the outward unit normal t o the boundary of ax; the corresponding domain. Consider the following boundary value problems
111. Spectral problems
318 1
where a0
I'
=
= const
J (1 + (+'(s)l2)ll2ds,a ( x ) is a smooth function in R2,a ( x ) 2 0
> 0.
Our aim is t o estimate the difference o f solutions o f problems ( 4 . 1 ) , ( 4 . 2 )
in terms o f fO, f',and after that, following the general method developed in Section 1.2, t o evaluate the closeness between eigenvalues and eigenfunctions of operators corresponding t o problems (4.1), ( 4 . 2 ) . Set
where
ds, is the element o f curve length on a R c .
Weak solutions of problems ( 4 . 1 ) , (4.2) are defined as functions
uc E
H 1 ( R c ) ,u0 E H 1 ( R ) which satisfy the integral identities
for any v
E H'(Rc), w E H1(R).
Theorem 4.1. Let
uc and uO be weak solutions o f problems ( 4 . 1 ) , ( 4 . 2 ) respectively. Then
the following estimate is valid
where c is a constant independent of
E, fO,
f'.
We first outline some auxiliary results t o be used in the proof of Theorem
4.1. Note that the existence, uniqueness and estimates (uniform in
H1(R') norms o f solutions o f problem ( 4 . 1 ) in terms o f
11 f'llL~(nc,,
easily obtained from Theorem 1.3, Ch. I, and the following
E)
for the can be
54. Third boundary value problem for second order elliptic equations 319 Lemma 4.2. There is a constant M independent o f
E
and such that
for any u E H 1 ( R L ) .
Proof. Since the diameter of
RE is bounded by a constant independent of E , one can find a constant b such that b does not depend on E and 1 5 2-ebzl 5 2 for all x E R E . Set u = ( 2 - ebXl)v.Then
/ b(2 - ebz1)ebz1v2vldsE.
-
an* Therefore
/ b(2 - ebz1)ebz1v2hdsE.
+
an* The estimate (4.4) follows from this inequality and the conditions imposed on a i j ( x ) ,a ( x ) , b. Denote by G6 the 6-neighbourhood o f d R and by
the 6-neighbourhood
of the domain R , where 6 is sufficiently small. In terms o f the coordinates ( s , t ) , introduced above in a neighbourhood of
80, one can write
The parameter
d R E c GbI2.
E
is assumed t o be so small that 0
S
< E $ ( - ) < 612. E
Thus
III. Spectral problems
320 Lemma 4.3.
For any v E H 1 ( R c )there is an extension Pcv E H1(R(6))such that
IIpcvII~l(n(~,) 5 IIvIIH1(ne) Moreover
IIvII~z(nqn) I c ~ E "IIvII~lcnr) ~ , where the constants GI, cl do not depend on E , v.
Proof. Fix v E H 1 ( R e ) .Let us consider v on G6 n Rc and extend it t o the set
G6 as follows. First we pass from the coordinates s, t in G6 t o the coordinates s S' = S , t' = t - E$(-). In the variables s', t' the sets G6 n Rc, G6\RC have E
the form
(G6\Re)' = { ( s f t') , : 0
< s' < 1 , 0 5 t' < 6 - ell(:)}
.
Set
w(sl,t') = v ( s ( s f t'), , t ( s f t, ' ) ) = v ( s ,t ) for t
< E$(:)
, t' I O .
According t o Proposition 2 o f Theorem 1.2, Ch. I, the function w(sl,t') can
Go = {(s',tl) : 0 5 s' I 1 , -26 5 t' 5 0 ) t o the set G = {(sl,t') : 0 5 sf 5 1 , -26 5 t' 5 26) as a function Pw E ~ ' ( 6 ) such that Pw is 1-periodic in sf and I I P w I I ~ ~5( ~c )I I w I I ~ ~ ( ~ ~ where ) , c is a constant independent o f w. Setting ( P c v ) ( s , t )= v ( s , t ) for ( s , t ) E R E , S (P.v)(s,t) = ~ w ( s , -t E$(;)) for E$(:) < t < 6, we obtain the needed be extended from the set
extension. Let us prove estimate (4.6). The set Rc\R
lies in the 6-neighbourhood
of 8 0 , 6 is of order E . Therefore applying Lemma 1.5, Ch. I, in the domain
R(6)\0 we get
Lemma 4.3 is proved.
54. Third boundary value problem for second order elliptic equations 321 Lemma 4.4.
1
Let 7 ( 7 ) be a smooth 1-periodic function o f 7 E
R1, such
that
/
7(q)dq
0
= 0. Then for any u E H 2 ( R ) ,v E H 1 ( R )the following inequality is satisfied
where c is a constant independent of E , u , v. The proof of this lemma can be obtained by the same method as the proof o f Lemma 2.9, Ch. I; however, in the case under consideration we should take
G = { ( s , t ) : 0 5 s 5 1 , -6 < t < 01, and instead o f the sets 07 consider the sets a, = { ( s , t ) : t = 0 , ~ ( -m1 ) 5 s 5 Em). as
0 1
the domain
Proof o f Theorem 4.1. We can assume that the function u0 is extended t o the domain
R(&)in such a way that
The possibility of such extension is guaranteed by the smoothness o f
dR.
Let us write the integral identities for uc, uO:
and set v = uE- uO. Subtracting the second equality from the first one we obtain
III. Spectral problems
322
Passing from the coordinates x to s, t in the 6-neighbourhood of 80 and S = a(s,t)uO(s,t ) v ( s ,t ) ; g ( E- ) = (1 + l$t(-)12)112, we have E S
setting w ( s , t )
Applying Lemma 4.4 in the case of u = a(s,t)uO(s,t ) , v = v ( s , t ) , ~ ( q=) g(q) -
r, we get
It is easy to see that
4:) w (s, E $ ( : ) )
- w(s, 0 ) =
/
dw
; i l( s , t)dt
=
0
Therefore
L
CI
lluOllH~(nqn) IlvII~l(n*\n).
From the estimate (4.6) we have quently 1121
5~
5
~
I l ~ ~ l l ~ l ( ~ c \ ~ )
3 I I ueO I I~~ ( n IIvlI~l(n*\n) ~) ~ .
Therefore from (4.9), (4.10), (4.11) we get
~ llu0ll~2(na). e ' ~ Conse~
$4. Third boundary value problem for second order elliptic equations 323
Let us estimate the remaining terms in the right-hand side of (4.8).By vritue of (4.6) we find that
5 ~5
[I~uO~Iil(n*\n) IIvIIilcn*\n)
+ IIY- P I I LIlvII~lcn) ~ ~ ~+,
+ IIPllLa(fic\f2)I I ~ I I L ~ ( ~5* \ ~[rl" ) ] IIuOIIi.(n)IIvIIi1(nC)+ + IIP- ~ ' I I L . ( ~I I)V I I H ~ ( W ) + I I Y I I ~ ( ~ * \ ~I I )V &I I W' (/ ~~* ) ]
+
(4.13)
Taking into account (4.12), (4.13) we deduce from (4.8) and Lemma 4.2 that lluC- u0llLl(n*)
5
c5
< + Ilf' - f011Z2(n,] .
+
[a IIuOIIL~(n) E IIYllbcn*\n,
This inequality implies (4.3). Theorem 4 . 1 is proved.
0
4.2. Estimates for Eigenualues and Eigenfunctions Consider the following spectral problems
+ Atut = 0 in fie, ut E H1(fie) , u(ut) + a(x)ut = O on dRe , L(ut)
Jufu:d~=6k{, n=
O < A ; < . . . ~ A ~ ..., <
I
III. Spectral problems
324
where the eigenvalues are enumerated in increasing order and according t o multiplicity, as in §2. To study the closeness o f
A t t o A;
we apply the general method described
in Section 1.2. Set 'He = L 2 ( n ' ) ,'Ho = L 2 ( R )= V ;
L 2 ( R )+ L2(Rc)setting REf = f ( x ) for x E R, R, f = 0 for x E RE\R. It is obvious that Condition C1 holds with y = 1. Let us introduce the operators A, : 'H, -+ 'H,, : 'Ho + 'Ho setting AEfE = u E ,dof 0 = uO,where uc, u0 are solutions of problems (4.1), (4.2) respectively. It is easy t o verify that A,, are positive compact and selfadjoint operators and that due t o Lemma 4.2 the norms IIAcll are bounded by Define the operator R,
:
a constant independent of
E.
E L 2 ( R ) . Then ACREf 0 = uc is the solution of problem (4.1) with f' = f 0 in R, f" = 0 in RE\R. We clearly have Consider Condition C3.
Let f 0
The first term in the right-hand side of this equality converges t o zero as E
+0
due t o estimate (4.3), and the second term converges t o zero since the norms
lluEIIHl(n.)are bounded by a constant independent o f as
E
E
and
mesRE\R
+
0
--+ 0.
Let us prove the validity o f Condition C4. Suppose that
sup 11 f l l L ~ ( n c ) E
<
co. Then sup
I I u ' I I ~ I ( ~ ~ ) < co,uC= A, f " .
E
P,uc E H1(R(&)) o f the functions uc, constructed ) L2(R(6)) in Lemma 4.3. Due t o the compactness of the imbedding H 1 ( R ( s )c there is a function U E H 1 ( R ( & and ) ) a subsequence E' + 0 such that Consider the extensions
IIP,IU"
Then
-U
I I L ~ ~-t~0~ as ~ , E')
+
0
54. Third boundary value problem for second order elliptic equations 325 This equality, together with (4.16) and Lemma 1.5, Ch. I, implies Condition
C4. We have thus established that Conditions C1-C4 are satisfied and therefore Theorems 1.4, 1.7 can be applied t o estimate the closeness of eigenvalues and eigenfunctions of problems (4.1), (4.2) in exactly the same way as it was done in 52 for the elasticity problems. Theorem 4.5. Let X,k, Xgk be the k-th eigenvalues of problems (4.14), (4.15) respectively. Then
where ck is a constant independent of
E.
Suppose that the multiplicity of the eigenvalue Ah+' = Xo is equal t o rn, i.e. X i
< A?' = ... = A+;"
< A;+"+,'
problem (4.15) corresponding t o Xo,
I
XE = 0, and uo is the eigenfunction of I ~ ~ l l = ~ 21.~ Then ~ , there is a sequence
{ti,) such that
where MI is a constant independent o f E ,
UO;
21, is a linear combination o f eigen-
functions of problem (4.14) corresponding t o the eigenvalues A:+',
...,A:+".
Remark 4.6. The case G(7l)
> 0,
i.e.
R C Re, has been considered merely for the sake of
simplicity. With the use of slightly more complex calculations, theorems on the closeness of solutions and spectral properties of problems (4.14), (4.15) can also be proved i f $(q) changes sign. Remark 4.7. Constructing suitable boundary layers we can also obtain estimates o f order for the difference of solutions of problems (4.1), (4.2).
E
III. Spectral problems
326 Remark 4.8.
Methods used in this paragraph can also be applied in the case o f n independent variables, when the boundary
dR
in local coordinates has the
{x : x, = $(?)I, and the perturbed boundary dRr has the form x, = $(2) + ~ ~ ( 2 ) ~ ( ! ) )where , 2 = (zl,...,x,-1) varies over a bounded open set G c Rn-', g(2) E C,"(G), ~ ( 6 is) a smooth function
form
{x
:
1-periodic in q. Remark 4.9.
A similar problem can be considered for the system o f linear elasticity. The main results o f this paragraph were obtained by another method in [4] (see also
[110]).
55. Free vibrations of bodies with concentrated masses $5. Free Vibrations o f Bodies with Concentrated Masses
5.1. Setting of the Problem We consider an eigenvalue problem for the Laplace operator with the Dirichlet boundary condition and with a density function which is constant everywhere in a domain
R c IR",n 2 3,
except for a small neighbourhood o f one
o f its interior points, say 0. It is assumed that O is the origin o f Rn and
R
is a bounded smooth domain. Here we study the following eigenvalue problem
where
E
> 0,
form an increasing sequence and each eigenvalue is counted
as many times as its multiplicity; x ( ( ) is a bounded measurable function such that x ( ( ) > M = const
> 0 for ( E G, x ( ( ) = 0 for ( $Z G, G is an open set
o f positive Lebesgue measure such that G c R, 0 E G. Our aim is t o study the asymptotic behaviour o f eigenvalues and eigenfunctions o f problem (5.1) as E -+ 0 for n >_ 3 and various real values o f m. There are three qualitatively different cases.
1.
-00
< m < 2.
For such values o f m the k-th eigenvalue o f problem (5.1)
converges t o the k-th eigenvalue o f the Dirichlet problem for the Laplace equation in R.
2 . rn
> 2.
In this case X ~ E ~ - " , where X,k is the k - t h eigenvalue of problem
(5.1), converges t o the k-th eigenvalue o f the following problem for the Laplace operator in
Rn
III. Spectral problems
328
3. m = 2. The set o f the limiting points (as E -+ 0) o f the spectrum of problem (5.1) is the union o f the spectrum o f the Dirichlet problem for the Laplace operator in R and the spectrum o f problem (5.2). The behaviour o f the eigenvalues of problem (5.1) will be studied on the basis of the general method suggested in $1. To this end we make a suitable choice o f spaces
?lo,%, V and operators &, A,, R,, and check that
Conditions C1-C4 are satisfied. Another approach t o the problem o f free vibrations of bodies with concentrated masses is described in papers [log], [82], [72]-[74], [25]-[29], [156] (see also [ I l l ] , [125]). We shall need the following auxiliary propositions. For any u E CF(lRn)( n 2 3) the Hardy inequality
holds with a constant c independent o f u (see [42]) Lemma 5.1. Let n >_ 3. Then for any u E HA(R)
Moreover the following inequality is satisfied:
/
lu12dx 5
CE'
J
lVzu12dx ,
(5.5)
n
EG
where c is a constant independent o f
E,
u ; the sets G , R are the same as in
(5.1).
Proof. The
estimate (5.5) follows directly from the Hardy inequality (5.3).
Let us establish the convergence (5.4). For any 6
v611H;(n)
< 6,
>
0 consider a function vs E C,OO(R) such that c1I211u -
where c is the constant from inequality (5.5).
ing estimate (5.5) t o u - va, we obtain
Then apply-
55. Free vibrations o f bodies with concentrated masses
This inequality implies (5.4), since n
> 2.
Lemma 5 . 1 is proved.
Lemma 5 . 2 . Let ue(x)b e a solution o f the problem
u ~ E H ~ ( Ra )P €,[ O , l ] ,m > - o o , n 2 3 . Then
where c is a constant independent o f a ,
Proof. T h e integral yields
P , m.
identity combined with (5.5) and t h e Friedrichs inequality
III. Spectral problems
Hence the inequality (5.6). Lemma 5.2 is proved.
5.2. The case -oo < m < 2 , n Denote by 3-1, and
23
?the lo space L2(R)equipped with the scalar product
and
respectively. We take f0
Hi(R)as V.
Set Ref0 =
f0
for any
f0
E 3-10 For
E V we have by Lemma 5.1
This means that Condition C 1 holds with 7 = 1. Denote by
A, : 'He-+ 3-1,
the operator which maps a function
f' E 3-1,
into the solution ueof the Dirichlet problem
By
&
: 3-10 -r 3-10 we denote the operator mapping
f0
E
into the
solution u0 of the Dirichlet problem
One can easily verify that
A,
and
&
are positive compact self-adjoint
operators defined on 3-1, and 3-10 respectively. The inequality sup IIdclltciy., C
<
m follows from (5.6) since for m
inequality we have
<2
by virtue o f (5.5) and the Friedrichs
$5. Free vibrations of bodies with concentrated masses
9 c,
/
IVuc12dx5 c2
J
(1
+ s-"x) 1 f.I2dx .
n
a
Thus Condition C2 is established. Let us show that Condition C3 is also satisfied. Set
f0
E XO. Then
where
Auc = - (1
+ c-"~(:))
fO
in R
,
uc E H: (R)
,
According t o Lemma 5.2 with a = 0 we have
J 1v.(uc - uO)12dx 5
J
C E ~ - ~ ~
n
lf012dx .
CG
By virtue o f (5.5) and the Friedrichs inequality we find
5 c2 J lv(uC- u0)12dx . n From this inequality and (5.11) we deduce that
1
IIdcRcfO- RclbfOIIk. 5 c3&2-2m If012dx
(5.12)
CG
for any
f0
For
f0
E
E XO,where c3 is a constant independent o f E and fO. E V = Hi(R) Lemma 5.1 implies that E-' 11 f O l l ~ z ( , ~ )+ 0 as
+ 0. Therefore convergence (1.5) follows from (5.12), since m
< 2.
This
shows the validity of the Condition C3. Let us prove that the Condition C4 is also satisfied.
If sup
11 f . 1 1 . ~ ~ < cm, i t
f ~ l l o w sfrom (5.6) that
E
where ue is the solution of problem (5.9).
sup
I I u ' I I ~ ; ~ ~ , < cm,
C
Therefore there exist a vector
w 0 E H,'(R) = V and a subsequence E' + 0 such that
111. Spectral problems
uc' 1 wO weakly in H,'(R) and strongly in L 2 ( R ) .
(5.13)
Thus due t o the inequality (5.5) we have
5
J
+
lur - ~
~
l
C ~ E ~ - "
n
1
~ IV(U' d - w0)12dl. ~ , n
where uE = AcfE and c2 is a constant independent of
E.
From the above
inequality we obtain (1.6) by virtue o f the convergence (5.13) and the fact that m
< 2.
Thus the Conditions C1-C4 are valid and we can apply Theorems 1.4, 1.7. The eigenvalue problem associated with the operator dohas the form
Theorem 5.3. Let m
< 2, n 2 3, and let Xi, X,k
be the k - t h eigenvalues o f problems (5.14),
(5.1) respectively. Then
where ck is a constant independent of
E.
Suppose that the multiplicity o f the eigenvalue Xo o f problem (5.14) is equal t o r , i.e. Xo = A;+'
=
...
= A;+'.
Then for any eigenfunction u0 of
problem (5.14) corresponding t o X 0 and such that I l ~ , , ( ) ~ z ( ~= ) 1, there is a linear combination iic o f eigenfunctions of problem (5.1) corresponding t o the eigenvalues A:+',
...,
and such that
$5. Free vibrations of bodies with concentrated masses where cl is a constant independent of
E
and uo.
Proof.
It has been shown above that operators A,, C1-C4 and therefore Theorems 1.4, 1.7 are valid.
do satisfy
Conditions
To obtain estimates (5.15), (5.16) from (1.13), (1.26) one has only t o note that
p,k
= (A:)-',
d,,is smooth.
p0 = (Xi)-1 and that each eigenfunction o f the operator Therefore, by virtue o f (5.12) for
f0
E N(&,d o ) we have
5.3. The case m > 2, n 2 3 Let us pass t o the variables ( = E-'x in problem (5.1), setting
Then problem (5.1) reduces to the following one
Let us study the behaviour of eigenvalues and eigenfunctions o f this problem First we introduce an operator whose spectrum is formed by the limits of eigenvalues of problems (5.18) as Denote by
E
+ 0.
H the completion of C,"(Rn) with respect t o the norm
I I U I I ~= J 1'.(
I C I - ~ + IVCUI')~C.
(5.19)
Rn
By virtue o f the Hardy inequality (5.3) we have for any u E
H . Therefore the norms (5.19) and
in H . Consider the following problem
(IuIIHI co I I V [ U I I ~ Z ( ~ ~ )
llVEu11~2(~n) are equivalent
111. Spectral problems
334
We define a weak solution of problem (5.20) as a function
uo E H which
satisfies the integral identity
By Theorem 1.3, Ch. I, this solution
u0 exists and satisfies the inequality
The estimate (5.22) follows from (5.21) for v = u0 and the Hardy inequality. Define the space 'Ho as
L2(G)with the scalar product
'Ho are defined Rn\G. Therefore we can consider each function from L:,,(Rn)vanishing outside G as belonging t o No. Let us define the operator & : 3i0 -t 'Ho setting & f o = nG(()uO, where K G is the characteristic function of the set G, u0 E H is the solution of In what follows we shall assume that all functions from
on
Rnand vanish
on
problem (5.20). First we show that
is a positive self-adjoint operator. Indeed, let dof 0 =
K ~ ( < ) u&go ', = nG(<)vO, where u0 is the solution o f problem (5.20) and v0 is the solution o f problem (5.20) with f 0 = By the integral identity (5.21) we have
These inequalities imply that
is positive and self-adjoint. Let us prove
its compactness. Suppose that sup
(Ifd((no< co,fS = 0 outside
G. Let ua be solutions
8
of the problems
Acua = - x ( t ) f V n Rn, "u
Af"= K ~ ( [ ) U It' . follows from estimate (5.22)
H . By definition we have and the Hardy inequality
335
$5. Free vibrations of bodies with concentrated masses
that sup I l ~ ' l l ~ l ( ~<, )oo for any bounded measurable set GI containing G. 8
Therefore there exist a subsequence s' -+ 0 and an element u0 E L2(G) such that u" + uOin the norm o f L2(G), and thus
&f"'
+ m(t)uOin the norm
of 'KOass1+ 0. Consider the following Dirichlet problem
Lemma 5.4. Let m
2 2. Then for any uCwhich is a solution of problem (5.23) the estimate
J R*
1
I V ~ U ' I5 ~~ C C (.ern
+ X(O)
(5.24)
1f'l2dt
n
holds with a constant c independent o f e , a .
Proof. It follows from the integral identity for u-hat
Hence, choosing 6 small enough and taking into account the Hardy and Friedrichs inequalities, we obtain (5.24). Lemma 5.4 is proved. We define the space 'Kc as L2(Rc)with the scalar product
By Re : 'KO-+ 'KCwe denote the operator extending
Rc\G. Set V = 'KO.Let us verify Condition It is easy t o see that
CI.
f0
E L2(G)as zero t o
111. Spectral problems
moreover llRcf01In,
--t
11 fO1lno
as E
We introduce operators A,
0.
--t
: 'H,
-t
H ' ,, setting A, f' = u c , where u" is
a solution o f the problem
We can easily check that
A,
is a positive self-adjoint and compact operator in
'He. It follows from the estimate (5.24) that sup lldcllt(ne, < m, since by E
the Hardy and Friedrichts inequalities we have
Consider now the Condition C3. Let f 0 E 'Flo. Then where u0 is the solution o f problem (5.20); RE& f 0 =
dofO= nc(<)uO,
KGUO,
d E R E f= O uC,
where us is the solution of the problem
ACuc= - (E"
+ X ( ( ) ) ~ G (,C )uC~ E H;(Rc) .
Therefore
IldcRcP - R c k f O l l f .=
/
(E"
(5.26)
+ x ) luc - u 0 ~ ~ ( € ) l '. d(5.27) l
n' For uO- U' we have
Denote by w' a solution o f the problem
Since u0 is a harmonic function in
Rn\G and u0 E H , it follows from the
results of [44], [45] that for sufficiently large
we have
55. Free vibrations of bodies with concentrated masses This inequality is based on the representation of
u O ( [ )= cn
/ fO(n)
- s12-ndfl ,
u O ( ( )in the form cn = const
.
G
By the maximum principle we have
Then
v" = u0 - u' - wc E H,'(Rc),
Applying Lemma 5.4 with
uc = v E , CY = 0,
KG
=X,
fC = foem and using
the Hardy and Friedrichs inequalities we get
From (5.32) taking into account (5.30), (5.31) we deduce
Due to (5.30) we have
III. Spectral problems
am
J
<
I U O ~ ~ ~ %sm E
II.f"l $(G,
R*\G
7'
,.4-2n,.n-tdr .
1
It is easy t o see that
1
,
for n = 4
,
for n
where MI, M2, M3 are constants independent o f
- R,A~OIIL 5 c [eZn-'
for n = 3
(5.33)
>4 ,
Therefore
E.
+
IlfOllb(nl7
(5.34)
where
y3 = 1 , y4 = const E (O,1] , y, = 0 for n
>4 .
(5.35)
Hence the validity of Condition C3. Let us verify the uniform compactness of operators A, (Condition C4). Suppose that sup
(IfcllX,< M. It followsfrom (5.24)
and the Hardy inequality
C
(5.3) that sup c
and uc
I I U ' I I ~ ~ ( ~ ~ <) M, where Q1 is any
ball containing the set G
= A, f'. Due t o the compactness o f the imbedding H1(Q1)C LZ(QI)
there exist a subsequence E' + 0 and a function G such that
0 as
E' -+
-+
0. Setting wO(() = G(() for ( E G, wO(() = 0 for ( E R n \ G we
obtain that
By the Friedrichs inequality we get
55. Free vibrations of bodies with concentrated masses
339
The first term in the right-hand side o f this inequality converges t o zero by virtue o f (5.36), and it follows from ( 5 . 2 4 ) that the second term also converges t o zero. This means that Condition C4 is satisfied. Similarly t o Theorem 5.3, on the basis o f the estimate ( 5 . 3 4 ) and Theorems
1.4, 1.7 we can establish a theorem on the asymptotic behaviour of eigenvalues and eigenfunctions of problem (5.18). The limit eigenvalue problem has the form
A e U k = -A,kX(E)Uk in Rn ,
J
UkE H
,
x ( O u k ( O u l ( O d E=
G
....
o
,
1
(5.37)
It follows from the estimate (5.34) and Theorem 1.4 that eigenvalues o f problems (5.18) and (5.37) satisfy the inequalities
where ck is a constant independent of
E.
Theorem 1.7 implies that if U is an eigenfunction o f problem (5.37) such
1~ ( IU12d( t)
that
= 1 and U corresponds t o the eigenvalue A. o f multiplicity
G
r (A0 = A:+' = ... = A:+r), then there is a sequence Vc such that
and
Vc is a linear
combination of eigenfunctions o f the problem (5.18) corre-
sponding t o the eigenvalues A:+',
...,A:+',
the constant c, does not depend
and U .
on
E
m
> 2 are related
Since the eigenvalues and eigenfunctions o f problems ( 5 . 1 8 ) and ( 5 . 1 ) for by (5.17), we have actually proved
Theorem 5.5. For m
> 2, n 2 3 the eigenvalues of
problem ( 5 . 1 ) have the form
111. Spectral problems
340
where ,8,k 5 C ~ ( E +* E- (~~ - Y ~ ) / ~Agk) , is the k - t h eigenvalue of problem (5.37), y, is defined by (5.35). Moreover, for any eigenfunction U o f problem (5.37) corresponding t o the eigenvalue A. o f multiplicity r (Ao = A:+' = ... = A:+') and such that IIJTSUIIL2(c,= 1 there is a sequence of functions i i c ( x ) such that each i i c ( x ) is a linear combination of eigenfunctions of problem (5.1) corresponding t o the ) estimate (5.39) is valid. eigenvalues A:+', ...,A:+', and for V c ( [ )= i i c ( ~ [the
The case
5.4.
m = 2, n / 3
Consider the problem (5.1) f o r m = 2, n 2 3. The asymptotic behaviour of eigenvalues of this problem as E
-t
0 is determined by eigenvalues of problems
(5.37) and (5.14), namely by the eigenvalues of the following system
It is easy t o see that in fact we have an eigenvalue problem in the Hilbert space ?lo = L2(G)r L2(R)whose elements are pairs o f functions
( ~ (u0( x,) )
and the scalar product is given by the bilinear form
J X ( O U ( C ) V ( O ~+CnJ
.
U(X)V(X)~X
G
do : 'Ho + 'Ha associated with the problem (5.40) and mapping each element ( U ( [ ) , u ( x ) )e ?lo into the element ( K ~ ( O V ( vO(,x ) ) ,where V ( ( ) ,v ( x ) are solutions of the following problems Let us introduce the operator
Here
K G ( [ )is the characteristic function o f the set G .
It is easy t o verify that & is a positive compact self-adjoint operator in ?lo.
34 1
85. Free vibrations o f bodies w i t h concentrated masses
.e the space 'Kc as L 2 ( R ) with the scalar product (5.7) for m = 2. As
Vc
rbb
Let
\takethespace L 2 ( G ) x H i ( R ) .
U E ,lo, U = ( U ( [ ) , u ( x ) ) .We introduce the operator Rc
:
?lo --+
?lc setting x x R,U = u ( x ) + K G ( - ) E ' - " / ~ U ( - ). E
E
Then
For any
U E V we have u ( x ) E H:(fl).
Therefore by Lemma 5.1 the first
integral in the right-hand side of the last equality converges t o E
IIuIIL~(~,as
+ 0. Obviously the second integral converges t o
/ x(O lU(()12dt
G
fore
as E + 0, and the third integral converges t o zero. There-
I I R ~ U+~ IIUllwo ~ ~ ~ for
any
u u V, which
means that Condition C1 is
satisfied. Define the operators
A.
:
?lc + 'He
setting
A, f' = uc, where u' is A, are
the solution of the problem ( 5 . 9 ) with m = 2. It is easy t o see that compact positive self-adjoint operators. If sup c
Lemma 5.2 with m = 2, that
sup
5
1 1 v 2 ~ C ( I ~ z ( nc)
c
SUP c
I( ff 117-1, < m, it follows from
llfc117-1.
< 00 .
From ( 5 . 5 ) and the Friedrichs inequality we deduce that
Therefore due t o ( 5 . 4 1 ) we have
(5.41)
111. Spectral problems
and thus the Condition C2 is also valid. Let us consider the Condition C3.
) )e have For f0 E X O , f 0 = ( B O ( < ) , G O ( xW
= (KG(c)u(E),~(x))I
&fO
where
A,, = -$O(x) in R ,
A,u(C) = -x(0Q0(C) R,&~O
= u(x)
u E H,'(Q) ,
uEH
+ n G (EE ) s 1 - n / 2 xE~ ( -.)
On the other hand
I
(5.42)
j
w CE Hi(Q) . Denote by vc a solution o f the Dirichlet problem
Since U(E) is a harmonic function outside G, by analogy with (5.30) we have
and therefore
Iu(;)( x
< en-'l l Q O l ( t . ( ~for )
x E
.
It follows from the maximum principle that
where cl is a constant independent o f E . The function W c ( x )= u ( x ) lem
+ E ~ - ~ / ~2 U-( ;v') is a solution o f the prob-
$5. Free vibrations of bodies with concentrated masses
343
Subtracting the equation (5.43) from (5.46) we obtain
Let us apply Lemma 5.2 with a = 0,
x
x(-) E
5
= K G ( - ) . Then we have &
Taking into account (5.42), (5.43) we establish the following relations
To obtain the last inequality we made use o f the estimates (5.5) and (5.45). Since the function U ( ( ) is harmonic outside G, by the same argument as in Section 5.3 we conclude that
III. Spectral problems where a,(&)is defined by (5.33). Thus from (5.48)-(5.50) we deduce
If f0 E V, then q0 E H i ( R ) , and by Lemma 5.1 the first term in the right-hand side of (5.51) converges t o zero as E + 0. It thus follows from (5.51) that Condition C3 is satisfied, i.e. relation (1.5) holds. Note that if $O(x) is a smooth function, the inequality (5.51) implies that
where the constants cl, c2 depend on same as in (5.35).
f0
but do not depend on
E;
y, is the
Let us establish now that Condition C4 is also valid. Suppose that
Due t o (5.53), the compact imbeddings H 1 ( R ) C L 2 ( R ) ,H 2 ( R ) c H 1 ( R ) and the estimate I I v ~ I I ~ 5 ~ (c~11 ,f L l l L z c n ) with a constant c independent of E (see [9])there exist a subsequence E' -+ 0 and functions uO,vOE H,'(R) such that
ucr -t u0 weakly in H i ( R ) and strongly in L 2 ( R ) , vcr+ v0 strongly in H,'(R) as e' + 0
.
Taking v E C,"(R) in the integral identity for wc we get
I
(5.54)
85. Free vibrations of bodies with concentrated masses dw'
dv
-dx = E-'
n
/ x(:)/.v dx 5 n
The first factor in the right-hand side of the above inequality is bounded uniformly in E , and the second one tends t o zero by virtue o f (5.4). Therefore
wE1= uC1- vC1+ 0 as c' -t 0 weakly in HA(R). It follows that v0 = uO. The function W e ( ( )= E"/~-'w'(E() is a solution o f the problem
and
I E " / ~f'(c0l2d( -' = E-'
/ I f'12dx . CG
G
Therefore sup IJWellH< w and there exist a subsequence E'
-t
0 and a
C
function W
E H such that
W c 1 ( ( -t ) W ( ( ) weakly in H and strongly in L 2 ( G 1 ) (5.55) for any bounded measurable set G1 c
Rn. Obviously we can assume that the
subsequence E' + 0 in (5.54), (5.55) is the same one. Denote by wO in the Condition C4 the pair ( K G ( ( ) w ( ( ) , u o ( x ) Then. ). taking into account that uc = v' we find
+ w'
and applying Lemma 5.1 t o vE- uO,
111. Spectral problems
Passing in this inequality t o the limit with respect t o the subsequence E' -, 0 we see that, due t o (5.54) and (5.55), (1.6) holds. This means that Condition C4 is satisfied and therefore we can use Theorems 1.4, 1.7 t o compare the eigenvalues and eigenfunctions of problems (5.40), (5.1) for m = 2. Theorem 5.6. Let A,k and Ak be the k-th eigenvalues of problems (5.1) and (5.40) respectively; m = 2. Then
where y, is defined by (5.35), ck is a constant independent of
E.
Let A0 be an eigenvalue o f problem (5.40) of multiplicity r , As+' =
As+' = AO. Let ii = to A0 such that
... =
(~(c),u(x))be an eigenfunction o f (5.40) corresponding
llullRo= 1. Then for any
E
there is a linear combination iic
of the eigenfunctions of the problem (5.1) with m = 2, corresponding t o
A;+1,
...,A:+r,
such that
where M, is a constant independent o f
E,
0.
Note that in deriving the inequalities (5.56) from (1.13) we have taken into account the inequality (5.52) and the smoothness o f eigenfunctions of the Dirichlet problem for the Laplace equation.
55. Free vibrations of bodies with concentrated masses Remark
347
5.7.
In the same way we can consider the cases n = 2, m
E lR1 and n = 1,
m E R1. Another approach t o the problems studied in this section was suggested in
[72]-[74],[82],[27]-[29],[log],[I111, [155].
III. Spectral problems
348
$6. On the Behaviour of Eigenvalues of the Dirichlet Problem in Domains with Cavities Whose Concentration is Small Let
R
be a smooth bounded domain o f R3and let Go = { x : 1x1 < 1)
be the unit ball. Set R, = R\
IJ
(s3Go
+ 2 ~ 2 ) .Thus RE is a perforated
zcz3
domain with ball-shaped cavities of radius s3 forming a 2s-periodic structure. Consider the following boundary value problems
Let us estimate the norm
Ilu,
- ~ 1 1 ~ To 2 this ( ~ end, ~ ~ in accordance with
the method suggested in [lo], we define an auxiliary function w,(x),
x E IR3
as follows
Theorem 6.1. Solutions u, and u of problems (6.1) and (6.2) respectively satisfy the inequalities
IIuC- uIIL~(R,)
where C,a = const
I CEIlfllca(n) ,
> 0, C and a do not depend
Proof. It is easy t o see that W,
= (r-I - E - ~ ) ( E - '
origin. Let us show that
on s, f(x).
in EG~\E~GO the function w,(x)
- E-~)-',
has the form
where r = r ( x ) is the distance from x t o the
349
$6. On the behaviour of eigenvalues of the Dirichlet problem
where Co > 0 is a constant independent of E. We have
where the constant C1 does not depend on E. The estimate (6.3) follows from the last inequality, since the number of
+
the domains ( E G ~ \ E ~ G ~~E) Z z, E iZ3, belonging t o 0, is of order E - ~ . Taking into account (6.1) and (6.2) we obtain the equalities
A(u, - UW,) = f - AUW, - ~ ( V UVW,) , - uAw, = = f(l
- w,)
- (Aw,
- p)u + p(w,
- 1)u - ~
( V UVwe) , =
which hold in the sense o f distributions. Using (6.3) and the well-known Schauder estimate
llullc2+a(n) l C2 Ilf llcqn) for solutions of problem (6.2) , we get
Let us estimate A4 in the norm of H-'(Re). We have
5 C
sup ZER la19
IDaul
I~W,
- lIILz(n,) 5 Cs
sup XER la152
-
III. Spectral problems By virtue o f the Schauder estimate it follows that
In order t o estimate the norm IIA211H-~cn,) we introduce an auxiliary function q, which is a solution o f the problem
where u is the outward unit normal t o edGo. Obviously q,(x)
is defined t o
within an additive constant. Choosing the suitable constant, we can assume that q, = 0 on &dGo,since Go is a ball. Let us extend q, as zero t o the cube &QO =
{x :
-E
< xj < E,
j = 1, ..., n) and then t o the whole R3
+
r2 E~ as a 2~-periodicfunction. Then q, = - - - in &(GO 22), z E Z3, where 2 2 r = r ( x ) is the distance from x t o the centre of the ball &(GO 22) and
IVq,l
< E.
Moreover, q, satisfies the equation
U &(Go+ is a distribution with support on the surface o f the ball &(Go+ 22) and U
where x,(x) = 1 for x E
+
&(GO 2z), x,(x) = 0 for X E
a€Z3
2z), 6;
+
seZ3
such that
Indeed, setting T, =
U
+
&(Go 22) we have for any cp E C,"(R3)
ZEZ~
where
v
is the unit outward normal t o aT,.
Since Aq, = 3 in
aqc T,, Aq, = 0 in R3\T,, - E, q, = 0 on dT,, we have avl
§6. On the behaviour of eigenvalues of the Dirichlet problem
( ~ q . , p ) = 3J
c
J
~ c - E
T.
351
V~S=(~X.,V)-&
(J:,P).
t€Z3
aT*
Therefore
On the other hand
U
where ye is a distribution with support in
+
e3(G0 2 z ) , i.e. ( y e ,4) = 0 ,
z€Z3
if 4 E C?(IR3) and vanishes on the set
U
E ~ (+G2 ~ )The . explicit form
z€Z3
of yc is unnecessary for what follows. Thus we have
Since the mean value of the function 3xe
+ p over the cube eQO is equal t o
zero, Lemma 1.8, Ch. I, provides the representation
where
f,(() are
bounded functions in
The inequality (Vq.1
a
<E
-Age = E -qf ax,
R3.
shows that
, IR3
where functions qf are bounded in
uniformly with respect t o e . Therefore
a h f ( x ) )u + 7.u + E2 PU ax, 1- E ~
V. = ( A w . - p)u = a (-
9
III. Spectral problems
<
C7 and C7 does not depend on It follows that
where lhfl
I C EIIuIIH;(R) From (6.5)-(6.8)
E.
.
and the inequality
which holds for solutions o f the problem Au, = f , u, E HA(R,), we obtain the estimate 1Iuc
Ilf
- W~UIIH~(CI,) 5 CE
llc0(n).
This inequality together with (6.3) yield the estimates asserted in Theorem 6.1. The theorem is proved. Now we can obtain estimates for the difference o f eigenvalues and eigen-
(6.1),(6.2). Set 'H, = L 2 ( R , ) , 'Ho = V = L2(R).As R, we take the operator restricting f € L2(R)t o the domain 0,. Define the operators A, and A0 by the formulas: A, f = -u,, .&f = -u, where u,, u are solutions of the problems functions related t o problems
(6.1), (6.2) respectively. Using Theorem 6.1 and the methods o f $2.2 we can easily check that the Conditions C1-C4 are satisfied. Therefore we can apply Theorems 1.4 and 1.7 t o estimate the difference of the eigenvalues and the eigenfunctions of the problems
$6. On the behaviour of eigenvalues of the Dirichlet problem
353
where A$, Xk form increasing sequences and each eigenvalue is counted as many times as its multiplicity. Theorem 6.2. Let X,k, Xk be the k-th eigenvalues of the problems (6.9), (6.10) respectively. Then
where c ( k ) is a constant independent o f
E.
Moreover, if the multiplicity of the eigenvalue A'+'
Xl+1 =
... = Xl+m , and uo is an eigenfunction of
= Xo is equal t o m ,
(6.10) corresponding t o Xo,
= 1 then there is a sequence {fie) such that
) ~ U ~ ) ) ~ Z ( ~ )
where
Ml
is a constant independent of
E,
uo; u, is a linear combination of
eigenfunctions of problem (6.9) corresponding t o A;+',
...,
The method suggested in this section can also be applied t o the case
n 2 3, as well as t o the general second order equations and the system o f elasticity, and for the case when Go is an arbitrary open set such that
Go C QO = {x : -1 < xj < 1, j = 1, ...,n) (see [154]).
354
111. Spectral problems
57. Homogenization of Eigenvalues of Ordinary Differential Operators We consider here a sequence of operators
{Lk)such
that
Lk =$ J?
as
k -t m, where Lk, E areordinary differential operators having the form (8.32), (8.33), Ch. II, and satisfying the Condition N' of 58.1, Ch. II.
It is also assumed that
and that the problems
have unique solutions which satisfy the estimates
with constants
Q,
cl independent o f k and f ; the first eigenvalues of
are bounded from below by a positive constant independent o f
Lk,k
k.
These assumptions imply that we can take p = 0 in Theorem 8.1, Ch. II. Consider the eigenvalue problems for the operators
where A:,
Lk,E :
A' are enumerated in an increasing order and according t o their
multiplicity. It is also assumed that
$7. Homogenization o f eigenvalues of ordinary differential operators 355 where constants cz, c3 do not depend on
k, the
norm o f H-"1"
is defined in
$9.2, Ch. I. Theorem 7.1. Let
X i , A'
be the I-th eigenvalues o f problems (7.2), (7.3) respectively. Then
where c, is a constant independent of k ; Ak are given by the formula (8.37), Ch. II. Let u be an eigenfunction of problem (7.3) such that llullL~= 1 and u corresponds t o the eigenvalue
X0
o f multiplicity r
(A8+' = ... = A"+' = X0 1.
Then for any k there is a function iik such that
and iik is a linear combination o f eigenfunctions of problem (7.2) corresponding t o the eigenvalues
Xi+', ...,Xi+',
c, is a constant independent o f
k and u .
The proof o f this theorem is based on the abstract results obtained in 51.2, and is carried out in a similar manner t o the proof o f Theorem 2.3. In the case under consideration N o , ('He = 1
the scalar product
'HFlllk),is the space L2(0,1 ) with
1
1 f g i d x . ( 1 fgpr d r ) . respectively, V = N O , Re is the
J
J
0
0
identical operator, &fO = uO,where u0 is a solution o f the Dirichlet problem
f?(uO)= / ; f Oon ( 0 , l), u0 E H r ( 0 , l ) , Akfk = uk, where uk is a solution of the problem Lkuk = pk f k on ( 0 , I ) , uk E H r ( 0 , l ) . Due to the relations (7.1) for the coefficients o f C k and
dk,
E , the operators
satisfy the Conditions C1-C4 of $1,and therefore we can use Theo-
rems 1.4, 1.7 to estimate the closeness o f the corresponding eigenvalues and eigenfunctions.
111. Spectral problems
356
$8. Asymptotic Expansion of Eigenvalues and Eigenfunctions of the Sturm-Liouville Problem for Equations with Rapdily Oscillating Coefficients In this section we shall construct complete asymptotic expansions for the eigenvalues and the eigenfunctions of the following Sturm-Liouville problem (see [143])
d
x duk
dx ( a E-
)CEX + b x
)
+ k ( ) p x( ) u= 0 ,
x E ( 0 , I] ,
I t is assumed that a ( < ) ,b ( < ) , p ( < ) , a - ' ( f ) , a 1 ( J )E of bounded continuous functions of [ E
K , where K
is a set
R1,and K satisfies the following
conditions:
1) K: is a ring containing all constants; 2) For any f E
j
f(t)dt
K
there is a constant c f such that the function g ( x ) =
+ e l x belongs t o K .
0
Let us give some examples o f the sets which satisfy conditions
I.
The set o f all continuous T-periodic functions in
I), 2).
R1
II. The set of all continuous functions which can be represented in the form
M N.
+ cp(x), where M = const, Icp(x)l 5 C N (+~
for any integer
Ill. The set formed by restrictions t o the line x; = p;t (i = 1, ...,n ) of smooth functions, 2a-periodic in x l , ...,x,, where the constants p1, ...,p , are such that
88. Asymptotic expansion of eigenvalues and eigenfunctions
C > 0,
s
> 0 are constants independent
integers, m:
+ ... + m: # 0 .
357
of m l , ..., mn; mj are arbitrary
It is obvious that conditions 1) an&2) hold for classes I and II. Let us show that class Ill also satisfies these conditions. To this end consider the Fourier series
( m ,x) =
C
mixi
.
i=l
The restriction of F t o the line x, = pit,
i = 1 , ...,n , is F ( p l t ,..., p,t), and
the primitive function corresponding t o F has the form
The inequality (8.3) and the smoothness o f F guarantee the convergence of the series (8.3). Therefore condition 2) is satisfied. To construct asymptotic expansions for the eigenvalues and the eigenfunctions of the problem (8.1) we shall need the following auxiliary propositions. Lemma 8.1. For each
f E K:
/ f ( t ) d t + 2ejx x
Proof.
According t o condition 2) the function h(x) =
be-
-2
longs t o
K
and therefore is bounded in x E
R1.This
convergence o f Lemma 8.1. Lemma 8.2. Let
M ( ( ) E K: and (M) = 0. Then the equation
obviously implies the
III. Spectral problems
has a solution N ( ( ) which belongs t o
K
and can be represented in the form
where
dN
Moreover - E
K.
dE Proof. Since (M) = 0, the primitive function L ( ( ) corresponding t o M(E) also belongs t o K. Due t o conditon 1) the set K is a ring, and therefore L ( J ) a - I ( ( )E K since a - I ( ( ) E K by assumption. The primitive o f a - ' ( ( ) L ( ( ) has the form P ( ( ) + ( L ( ( ) a - ' ( ( ) ) ( ,where P E K . The primitive o f a - ' ( [ ) has the form Q ( ( ) (a-I(())( where Q E K . Choosing the constant C given by
+
(8.5) we see that the linear terms in the integrals entering (8.4) are mutually reduced, and therefore N ( ( ) E
K. Lemma 8.2 is proved.
Direct calculations show that the following lemma is valid. Lemma 8.3. The boundary value problem
d2u h - Au = w ( x ) dx2
+
+ Xwo(x) on
[0,11 ,
u(O)=a, u ( l ) = P , where A = ( ~ k ) ~wO(x) h , = sin ~ k xh, > 0, A, a , P are constants, admits a solution. if
+
X = 2xkh [ ( - l ) k + l p a] - 2 The solution is given by the formula
1
sin xky w(y)dy .
$8. Asymptotic expansion o f eigenvalues and eigenfunctions
u ( x ) = a cos n k x
+
1
sin n f k - y)
359
+
[w(Y) X W O ( Y ) ] ~ Y
+
0
+ C sin n k x ,
C = const
Now we formally construct asymptotic expansions for the eigenvalues and the eigenfunctions o f the problem (8.1). Strict mathematical justification of these expansions will be given later. We seek the expansion of the eigenvalue X k ( & ) and of the corresponding eigenfunction u $ ( x )o f problem (8.1) in the form
(the index k is omitted for the sake o f convenience). Here M
> 2 is an integer,
N(',")((),v,(x) are the functions t o be determined, A,(&) are unknown real numbers. In what follows it is assumed that N('") are defined for all integers and
~ ( ' 1 " )
= 0 for i
< 0 or s < 0,
or s
> i.
i,
s,
When the range o f summation
is not indicated, the sum is assumed t o contain all terms with non-vanishing functions
~ ( ' 9 " ) .
Let us substitute expressions (8.6), (8.7) for Xk(&),U $ in equation (8.1). We get
d
+ q(~(ON'O*)
dN(OtO)dv, dt -jE-
+ a ( [ )1-
+d
d ~ w " ( a ( [ )-) d t v,)
+
111. Spectral problems
360
dN('y0) dv,
dlv, F,O(x) is a sum of terms having the form stcp(()-, and 15 M + 2, dx' t 2 0; y ( t ) is a bounded function. Set N(OlO)r 1 , N('t0) 5 = 0 and define N('gl)(() as a solution of
where
~
(
~
1
'
)
the problem
N('sl) follows from Lemma 8.2. Define N ( ~ ? ~ )as( (a )solution of the problem
Existence of
where
h(2r2)is a constant given by the formula
$8. Asymptotic expansion of eigenvalues and eigenfunctions
361
Note, that due t o Lemma 8.2 the right-hand side o f equation (8.8) belongs t o class
K. We also define N ( ~ I O ) (as( )a solution o f the problem
where h(210)is a constant such that
h(290)= X O ( P ( E ) )
+ ( b ( € ) ).
For the values of 1 larger than 2 we define functions N('.")(()as solutions of the problems
where
Let us introduce the following notation
i, s we can successively find N('~") from (8.12). As a , ( ~ J ) , basis of induction we take the functions N(Oy0),N ( ' - O ) , N ( ' ~ ' ) N, ( ~ ~ O ) N Using induction over
~
(
~
defined above. It is easy t o see that 9
~
)
C i=l
s=o
ddv, 6('+2.s)+ X i ( E ) ( p )v6(x)
dxs
III. Spectral problems (8.14) where F,'(x) has a form similar t o that of F f ( x ) . Let us seek v,(x) in the form
Substituting this expression for v,(x) in (8.14) we obtain
-
y i=o
Ei
(=
'-9' X(i-p+2,8)dS
p=o
'Up
s=o
dxS
+
~ = oi - p p v p x )
+
Let us now define v p ( x ) ,p = 0, ...,M - 2, as functions satisfying the equations
and the boundary conditions
Let us single out those terms of (8.15), (8.16) which contain vi and rewrite (8.15), (8.16) as follows. For
i = 0 we have
For i= 1,2,
...,M
- 2 we obtain the following boundary value problems
$8. Asymptotic expansion of eigenvalues and eigenfunctions
d% ( 0 ) N('-P-")(o)A . dxs '
i-p C C i-1
v i ( ~=) -
s=o
p=o
363
Assuming X o , ..., X i - 1 , vo,...,v i - l , N(OtO),...,N(""" to be known, let us choose A;(&) such that the problem (8.18), (8.19) has a solution. Consider the k-th eigenvalue A. = X i of the problem (8.17). Then A0 = (p)-' ( ( ~ k ) ' h ( ~ > ~ ) -
(4). Setting
in Lemma 8.3, we get r
-
i-1
C
i-p
p=o
a=O
-
P
(
)
I
m dx
+
III. Spectral problems
12 ic2
sin n k ( x - y )
nkh(2J)
0
p=o
s=o
~ ( i - p + ~ ,@ s )v P ( y ) dxs
Using the formulas (8.21), (8.22) and (8.18)-(8.20) we can easily construct by induction the constants A , ( € ) and the functions v i ( z ) , ~
Xo, ...,X i - 1 , 00, ..., 0,-1,
(
~
~
provided that ~
7
"
)
..., N('+'V" are already known.
Thus we have constructed a formal asymptotic expansion for the eigenvalue
,Ik(€)o f the Sturm-Liouville problem (8.1) and the corresponding eigenfunction u,k(x). Remark 8.4. Formula (8.21) for
X I is reduced t o
1
-
N(
11)
0 dvo(0) - 2 ( )
/
[
sin ~ k y
0
where v o ( y ) = sin ~ k y Since .
&(3*0)=
hl = 0 for any k , provided that
E-'
j1(332)
3
C
L ( 3 1 s )
I
dSvo(y) dy dx8
,
= 0 due t o (8.13), it follows that
is an integer and a ( ( ) is 1-periodic in (.
Note that the sequence of operators
365
58. Asymptotic expansion of eigenvalues and eigenfunctions
is strongly G-convergent t o the operator
d2 h(212) dx2
+ (b) .
This fact was established in $8, Ch. II. It was also shown in 58, Ch. II, that the eigenvalues o f the problems (8.1) and (8.17) satisfy the inequalities
[(A:)-'
- ( A n ) - ' [ 5 ~ ( A +c IIPc(x)
C = const
,
A: = A n ( & )
- ( P ) I I H - ~ ~ ,~ )
(8.24)
,
where
A, = max ~E[0,11
)ldt +
[!(T-I h(212)(t)
j((h)-h(l))dx E
O
and by the definition of the norm in H-'9"
Due to the assumptions on the coefficients o f equation (8.1) the right-hand side o f (8.24) converges t o zero as E + 0 . It follows from (8.24) that A: -+ A: as
E +
0 for any n.
On the other hand the formal asymptotic expansion constructed above satisfies the following equalities
u i " ) ( o ) = E ~ - ~ $ ~ ( E u) L, " ) ( l ) = E
~
$ I (-& )
~
,
where I I F E I I ~ 2 ( o ,I~ )~o
,
l$o(~)I
+ I$i(&)I
I
CI
% , C i = const
and therefore according t o Lemma 1.1there is an eigenvalue X i ( & ) of problem (8.1) such that ~ A ' ( E )smooth function such that
5 fiM-l,
c
= const. Indeed, let $ , ( x ) be a
111. Spectral problems
Then 1$,(x)l
I C 2 ,where C2 does not depend on E
by virtue of the maximum
principle. Therefore Lemma 1.1 can be applied t o the function uLM)(x)-
E ~ $ , ( x )and the operator A, defined on the space L2(0,1) with the scalar 1 x product ( u , v ) = p(-)uvdx and mapping f E L2(0,1 ) into uc = A, f.
/ 0
E
where uc is a solution of the problem
It follows that, for sufficiently small E , 1 = k since X k ( & ) -+ X i , x ( ~ ) ( E )-i X,k as E -i 0 and X i has unit multiplicity, which implies that for sufFiciently small E a neighbourhood of X k contains only one eigenvalue of operator LC with homogeneous Dirichlet conditions. The above considerations provide justification for the formal asymptotic expansions (8.6). Thus we have proved Theorem 8.5. The eigenvalues and the eigenfunctions o f the problem (8.1) satisfy the following inequalities:
where c l ( k ) ,c2(k)are constants independent o f E .
$9. Eigenvalues and eigenfunctions o f a G-convergent Sequence
$9. On the
367
Behaviour of the Eigenvalues and Eigenfunctions o f a
G-Convergent Sequence of Non-Self-Adjoint Operators In $8, Ch. II, we introduced the notion of G-convergence o f operators having the form
and belonging t o the class E(Xo, XI, X z ) .
In general the operators o f a
G-
converging sequence are not necessarily self-adjoint. The aim o f this paragraph is t o study the behaviour o f the eigenvalues and eigenfunctions of a G-convergent sequence of non-self-adjoint elliptic operators and t o extend t o this case the results of $2 on G-convergent sequences o f elasticity operators which are self-adjoint. We shall need some well-known (see [128]) results on the convergence of the eigenvalues and eigenfunctions o f a sequence o f compact operators in a Hilbert space. It is sufficient for our purposes here t o formulate and prove the corresponding theorems in a less general form as compared with [128]. Let A E C ( H ) be a bounded linear operator in a separable Hilbert space
H with a complex valued scalar product. By u ( A ) we denote the spectrum of the operator A , i.e. the set o f all points p o f the complex plane C 1 such that there is no bounded operator inverse t o A - p I . Here I stands for the identity operator. If pO E u ( A ) and there is an element x E H such that x # 0, ( A - p O I ) x = 0, then po is called an eigenvalue of A and x is an eigenvector corresponding t o Po. If for some integer m
>
1 we have ( A - poI)x
#
0, ( A - poI)"x = 0 ,
then x is called a root vector corresponding t o po. By Ker A we denote the set { u E H , Au = 0 ) , Im A is the set consisting o f such u E H that the equation A w = u admits a solution w E H . Let R ( p ) be a holomorphic function of p defined in a domain w
c 6"
and taking values in the Banach space C ( H ) o f bounded linear operators. Let
r
be a closed curve limiting a subdomain w l , w l
c
w. Then the following
maximum principle holds for holomorphic functions with values in C ( H ) . The
III. Spectral problems
368
norm R(p) for p E w l is not larger than the maximum o f the norm o f R(p) on the curve
r = awl.
We shall also use the following well-known results. Theorem 9.1. Let T E L ( H ) be a compact operator. Then
1) the conjugate operator T* is compact; 2) the set a ( T ) is discrete; if a is a limiting point of a ( T ) ,then a = 0; the set o(T)\{O) consists o f the eigenvalues o f the operator T ;
3) a ( T * )is formed by the points complex conjugate t o those of u ( T ) ;
# 0 and T has no root vectors corresponding t o p , then H can be represented in the form
4) if p E a ( T ) , p the space
the dimension o f Ker(T - P I ) is finite and equal t o the dimension of
Ker(T' - P I ) ; 5) for each p
# 0 the operator T - pI
is of Fredholm type (see [40], p. 1071).
The sign $ denotes the direct sum of spaces, and
6
denotes the direct
sum of orthogonal spaces. We say that the operator B E L ( H ) is o f Fredholm type, if the dimension of KerB is finite and equal t o the dimension of the orthogonal complement of I m B in H. Let {A,)
be a sequence of compact operators in
H , and let A E L ( H )
be also a compact operator. Definition 9.2. The sequence {A,) is called compactly convergent t o operator A as m -+ m,
if the following conditions are satisfied.
$9. Eigenval ues and eigenfunctions of a G-convergent Sequence 1. Amu + AU strongly in
369
H for any u E H .
2. If {urn) is a sequence such that urn E H , IIumII {Amurn) is a compact set in H .
< 1 , then the sequence
Definition 9.3.
A sequence of operators {B,) (nor necessarily compact ones) is called properly convergent t o operator B E C ( H ) as m -t oo,if the following conditions are satisfied.
1. B m u -+ B u strongly in H for any u E H . 2. If {urn) is a sequence such that urn E H , llurnll = 1 and {Bmum) is a compact set in
H , then {u,) is also a compact set.
Lemma 9.4.
L ( H ) is strongly convergent t o L ( H ) , i.e. Amu + Au strongly in H for any u E H as m + KI.
Suppose that a sequence o f operators Am E operator A E Then
where C is a constant independent o f m. The proof o f this lemma follows from the Banach-Steinhaus theorem (see (1071, [134]), according t o which
sup I I A r n l l ~ (<~ )KI, if m
< oo for
any u E
SUP
llAm~ll
m
H.
Lemma 9.5. Suppose that {B,) vergent t o
is a sequence o f operators of Fredholm type properly con-
B E L ( H ) as m + oo,and B is an invertible operator. Then for B, are invertible and IIB,'~~.c(H) 5 C, where C
sufficiently large m operators
is a constant independent of m.
Proof. First we show that
operators
B;'
exist for large m. Let us suppose
the contrary. Then there is a subsequence m' + oo such that
B,I do not
111. Spectral problems
370
admit bounded inverse operators. Since operators B, are o f Fredholm type there is a sequence of vectors {x,,), lIx,,II
= 1 , such that B,,x,t
= 0. By
B , we can extract a subsequence {x,rr) of {x,,) which strongly converges t o an element x E H such that 11x11 = 1. It follows that Bx = 0, since Bx = B,t(x - x,,) + ( B - B,I)X and IIBrnllLcH,C due t o Lemma 9.4. This fact is inconsistent with B being
virtue o f the proper convergence o f B, t o
<
an invertible operator.
5 C. Suppose the contrary. Let us prove now the inequality IIBm1llccH, Then we can choose a subsequence {xmj)such that llxrntll = 1 and B,,x,I
0 strongly in H as m'
+
+
oo. Indeed, if IIB;fII > C(ml)and C(ml)+ m such that IIy,~ll = 1
as m' + oo, then there exists a sequence {y,,) and (IB,3ymll( 2 C(ml). Set z , ~ =
B G ? ~ , ~ Then . B r n , ( l l ~ ~ , l ( - ~=~ r n , ) I j z , ~ I I - ~ y ~ l+ 0 as m' + oo and we can take as X,I the vectors llz,111-'z,1. Due t o the proper convergence of B, t o B there is a subsequence m" such that X,II + x strongly in H as m" + co. Then B,,,X,I~ + Bx, since Bx - B m ~ ~ m = fB,rr(x r - x,I,) + ( B - B,r,)x and therefore Bx = 0 which contradicts the invertibility of B. Lemma 9.5 is proved. Let w
c 6''
be a subset o f the complex plane. Denote by N(w,A) the
span of all the eigenvectors o f A which correspond t o the eigenvalues o f A belonging t o w . For example, if w =
and po is an eigenvalue o f A, then
N(po,A) is the linear space of all eigenvectors o f A corresponding t o po. Theorem 9.6. Suppose that A,
+
A compactly as m
+
oo, and A,,
A are compact
operators of Fredholm type. Then the following assertions are valid.
1) If Po E u ( A ) ,PO # 0, then there is a sequence {P,), that pm + po as m + oo. 2) If Pm E ,-J(Arn) and pm
+
p
P , E ,-J(A,),such
# 0, then p E u ( A ) .
3) Ifurn E N(~rn,Arn) and p, + p # 0 , urn+ u in H a s m + oo,then E N(P,A).
"
59. Eigenvalues and eigenfunctions o f a G-convergent Sequence
371
Proof. Let us first
>0
establish 1). Suppose the contrary. Then there is a 6
and a sequence m' + oo such that the 6neighbourhood o f po contains no spectral points o f operators A,, for sufficiently large m'. Let
= {p E C1, Ip
- pol = 6)
be a circle such that the ball Ip
- pol 5 6
contains only one spectral point po of the operator A and does not contain any spectral points of operators A,,. By the same argument that was used in the proof o f Lemma 9.5, and taking
r6,we can easily show that for sufficiently large m' there exist the inverse operators (A,, - pI)-', p E r6,and
into account the compactness o f
r6. r6,there are no spectral points o f the operators
where C is a constant independent o f m' and p E Since, inside
A,,, the
above mentioned maximum principle for holomorphic operator valued functions guarantees that the inequality (9.2) holds inside
r6and in particular
On the other hand, since po
E u(A), there is an element uo such that lluoll = 1 and (A - poI)uo = 0. Then (A,, - poI)uo + 0 strongly in H , which contradicts (9.3). Therefore for any 6 integer N such that for any m
>N
> 0 sufficiently
the inside o f
small there is an
r6contains a spectral point
pm of operator A, and Ip, - pol I6 Therefore p, + po as m -+ m. Now we prove assertions 2) and 3). Let p, + p and p # 0. Then there are elements u,
E H such that llurnll = 1, (A,
- pmI)um = 0. Let us
pass t o the limit in the last equality for a subsequence m' + m such that
urn)+ u strongly in H. Such a subsequence exists due t o the proper conver- p I as m -+ m. Moreover A,Iu,, + Au, since Amurn= A,(u, - u) A,u and IIArnllc(~) 5 C by Lemma 9.4. It follows that (A - pI)u = 0 and therefore p E u(A). Theorem 9.6 is proved. gence of A, - p,I t o A
+
In order t o estimate the difference between eigenvalues and eigenvectors of operators A, and A we shall need the following
III. Spectral problems
372 Lemma 9.7. Let A,
-t
A compactly as m
-+
oo and p , -+ pop where p,, po are eigenA respectively, pm # 0, po # 0. Then
values of the compact operators A,, the following assertions are valid.
1. Let P : H
H be an orthogonal projection on a finite dimensional subspace V c H . Then B, = A, + P - p,I -, B = A + P - poI properly as m
-+
-+
co;the operators B, are of Fredholm type.
2 . Suppose that operator A does not admit root vectors corresponding po and {g,)
is a sequence of vectors of H such that g ,
II(Ak - /imI)gmll -+ 0 as m m -+ oo. Proof. The convergence B,u
-+
oo. Then g ,
-+
-+
0 weakly in H ,
0 strongly in I3 as
Bu as m -i oo follows from the definition of the compact convergence of A, t o A. Obviously -+
where f l ,...,f 5 s an orthonormal basis of V = Im P. If the sequence {B,u,) is compact and llurnll = 1, it follows from (9.4), by virtue of the uniform compactness of A,,
and the convergence p ,
#
0, that the sequence {u,) is also compact in H. Operators B, are of Fredholm type, since B, = ( A , P ) - p,I and operators A,, P are compact. Let us prove the assertion 2. Denote by P E L ( H ) the operator o f orthogonal projection on the space Ker(A*- pol). Then according t o assertion 1of -+
po
+
this lemma we have
properly as m
-+
co,and operators B, are o f Fredholm type. Using assertion
4 o f Theorem 9.1 let us show that B is invertible. Indeed, if Bx = 0, then
x E Im(A - p o l ) n Ker(A - p o l ) = (0). I f x is orthogonal t o Im B , then x E Ker B* = ~ e r ( (- p~o *l ) + P * ) . Note that P is a self-adjoint operator. Therefore (A*- poI)x + Px = 0 and thus
59. Eigenvalues and eigenfunctions of a G-convergent Sequence
373
H, we have I m B = H. Thus, the B is injective and by virtue of the Banach theorem (see [134]) B is
Since I m B is a closed subspace of operator
invertible.
It follows from Lemma 9.5 that for sufficiently large m there exist the inverse operators B;'
and J(BL-llJcH= IJBt;llJIqH)I
C, where C is a
constant independent o f m. Let us now prove the strong convergence g, -+ 0 as m -+ m. We have
as m + co,since by assumption //(A; - p,I)g,II 8
(gm,
?)fit where f l , ...,f a
-+ 0 and Pg,
=
isanorthonormal basisin Ker(A*-pol) and
i=l
therefore Pg, + 0 due t o the weak convergence t o zero o f g,
as m -+ m.
Lemma 9.7 is proved. Let M be a subspace o f H. For u E H set p(u, M) = inf
Ilu - 1111.
vEM
Theorem 9.8. Suppose that A,,
A are compact operators and the sequence {A,) is compactly convergent t o A as m + m. Suppose also that p, -+ po, pm # 0,
PO # 0, p,
E a(A,), po E c ( A ) , and the spectral point po corresponds t o eigenvectors only (not root vectors) o f operator A, urn E N(p,, A,), llurnll= 1. Then the following estimates are valid
III. Spectral problems where the constants
Proof.
C1,C2do
not depend on m
Let PO E a(A), PO # 0, and let el, ..., eS be a basis of the eigenspace
N ( p 0 , A ) It is easy t o see that ((A, - p m ~ ) e i , g m=) 0 for any sequence 9, E H such that gm E N(pm,A',), llgml(= 1. Fix any such sequence. We ' , = ((A - ~ , ) e ' ,9,). It follows that have ((A - p m ~ ) e 9,)
Let us show that we can choose the elements ei(") E N(po,A), so as t o have, for sufficiently large m , the following inequality:
where a is a constant independent of m. Suppose that such a choice is impossible. Then for some subsequence {g,~) we have
Due t o the compact weakness of a ball in a Hilbert space we can assume that
g , ~ -t g weakly in H as m' -+ m. Let us show that g E N(po,A*). We have ((A,. - p,11)~,1,2) = 0 for any s E H. Hence (g,,, (A,, - pm,I)x)= 0. Passing here t o the limit as m' -r co we get (g, (A - poI)x) = 0 for any x E H, and therefore g E N(pO,A*). Let us show that g # 0. Suppose that g = 0. Then according t o the = O implies strong assertion 2 of Lemma 9.7 the equality (A; - p,I)g, convergence g , ~ -t 0 as m' --+ co,which contradicts the fact that I(g,(( = 1. Therefore g # 0. Due t o conditions (9.9) we have
Since A* has no root vectors corresponding t o popthe decompositions (9.1) hold with
T
= A*. It follows from (9.10) that g E Im(A* - pol). We
have shown above that g E N(p0, A*) = Ker(A* - POI),and therefore g E
Im(A* - POI)
n Ker(A* - POI).This
inclusion contradicts the absence o f
root vectors of operator A' corresponding t o jio. Thus the inequality (9.8) is proved.
§ 9. Eigenval ues and eigenfunctions of a G-convergent Sequence
375
The estimate (9.5) follows directly from (9.7) and (9.8). Now we consider the closeness o f the eigenvectors urn o f the operators A, t o those of A. Let
E N(pm, A m )
inf
llumll = 1 , pm =
IIum - uII
-
UEN(PO,A)
We first show that
where uO, are the eigenvectors o f A on which the infimum in the definition of
> 0 is a constant independent of m. It is easy t o see that
p, i s realized, a
Suppose that the inequality (9.11) does not hold. Then one can find a subsequence m' + co such that (A,, - pmlI)[11u$, - ~ , ~ l l - ~ ( u-; ~u,~)] + 0
H as m' + co. Due t o the proper convergence of the sequence {A, - pmI)we have Ilu:, - u m l l l - l ( ~-~urn,) l + u in H. Moreover it is
strongly in
easy t o see that u E N(pO,A). Since
21%
- Iluk -urn()u E N(po,A), we have
where ~ ( m + ) 0 as m = m' + co. This contradiction proves the validity of estimate (9.11). Since 11uk11 5 Ilumll = 1, the relations (9.11), (9.12) yield
where C Let
>0
is a constant independent o f m. Theorem 9.8 is proved.
{L,)be a sequence of differential operators
111. Spectral problems
(aap(E) are smooth 1-periodic in ( functions), which is strongly G-convergent to the operator
as
+ 0.
E
I t is easy t o verify that the results obtained in $8, Ch. II can be obviously extended t o the case o f operators with complex coefficients. In this paragraph the operators
LCare also assumed t o satisfy the inequal-
ities
for any u E C r ( R ) , where C is a positive constant independent o f
E,
u;and
2 is assumed t o satisfy a similar inequality. It is easy t o see that L,,2 considered in $8, Ch. II, if one adds the term pu t o LC and 2 with a sufFiciently large real constant p .
the operator
these conditions hold for the operators Inequalities of type (9.13) for
C,,
guarantee unique solvability o f the
problems
and the estimates
where the constants c l , cz do not depend on
f, E .
We also assume that solutions o f the problem i ( u ) = satisfy the following estimate
f , u 6 Hr(0)
$9. Eigenvalues and eigenfunctions o f a G-convergent Sequence
5 C l l f llHs(n) where
C
is a constant independent of
coefficients of
377
(9.14)
f . Estimate (9.14) is always valid if the
2 and the domain R are sufFiciently smooth.
Let us define
A,, A as operators from L2(R)t o L2(R)mapping a function
f E L2(Q)into the solutions o f the respective Dirichlet problems
where p,, ; ( x ) are bounded (uniformly in e) measurable functions.
L2(R)norm of the difference u, -u = ( A , -A)f . DeB,, B the operators mapping f E L2(R)into the respective solutions
Let us estimate the note by
of the problems
L(u)=f
, UEH~(R).
Then
(A, - A ) f = B,p,f - B ; f = = BE( ( P ,
- ~ ) f +) (BC- B ) b f
.
Let us estimate each term in the right-hand side o f the last equality. We have
I
C2 l l ~ e- bIIH-l,m(n)I l f llH1(n).
Theorem 8.1, Ch. II allows t o obtain an estimate for
( B E- B ) j f . In order
t o apply Theorem 8.1, Ch. II, we have t o check the Condition N' of Section
N,k(x) as follows: 1 k = - , 171 I m ,
8.1, Ch. II. Let us define the functions
1 N , ~ ( x=) - N,(kx) , km E where N , ( [ ) E H 1 ( Q )are 1-periodic in [ solutions of the equations
III. Spectral problems
378
The solvability o f these equations can be proved by the standard method based on the Lax-Milgram theorem. Let us verify Condition N' in our case. We have
1
D: N: ( x ) = N where
7
DaN,(() are smooth 1-periodic functions. Therefore
C
I(D6N,kllLm(n) 5 - for 161 5 m - 1. Thus Condition N 1 l is valid with k = ck-l.
and ak
la1 =
7
We further have
weakly in
L2(R) as k
-
m, where ( f ) =
1
f (()d(, f (() E L1(Q) is 1-
Q
6kp(x)- Gap = f a ~ ( ~ l c = where k z l f(0 are 1periodic in ( and such that ( f a p ) = 0. By virtue o f Lemma 1.8, Ch. I, we
periodic in (.
Therefore
have f a p ( t ) =
C
af'p , where fLp -
are bounded functions I-periodic in (.
&i
Therefore
Ik-I f:p(kx)l 5 k-'C, by the definition o f the norm in H-'tm(R) we 5 C k - l . It follows that &',&) 5 obtain the inequality I16kp - 6aPIIH-~,m Ck-'. Thus we have proved the validity of Condition N'2. Condition N13 follows from the equations for IVY(().We obviously have 7 k = 0. Let us estimate the norm llvkllo which enters the inequality (8.10), Ch. II. Since
We have
$9. Eigenvalues and eigenfunctions of a G-convergent Sequence
379
To obtain the above inequalities we used the following facts: the a priori estimate
for a solution w of the Dirichlet problem; the inequality
the trace estimates for functions in H 8 ( R )(see [117]), the a pm'ori estimate (9.14). Thus we have actually proved Theorem 9.9. For any
f E H 1 ( R ) and
the operators
A,, A
defined above the following
inequality is satisfied
where
C is a constant independent o f E .
Now we can consider the eigenvalue problems
where P,(x), @(x) are functions whose L W ( R ) norms are bounded by a constant independent of
E,
111. Spectral problems
as
E
--+ 0.
We say that an eigenvalue A. of problem (9.16) admits only eigenfunctions,
if the operator A has no root vectors corresponding t o the eigenvalue p o = A.', Theorem 9.10. Let A. be an eigenvalue of problem (9.16) which admits only eigenfunctions. Then there is a sequence{X,,} for the operators C,
o f eigenvalues o f the Dirichlet problem (9.15)
such that A,,
+ Xo as
k + oo and the estimate
holds with a constant C independent o f a. o f the Dirichlet problems (9.15) for operators L,
Eigenfunctions u,,(x)
corresponding t o the eigenvalues ,A,
satisfy the inequality
C1 is a constant independent of
where
E,
M ( x ~ , E ) is the space o f all eigen-
functions of problem (9.16) corresponding t o Xo. If A, eigenvalues of problems (9.15) for C,
-+
A.
# 0 and A,,
are
then Xo is an eigenvalue o f problem
(9.16).
Proof. Let
us first show that the strong G-convergence o f L, t o
J?
implies
the compact convergence of A, t o A . Indeed, let f E L2(R). Then there is a function g E H1(R) such that
11 f - g(lLz(n) 5 a , where a is an arbitrarily
small real number. Obviously
(A, - A)f = (A, - A ) g
+ (A, - A ) ( f - 9) .
Therefore II(A= - A ) f I l ~ z ( n I )
+
II(.A. - A ) g l l ~ z ( n ) Ca ,
C = const
.
The first term in the right-hand side of the last inequality converges t o zero as a --+ 0 due t o Theorem 9.9, and a can be chosen arbitrarily small. Therefore
1I(A,
- A ) fllL~(n) -, 0
f" E
L 2 ( R ) such that
as
E +
11 f'llvcn)
0 for any f E L2(R). For any sequence {A, f } is a compact set
= 1, the sequence
$9. Eigenvalues and eigenfunctions of a G-convergent Sequence
381
L2(R) since IJSt,fCJJHrn(n) 5 C, with C = const independent of E. Hence dE-+ A compactly as E -, 0, and estimates (9.17),(9.18) follow directly
in
from Theorems 9.9, 9.8.
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