Mathematical Problems in Elasticity and Homogenization

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00000___c2aa4dca735d2d884f9faa442f028048.pdf

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00001___425102020a351cc790acd654f6a44d51.pdfMATHEMATICAL PROBLEMS IN ELASTICITY AND HOMOGENIZATION

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00002___b2e53566ce3f55863905498b0aef7e61.pdfSTUDIES 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/0444884416/files/00003___26ddb11238ed5df0cf83ccdec78d85a5.pdfMATHEMATICAL 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

1992

NORTH-HOLLAND AMSTERDAM LONDON NEW YORK TOKYO

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00004___4cb4e71c31e02dcf958b7b5f76b18426.pdfELSEVIER SCIENCE PUBLISHERS B.V. SARA BURGERHARTSTRAAT 25

P.O. BOX 21 1.1000 AE AMSTERDAM, THE NETHERLANDS

Library of Congress Cataloging-in-Publication Data

Olelnik. 0. A. Mathematical problens in elasticity and hoaogenlzation / O.A.

and its:applications ; v. Olelnik. A . S . Shamaev. G . A . Yoslfirn

26 ) p . cn. -- (Studles in nathenatic

Includes bibliographical references. ISBN 0-444-88441-6 talk. paper) 1. Elasticity. 2. Homogenization (D1

I. Shamaev. A. S. 11. Yosiflan. G. A. P A 9 3 1 .033 1 9 9 2 631'.382--dc20

ferential equations) 111. Title. IV. Series.

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ISBN: 0 444 88441 6

0 1992 O.A. Oleinik, A S . Shamaev and G.A. Yosifian. All rights reserved

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Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00005___4e1fc666597519d24fb7fbb8bb4eaace.pdfV

CON TENTS

PREFACE xi

CHAPTER I: SOME MATHEMATICAL PROBLEMS OF THE THEORY OF ELASTICITY

$1. Some Functional Spaces and Their Properties. Auxiliary Propositions

$2. Korns Inequalities 2.1. The First Korn Inequality

2.2. The Second Korn Inequality in Lipschitz Domains 2.3. The Korn Inequalities for Periodic Functions

2.4. The Korn Inequality in Star-Shaped Domains

53. Boundary Value Problems of Linear Elasticity

3.1. Some Properties of the Coefficients of the

3.2. The Main Boundary Value Problems for the System

3.3. The First Boundary Value Problem

3.4. The Second Boundary Value Problem

3.5. The Mixed Boundary Value Problem

$4. Perforated Domains with a Periodic Structure. Extension Theorems

4.1. Some Classes of Perforated Domains 4.2. Extension Theorems for Vector Valued Functions

in Perforated Domains

Elasticity System

of Elasticity

(The Dirichlet Problem)

(The Neumann Problem)

1

1

13

13 14 21

23

29

29

32

33

36 38

42

42

45

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00006___cd6f4ad3771055e632f87e77fbc31c7d.pdfvi Contents

4.3. The Korn Inequalities in Perforated Domains

55. Estimates for Solutions of Boundary Value Problems

of Elasticity in Perforated Domains

5.1. The Mixed Boundary Value Problem 5.2. Estimates for Solutions of the Neumann Problem

in a Perforated Domain

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

6.3. Elasticity Problems with Periodic Boundary Some of the Variables

Conditions in a Perforated Layer

57. Saint-Venant's Principle for Periodic Solutions

of the Elasticity System

7.1. Generalized Momenta and Their Properties

7.2. Saint-Venant's Principle for Homogeneous Boundary

7.3. Saint-Venant's Principle for Non-Homogeneous

Value Problems

Boundary Value Problems

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 a t

Infinity

59. Strong G-Convergence of Elasticity Operators 9.1. Necessary and Sufficient Conditions for the Strong

9.2. Estimates for the rate of Convergence of Solutions of

G- Convergence

the Dirichlet Problem for Strongly G-Convergent Operators

51

55

55

56

59 59

61

64

67

67

71

73

84

84

87

93

98

98

111

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00007___723a9ed0ffd5b86f3ced92d715fdf060.pdfContents vii

CHAPTER II: HOMOGENIZATION OF 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 134

2.1. Homogenization o f the Neumann Problem in a Domain 52 for a Second Order Elliptic Equation with Rapidly

Oscillating Periodic Coefficients 134

for the System o f Elasticity in a Perforated Domain.

Formulation of the Main Results

2.2. Homogenization of the Neumann Problem

140

2.3. Some Auxiliary Propositions 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

3.1. Setting of the Problem

3.2. Formal Construction o f the Asymptotic Expansion 3.3. Justification o f the Asymptotic Expansion.

163

163

164

Estimates for the Remainder 171

54. Asymptotic Expansions for Solutions of the Dirichlet

Problem for the Elasticity System in a Perforated Domain

4.1. Setting o f the Problem. Auxiliary Results

178

178

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00008___3e5e0fd2ccac2b4d6ac1dd930792458f.pdf... Vlll

4.2. Justification o f the Asymptotic Expansion

55. 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 o f the Problem. Auxiliary Propositions

5.2. Justification of 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 of the Homogenized

7.2. Necessary and Sufficient Conditions for Strong

Equations. Estimates of Solutions

G-Convergence o f Operators Describing

Stratified Media

58. Estimates for t h e Rate of G-Convergence o f

Higher-Order Elliptic Operators

8.1. G-Convergence o f Higher-Order Elliptic Operators

8.2. G-Convergence o f Ordinary Differential Operators

(the n-dimensional case)

Contents

185

191

191

197

203

206 206

209

217

220

220

230

245

245

255

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00009___f3678777e8f6b8735d52175df25575f4.pdfContents ix

CHAPTER Ill: SPECTRAL PROBLEMS 263

$1. Some Theorems from Functional Analysis. Spectral Problems for Abstract Operators

1.1. Approximation of Eigenvalues and Eigenvectors of

1.2. Estimates for the Difference between Eigenvalues and

263

Self-Adjoint Operators 263

Eigenvectors o f Two Operators Defined in Different Spaces 266

$2. Homogenization of Eigenvalues and Eigenfunctions of Boundary Value Problems for Strongly Non-Homogeneous

Elastic Bodies 275

2.1. The Dirichlet Problem for a Strongly G-Convergent Sequence of Operators 275

Rapidly Oscillating Periodic Coefficients in a

Perforated Domain 279

2.2. The Neumann Problem for Elasticity Operators with

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

3.3. Homogenization o f a Second Order Elliptic Equation

3.4. Homogenization of Eigenvalues and Eigenfunctions

Order Equation with a Non-Negative Characteristic Form 296

Degenerate on the Boundary 308

of the Dirichlet Problem in a Perforated Domain 313

$4. Third Boundary Value Problem for Second Order

Elliptic Equations in Domains with Rapidly Oscillating

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00010___3ab9232bdc1f4739dfe97ec3e38447f0.pdfX Contents

Boundary

4.1. Estimates for Solutions

4.2. Estimates for Eigenvalues and Eigenfunctions

55. Free Vibrations of Bodies with Concentrated Masses 5.1. Setting of the Problem 5.2. The case --oo < m < 2, n 2 3 5.3. The case m > 2, n 2 3 5.4. The case m = 2, n 2 3

56. On the Behaviour of Eigenvalues o f the Dirichlet

Problem in Domains with Cavities Whose Concentration

is Small

57. Homogenization of Eigenvalues o f Ordinary Differential

Operators

317

317

323

327

327

330

333

340

348

3 54

58. Asymptotic Expansion o f Eigenvalues and Eigenfunctions

o f the Sturm-Liouville Problem for Equations with Rapidly

Oscillating Coefficients 356

s9. On the Behaviour of the Eigenvalues and Eigenfunctions o f a G-Convergent Sequence o f Non-Self-Adjoint Operators 367

REFERENCES 383

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00011___ba21816f9131b5bc8f622e771eb5ad9e.pdfXi

PREFACE

Homogenization o f partial differential operators is a new branch of the the-

ory 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

of non-linear mechanics.

In the field o f partial differential equations the development of the homoge-

nization 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 of purely mathematical problems which are very important for

applications.

The theory o f homogenization o f differential operators and its applications

form the subject of a vast literature. However, for the most part the material

presented in this book cannot be found in other monographs on homogeniza-

tion. 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 anal-

ysis, 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 of 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 of them

are new.

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00012___40815091d42313a88bf325e99f578175.pdfxii 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 c 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 of 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 non- homogeneous 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 to 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-

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00013___fe85e6b2ec18e1dfd885c91c0dfd636c.pdf... Preface Xlll

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 to non-stationary equations.

With the exception o f some well-known facts from functional analysis and the theory o f partial differential equations, al l 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 t h a t 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 dif-

fusion 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 to 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.

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CHAPTER I SOME MATHEMATICAL PROBLEMS OF THE 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 to 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

to the questions o f existence and uniqueness of solutions for boundary value

problems of elasticity in bounded and unbounded domains, and we obtain es-

timates 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 the-

orems can be found in various monographs and manuals (see e.g. [40], [106], [107], [117], [1081).

Points o f the Euclidean space R" are denoted by z = (zl, ..., z,,), y = (yl, ...,yn), E = (rl, ...,(,,) etc.; A stands for the closure in IR" of the set A .

Let R be a domain o f R", i.e. R is a connected open set in R". If not

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00016___d3b7188fcbd28b15bb390f123343e0a6.pdf2 I. Some mathematical problems of the theory of elasticity

indicated otherwise we assume R t o be bounded. For the main functional spaces we use the following notations:

Cr(R) is the space of infinitely differentiable functions with a compact support belonging t o R;

@(a) consists o f functions defined in a and possessing all partial derivatives up t o the order [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 such that the corresponding norms

are finite. For p = 2 we get the Hilbert space L2(R) with a scalar product

(u,'u)o = u(x)v(x)dx ; n

H"(R) (for integer m 2 0) is the completion o f Cm(n) with respect t o the norm

112

IIuIIHm(n) = ( ~ ~ D " u ~ ~ b ( ~ ) ) I (1.1) bl 0 and a family of smooth surfaces S,, T E [0,60], such that S, is the boundary of a domain 0, C 0, 0, 3 R,, if T' > 7, R,-, = 0, c1T L p(z,dR) I c27 if z E S,, T E [0,60], c1 ,c~ = const, R\R, 3 B,.

By virtue o f the imbedding theorem (see Theorem 1.2) we have

7 E [o, 601 7 J 1 v l 2 d ~ L cg llvllgi(n7) L c3 llvllL1(n~ 3 S,

where c3 is a constant independent o f T . Integrating this inequality with respect

to 7 from 0 t o 6, we get

II~II.u(B,) 2 5 ~ 4 6 IIvIIh(n,

This inequality implies (1.12). Lemma 1.5 is proved. 0

Let R be a bounded domain with a Lipschitz boundary. Denote by k(R" x 0) the set o f all functions f((, z) which are bounded and measurable in ( f , x) E R" x R, 1-periodic in f and Lipschitz continuous with respect t o z uniformly in f E R" i.e.

If((, $1 - !( 6 } , we obtain a p a v 1 ( p ( z ) - 6)' IVv12dz = - 2(p(x) - 6) - - vdx - J ax; ax;

'(8) n(,)

Lemma 2.3.

E L2(R). Then w E H'(R) and Let w E Cm(R) n L2(R), p - d'W ax;axj

where the constant C does not depend on w.

Proof. It is easy t o see that for any scalar function f E C'[O, b] we have

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00030___8297a9d1ee6345f37e0513578592c7b5.pdf16 I. Some mathematical problems of the theory of elasticity

Using the mean value theorem let us choose 7 such tha t

f2(r) 5 f 1 f2(t)dt , b / 2 5 r 5 b . bI2

This inequality together with (2.6) yields

where Cl is a constant independent o f f . Let us cover R by the domains Ri, i = 0 , l ) ...) N , such tha t Ro =

{z : p(x,dR) > 6}, 6 = const > 0, and Ri = {x : $i(x') < Xk, <

n, (possibly after an orthogonal transformation o f the variables z), where the

functions $i are Lipschitz continuous and dR n dRi = {x : xk, = $i(X'), x' E Ri}. By virtue of Lemma 2.2 we find that

?+hi(d) + b', 2' = ( 2 1 ,..., Zk,- l , Xki+1, ..., xn), x' E R;}, i = 1, ..., N , 1 I Ic; I

J (Vw12dx 5 no

where Rt'2 is the 6/2-neighbourhood o f Ro, the constant C2 depends only on 6.

Suppose that the domain R, is defined by the conditions: $(d) < xk < $(XI) t bi, x' E 0:. Setting b = bi, f = - , t = xk in (2.7) and considering a W - as a function o f xk, we get from (2.7)

a W

ax d X j

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00031___72849cce1b3d4aa34063fd4d49df755a.pdf2. Korn's inequalities 17

Since +(x') satisfies the Lipschitz condition, it is easy t o see tha t I+(x') + a - xi1 5 Cp(x), where the constant C depends only on the Lipschitz constant for +(XI). Therefore integrating (2.9) over R: and making a tend to zero we find

provided t h a t 6 is chosen sufficiently small. Summing up these inequalities with respect to i from 1 t o N and using (2.8) we obtain

It follows t h a t estimate (2.5) is valid since p(x) 2 6 > 0 in 0;. Lemma 2.3 is proved. 0

Theorem 2.4 (The Second Korn Inequality).

Let R be a bounded Lipschitz domain. Then each vector valued function u E H ' ( 0 ) satisfies the inequality (2.3) with a constant C depending only on 0.

Proof. Obviously we can restrict ourselves to the case of u E C"(fi). By the definition of the matrix e(u) we have - = 2 - eij(u) - - ejj(u)

(there is no summation over i , j) .

a2ui a a ax; axj dX;

Consider the following equations

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00032___1b059ff7af74ace7b049b72b5f988ec4.pdf18 I. Some mathematical problems of the theory of elasticity

Set Fi = 0 outside R, i , j = 1, ..., n. Let v; E Hi(Ro) be a solution o f the equation (2.10) in a smooth domain Ro such that i=l c RO. According t o the well-known a priori estimate we have

n

llvillw(n0) I Ci C IIf'jIIv(n) I Cz (Ie(u)IIL2(n) . (2.11) j=1

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 = (vl , ..., vn)*, w = u - v. Then

A w = O i n R , w E cyn) ,

A(eij(w)) = 0 in R , e;j(w) E Cm(R) , i , j = 1 ,..., n .

Due t o (2.11) we get

where the constant C, does not depend on u. Therefore using (2.4) we find t h a t

It is easy t o see that

aZwi a a a ax,axl ax, 3x1 ax; - e;I(w) + - eip(w) - - el,(~)

Therefore (2.13) yields the inequality

Combining this inequality with estimate (2.5) of Lemma 2.3 we establish

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00033___eca453ad7568fb40276b6e7cafaac005.pdf$2. Korn's inequalities 19

Since w = u - v the above estimate implies

IlvullLw I ca(lle(4ILzca) -t IIullv(u) + I lV l lH l cn ) ) *

Therefore owing t o (2.11) we find t h a t (2.3) is satisfied. Theorem 2.4 is proved. 0

In applications it is often important to have another version o f the Second Korn Inequality, namely the inequality

which holds for v belonging t o a subspace V of H'(R). Subspaces V of t h a t kind will often be dealt with below.

Denote by R the linear space of rigid displacements o f R", i.e. the set of a l l vector valued functions 17 = (ql, ..., 7,) such t h a t 7 = u + A s , where a = (u l , ..., u,,) is a vector with constant real components, A is a skew- symmetric (n x n)-matrix with real constant elements. Here 7 , a, z are

column vectors.

It is easy t o see tha t 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 n R = {0}, where R is the space of 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 t h a t

IlvmllHl(n) = 1, Ile(vrn)ll;2(n) -+ 0 as m + 00 ' (2.15)

Since the imbedding H'(R) c L2(R) is compact (see Theorem 1.2), it follows that there is a subsequence mj + 00 such that for some v E L2(R) we have urn' + v in L2(R). According to Theorem 2.4 the Second Korn Inequality (2.3) is valid in R , and therefore

IIurn+P - "rnllkl(n) I

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00034___a60a9e73aec0d51c97932b5b2bfdda6a.pdf20 I. Some mathematical problems of the theory of elasticity

This estimate and (2.15) show that urn) -+ v in N'(R) as mj + 00. Since V is a closed subspace of H'(R), by virtue of (2.15) we conclude that

The last equality implies that

avi avh -+ - = o , axh axi i ,h =1, ..., n (2.16) Les us show that any v satisfying (2.16) belongs t o R.

Consider the mollifiers for v:

tJC(X) = J p(?)v(y)dy , Ix -v l lc

where v = 0 outside R , p(6) E Cr(Rn), ~ ( 6 ) L 0, / cp(t)d( = 1, ~(6) = 0 for161 2 1. One can easily verify (see e.g. (1171, [311) that vc E C"(G) and oc + v in H'(G) as E -+ 0 for any subdomain G such that G c 52.

m

It follows from (2.16) that for sufficiently small E

avf av;, axh ax, - + - = O in G , i , h = l , ..., n .

Since the vc are smooth in G these equations imply azv;

in G . -~ - a2v;, - a 2 v i - a 2 q axbash axiaxh axiaxk axhaxk Therefore vf = a4.x. + bf, where at j , bf are constants such tha t atj = - a f i .

13 J

Due to the convergence o f vc t o v in H'(G) as E + 0 we have v E R. Thus v E V n R, I I v I I H I ~ ~ ) = 1, which is in contradiction with the condition

0 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 H'(R) : (v,q)Hl(n) = 0 vq E R} , or

v = {v E H'(R) : (?J,77)L2(*) = 0 vq E R} .

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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

to obtain estimates for these solutions.

Theorem 2.7.

Let R be a bounded domain with a Lipschitz boundary. Suppose t h a t the set y c dR can be represented in the form x, = cp(3), where 3 = ( 2 1 , ...,x,- 1) varies over an open subset of lan-', cp(3) is a Lipschitz continuous function. Then each vector valued function v E H'(R,y) satisfies the inequality (2.14).

Proof. If we show that H'(R,y) n R = {0}, then in order t o obtain (2.14) we can use Theorern 2.5 with V = H'(R,y).

Let r] E H'(R,y) n R. Therefore 77 = 0 on y. Every rigid displacement has the form r] = b + Ax, where A is a skew-symmetric matrix with constant elements, and b is a constant vector. Since the system Ax + b = 0 is linear, it is obvious that the ( n - 1)-dimensional surface y = {x : x, = cp(3)) must belong to a hyperplane, provided that A # 0. Therefore the dimension of the space formed by a l l solutions o f system Ax + b = 0 is not less than 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 a l l elements on the main diagonal of A vanish, the coefficients 0 by xl , ..., x, vanish, too. Hence r] = 0. Theorem 2.7 is proved.

2.3. The Korn Inequalities for Periodic Functions

Here we establish the Korn inequalities similar t o (2.14) for 1-periodic vec-

tor valued functions.

Theorem 2.8. Let w be an unbounded domain with a 1-periodic structure and let w n Q be a domain with a Lipschitz boundary. Then for any v E l@..(w) such that

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1 v d x = O , Q=]O,l[", (2.17) wnQ

the inequality

holds with a constant C independent of v.

Proof. Denote by V the linear space consisting of a l l restrictions t o w n Q of vector valued functions in w i ( w ) satisfying the conditions (2.17). It is easy t o

see that V is a closed subspace o f H ' ( w f l Q) and that any rigid displacement 1-periodic in x is a constant vector. Therefore if v E V n R then by virtue of (2.17) we have v = 0. Now Theorem 2.5 for R = w n Q yields the inequality (2.18). Theorem 2.8 is proved. 0

The Second Korn inquality of type (2.14) for functions 1-periodic in 3i. =

(xl , ..., x,,-') is the result of

Theorem 2.9. Le t w be an unbounded domain with a 1-periodic structure and le t the do-

mains w(u ,b ) , Lj(u,b) (0 < a < b < m) be defined by (1.6). Suppose t h a t 3 ( u , b ) has a Lipschitz boundary. Then for any vector valued function

v E &'(w(u, b ) ) such that 1 v dx = 0 the following inequality holds 44)

where c is a constant independent of v.

(2.19)

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 i is also a constant vector.

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2.4. The Kern Inequality in Star-Shaped Domains

In many applications it is important t o know the nature o f the depend

23

nce

of the constants in Korns inequalities on the geometric properties of the do-

main. 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.

Korns inequalities in unbounded domains and some more general inequali-

ties of that type for the norms in LP(R) and in weighted spaces were considered

in [42], [43], [68], [46].

A domain R is said t o be star-shaped with respect to a ball G belonging to R, if the segment connecting any two points x E G, y E 51 lies in R.

Theorem 2.10. Suppose that R is a bounded domain o f diameter R and R is star-shaped with respect to t h e ball QR1 = {x : 1x1 < Rl}. Then for any u = (uI , ..., u,) E H(R) we have the inequality

(2.20)

where C1, Cz are constants depending only on n.

Proof. Obviously it is sufficient to prove (2.20) for smooth vector valued functions ti(.). Let R1 = 1. By Cj we denote here constants which can depend only on n . Let v = (wl, ..., v,) be a solution o f the system

(2.21)

with the boundary conditions

v i = 0 on dR , i = 1, ..., n . (2.22)

Multiplying (2.21) by v, and integrating by parts in R the resulting equality, we find t h a t

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(2.23) n

Set w = u - v. For any smooth valid

V = (Vl, ..., Vn) the following identities are

a2v, - a a a - eiq(V) + - eip(V) - - ax; e,,(V) ,

ax,ax, ax, 8% i , p , q = 1 ,..., 12 I

Therefore due t o (2.24), (2.21) we have

A w = O in R ,

Aeij(w) = 0 in R , i , j = 1, ..., n .

It follows from (2.23) that

(2.24)

(2.25)

(2.26)

(2.27)

Therefore by virtue of (2.26) and Lemma 2.2 we get

where p = p ( x ) is the distance from x E R to dR. It follows from (2.28) and (2.24) that

(2.29)

Let us apply the following inequality

p f 2 ( t )d t I c ( I t 2 ( f W + a f 2 ( a ) ) , (2.30) 0

where C is a constant independent of a and f. The proof of (2.30) follows immediately from (2.6).

Let us apply (2.30) to the function f = dw;/dej and the segment AP

belonging t o the segment OP, where P is any point on aR, 0 is the origin. Considering P as the origin, we obtain

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00039___c3fead415b75c7b4e205aadbaa73da58.pdf52. Korn 's inequalities 25

+ I P - A ~ I V ~ ( A ) I ~ . 1 Let us choose the point A such that A E QR], (A1 = X E [i ,1],

(2.31)

J I V ~ ( A ) I ~ ~ ~ I ~9 J ~ v w ( t ) ~ ~ d s , (2.32)

where dw is the area element on the unit sphere. Such a choice of A is possible

due t o the mean value theorem. Obviously (2.31) implies

IAI=X Q R1

+ IP-AlIVw(A)I ' . (2.33)

Let us integrate (2.33) over the unit sphere. Since the domain R is star- shaped with respect to Q R ~ with R1 = 1 it follows that IP - t/ < p ( x ) R . Therefore (2.32), (2.33) yield

J lVwl'dx 5

1

Q\QR,

+ R" / (Vw(x)12dx . Q R1 1 (2.34)

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Estimate (2.20) with R1 = 1 follows from (2.23), (2.29), (2.34), since w = u - u. The inequality (2.20) with any R1 > 0 can be obtained from

0 (2.20) with R1 = 1, if one passes t o the variables y = x / R 1 .

Remark 2.11. The coefricient by the second term in the right-hand side of (2.20) i s asymptot-

ically exact and cannot be improved in the following sense. Let u = A x + 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/Rl)n, provided that R has the volume of order R.

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 coefkient 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

Korns type are valid

In order t o prove (2.35) we should use the inequality

(2.36)

3af4 p fZ ( t ) tPd t 5 c [j ( t - U ) Z t P ( f ) Z d t + up+l J ( f p p d t + 0 0

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1

+ ap+l J tPp(t)dt] , 0

a = const > 1, p = const > 1 , (2.37)

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

3a/4 j t f ' ( t ) d t 5 c ] (t - a)*t(f')'dt + a2 in4a J ( f ' ) ' t d t + 0 L4 0

+ a ' / t f 2 ( t ) d t ] , a = const > 1 , 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 vec-

tor valued function u = $ ( A z + B ) , where A is a constant skew-symmetrical matrix, B is a constant vector, $ E C m ( R n ) , G(z) = 0 in QR, , $(x) = 1 outside of Q ~ R ~ = {z : 1x1 < 2R1}, Q ~ R ] C R. Then (2.36) (in the form of an equality) holds for u(z ) with the coefficient C1(R/R1)" by the first integral in the right-hand side, provided that R has the volume of order R".

Theorem 2.14.

Suppose that R satisfies t h e conditions of Theorem 2.10 and u E H'(R). Then

(2.38)

where y is the distance of QR] from dR.

Proof. Let 'p E CF(R), 'p = 1 in Q R , , 0 5 'p 5 1 in R. Then according t o Theorem 2.1 we have

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It follows t h a t

IIV4zL2(QR,) I 2 ll+)IlzL2(*) + C3Y2 ll41zLqn) . (2.39) 0 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.

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$3. Boundary Value Problems of Linear Elasticity

3.1. Some Properties of the Coeficients of the Elasticity System

In a domain R c R" consider the differential operator o f linear elasticity a au

t ( u ) = - (Ahk(.) -) . ax h axk

Here u = ( u l , ..., u,) is a column vector with components u l , ..., u,, A h k ( x ) are (n x n)-matrices whose elements a f / ( x ) are bounded measurable functions

such t h a t

where { v i h } is an arbitrary symmetric matrix with real elements, x E 0,

K ~ , 1c2 = const > 0. We say t h a t a family of matrices Ahk, h,lc = 1, ..., n , belongs to class

E ( K ~ , t c 2 ) , if their elements uf' 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 ~ , K Z ) . The operator L defined by (3.1) can also be written in coordinate form as

follows

a au L i ( U ) E - (.I;"(.) -) , i = 1, ..., n .

axh axk (3.4)

In the classical theory of linear elasticity for a homogeneous isotropic body

the coefficients o f operators (3.4) are given by the formulas

= Xbihbjk + pbijbhk + pbikbhj where X > 0, p > 0 are the Lam6 constants, b;, is the Kronecker symbol: b,j = 0 for i # j , 6,j = 1 for i = j . In this case we have

af)qihr]jk = Xqhhvii + 2pqihvih (3.5) for any symmetric matrix { v i h } . Moreover, the family of the matrices Ah',

h,k = 1, ..., n, belongs t o the class E(2p,2p +nX). Indeed. it is obvious that K~ = 2p, and the estimate K~ 5 2p + nX follows from (3.5), since

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(a1 + ... + u,)2 5 n(u? + ... + u:) . Thus the elasticity operator corresponding t o a homogeneous isotropic body

has the form

where aZUr

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 fre-

quently used below.

With each family o f matrices Ahk(x) of class E(nl , K * ) for any fixed x we associate a linear transformation M of the space of (n x n)-matrices, which maps a matrix ( with elements ( j k into 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.

Let A h k , h , k = 1, ..., n, be a family of matrices o f class E(nl , K ~ ) . Then for any ( n x n)-matrices ( = { ( i h } , Q = { v ; h } with real elements the following conditions are satisfied

Proof. By virtue of the first inequality in (3.2) we obtain that

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00045___88ec8a722823e50b00ee09637b577f56.pdf$3. Boundary value problems of linear elasticity 31

hk kh (M[j 71) = a i j [ i h q j k = a j ; [ i h q j k = a;:[ jkVih = ([, M7) . Due t o (3.3) and (3.6) the bilinear form ( M [ , T ) 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

1 ( M t , v ) = 4 ( M ( t + t . 1 9 7 7 + v*) I y It + E * l h + 7*l *

It follows from (3.2) and (3.3) for 7 = ([ + [*) tha t

~i It + < * I 2 I ( M ( t + E * ) , E + E * ) = 4(MJ,t) . Lemma 3.1 is proved.

for a fixed [ # 0. If J ( 7 ) = 0 it follows from (3.9) that

Multiplying each of these equations by [ i v h and summing with respect t o i, h from 1 t o n we obtain Itiv;12 + 1[I2(vl2 = 0. Therefore 7 = 0. Thus J ( 7 ) > 0 for 7 # 0. Lemma 3.2 is proved. 0

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3.2. The Main Boundary Value Problems for the System of Elasticity

Let L: be an elasticity operator o f type (3.1) belonging t o class E ( n l , Q), nl, n2 > 0, and let R be a bounded domain o f R" occupied by an elastic body. The displacement vector is denoted by u = ( u l , ..., tin)*.

the theory of linear elasticity.

The following boundary value problems are most frequently considered in

The j rs t boundary value problem (the Dirichlet problem)

C(u) = f in R , ti= @ on 8 0 , (3.10)

involves finding the displacement vector u a t the interior points of the elastic

body for the given displacements u = @ a t the boundary and the external

forces f = (fl, ..., f,,) applied t o the body.

The second boundary value problem (the Neumann problem)

L(u) = f in fl , (3.11)

i.e. a t the points of the boundary the stresses a(u) = 'p are given. Here v = (vl, ..., vn) is the unit outward normal t o dR.

The third boundary value problem (the mixed problem)

(3.12) 1 L(u) = f in Q , u = @ on r , o(u )= 'p on S . It is assumed here that the boundary do of R is a union o f two sets r and S such that r n S = 0.

In order t o prove existence and uniqueness of solutions of these problems, it

is necessary t o impose certain restrictions on dR, r, S, which will be specified below.

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In 56 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 l , ..., un) be the displacement vector and let e (u ) be the corresponding strain tensor, i.e. e (u ) is a matrix with elements e i j ( u ) =

1 d ~ i duj 2 axj axi

= - (-+ -). Set

Then taking into account (3.7), (3.8) for t = Vu, [* = (Vu)*, we find

K;' l&(u)IZ I le(u)12 I K;' I&(u)12 . (3.13)

3.3. The First Boundary Value Problem (The Dirichlet Problem)

Let R be a bounded domain o f R" (not necessarily with a Lipschitz bound-

We say that u ( r ) is a weak solution o f the problem

ary), fj E L2(R), j = 0,1, ..., n, 'p E H'(R).

(3.14) af' ax;

L(u)=j+'+- in R , u = ' p on 80,

if u - cp E HA(R) and the integral identity

holds for any w 6 Ht(R).

Theorem 3.3. There exists a weak solution u(z) of problem (3.14), which is unique and

satisfies the est i m at e

(3.16)

where the constant %(a) depends only on K ~ , K~ in (3.3) and the constant in the Friedrichs inequality (1.2) for 7 = 30.

Proof. It follows from (3.15) t h a t 20 = u - 'p must satisfy the integral identity

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00048___b31343410a64f523313ddb1181082da4.pdf34 I. Some mathematical problems of the theory of elasticity

for any v E H,1(R). Note t h a t due t o the Friedrichs inequality (1.2), the First Korn inequality (2.2) and estimates (3.13) the quadratic form

au av a ( u , v ) = ( A h k ( x ) a, -)dx , u , v E Hi(sZ)

xk axh n

satisfies the conditions of Theorem 1.3, if we take as H the space o f a l l vector valued functions with components in H i ( f l ) .

Obviously the right-hand side of (3.17) defines a continuous linear func-

tional on v E H t ( 0 ) . Therefore by Theorem 1.3 there is a unique element w E Hi(R2) satisfying the integral identity (3.17). Setting u = w + 'p we obtain the solution o f the problem (3.14).

Let us prove the estimate (3.16). Set w = u - 'p, v = u - 'p 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 C, depends only on I C ~ , ~2 and the constant in (1.2). Since I llull - ll'pll I 5 IIu - 911, the estimate (3.18) implies (3.16). Theorem 3.3 is proved. 0

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The details, concerning the smoothness o f the solutions obtained in Theo-

rem 3.3, are given a thorough consideration in the article [17] which contains in

particular the proof o f the fact that the smoothness of do, the data functions f', 'p and the coefficients of C guarantee the smoothness of the weak solution u ( z ) of problem (3.14).

Denote by H-'(R) the space o f continuous linear functionals on the space o f vector valued functions with components in HA(R). As usual the norm in

H-'(R) is defined by the formula

I l f I l H - l ( n ) = {If(v)l,v E H:(R),IIvIIHA(n) = l} . U

It follows from the proof of Theorem 3.3 that

defines a continuous linear functional on H ; ( o ) , namely

(3.19)

for any v E H,'(R). We obviously have n

I l f ( l H - l ( n ) 5 c ( I fm\lL2(n) 7 c = const . m=O

On the other hand, for any f E H-'(R) there exist functions f" E L2(R), m = 0, ..., n, such that

af' f = f O t -

f 3 X i (3.20)

in the sense of the integral identity (3.19), and

c I l f i l l L 2 ( * ) I c1 IlfllH-l(n) 1 Cl = const . (3.21) Indeed, by the Riesz theorem (see [107]), every continuous linear functional

f(v) on HA(R) can be represented as a scalar product in H,'(R), i.e. there is a unique element u E Hi(R) such that

i = O

J (Vu, Vv)dz t (u, v)dx = f ( v ) J n n

(3.22)

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Setting v = u in (3.22) and taking into consideration the definition of the norm in H-l(R), we find that

(3.23)

Setting fo = u, f = - e , by virtue of (3.22), (3.23) we obtain the repre- sentation (3.20) and the estimate (3.21).

Remark 3.4.

In the special case when p = 0 in (3.14), we can consider the problem

L(u) = f , 11 E H i p ) (3.24)

for any f E H-(R), since f can be represented in the form (3.20). Then by Theorem 3.3, due to (3.21) we have

where the constant C depends only on K ~ , ~ 2 , 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 to be a bounded domain with a Lipschitz Let Sl be a subset of dR with a positive ( n - 1)-dimensional boundary.

Lebesgue measure on dR. Set

hk O(U) 2 v h A (Z) - . ax k

We say that U ( Z ) is a weak solution of the problem

af d X i

L (u) = fo + - in R , O(U) = cp + vif on Sl , ~ ( u ) = vif on dR\& ,

(3.26)

(3.27)

where fj E L2(R), j = 0, ..., n, p E L2(S1), if the integral identity

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00051___b0dfa65f33a1cbbe4b0d3e286119716b.pdf$3. Boundary value problems of linear elasticity 37

(3.28)

holds for any v E H1(R). Note that if dR, fj, cp, Ahk are not smooth, the boundary conditions in

(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

for any v E H'(R), which follows from Proposition 3 o f Theorem 1.2.

Theorem 3.5. S u p pose that

(3.30)

for any rigid displacement q E R. Then there exists a weak solution u(z ) of problem (3.27). This solution is unique (to within an additive rigid displace-

ment) and satisfies the inequality

(3.31)

Here the constant ~ ~ ( $ 2 ) depends only on nl, n2, the constants in (3.29) and in (2.14) when V is a closed subspace o f H1(R) orthogonal t o R with respect t o the scalar product in L2(R) or H1(R). Proof. Let 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

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the conditions of Theorem 1.3. Thus there is a unique element u E H such that the integral identity (3.28) holds for a l l w E H . For w E R the left-hand side of (3.28) is equal to zero due to the fact that L ( v ) = 0 in R , ~ ( v ) = 0 on dR; the right-hand side of (3.28) is also equal to zero for w E R, since we have assumed that conditions (3.30) are satisfied. Therefore the integral identity (3.28) holds for al l o E H1(R), which means that u(z) 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. 0

Remark 3.6. In Theorem 3.5 we can choose a solution u(x) orthogonal in L2(R) or H'(R) to the space of rigid displacements R. For such u ( z ) we have the following est i mate

(3.32)

where the constant C2(R) depends on the same parameters as the constant Cl(R) in (3.31). This fact is due to the Second Korn inequality (2.14) (see Theorem 2.5).

Remark 3.7.

Similarly to the case of the Dirichlet problem one can prove the smoothness of

weak solutions of the Neumann problem, provided that the coefficients a f / ( x ) , the boundary of R, and the data 9, p, i = 0, ..., R, in (3.27) are smooth (see ~ 7 1 ) .


In a bounded domain R c R" we consider the following boundary value problem for the operator C of class E(n l , n2), 61, n2 > 0:

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00053___5291fadf852c7a3ffe937595f857abe0.pdf53. Boundary value problems of linear elasticity 39

u = i p on y , J where f j E LZ(R), j = O , l , ..., n, cp E L Z ( S l ) , @ E H'/'(y), v = (v1, ..., v,) is the unit outward normal to dR.

Before giving a definition of a solution of the mixed problem we impose

the following restrictions on a R , y, S1, Sz.

1. dR = 7 U Sl U Sz and y, S1, Sz are mutually disjoint subsets of dR.

2. Q is a domain with a Lipschitz boundary XI, 7 contains a subset satisfying the conditions of Theorem 2.7.

Note t h a t al l further results are also valid under weaker assumptions on 80

We define a weak solution of problem (3.33) as a vector valued function

and y which guarantee the inequalities (1.2), (2.14).

u E H'(R) satisfying the integral identity

(3.34)

for any v E H'(R,y), and such t h a t u = ip on y (i.e. u - @ E H1(R,y)). Note that by the definition o f H1/'(y) we can consider ip as an element of

H'(R).

Theorem 3.8. There exists a weak solution u(z) of problem (3.33). This solution is unique

and satisfies the estimate

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00054___a3a185415766f990e027ec55879078e2.pdf40 1. Some mathematical problems of the theory of elasticity

where the constant C(R) depends only on 6 1 , 6 2 , the constant in (3.29) and the constants in the Korn inequality (2.14) for vector valued functions in

H'(R,y) (see Theorem 2.7).

Proof. From (3.34) we conclude that w = u - @ must satisfy the integral identity

(3.36)

for any v E H'(R,y). Due to Proposition 3 of Theorem 1.2 the inequality (3.29) holds for a l l v E H'(R,y), and according to 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 al l assumptions of Theorem 1.3 with H = H'(R,y). By virtue of (3.29) the right-hand side of (3.36) defines a continuous linear

functional on H'(R,y). It follows from Theorem 1.3 that there is a unique element w E H'(R,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

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00055___f3f794e2a0f42297aa5ba10761dae5ee.pdf$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 & such

that 9- 6 E H'(R,y), and this would not affect the constant C, which does not depend on 9.

Thus by the definition of the norm in H' / ' ( y ) we obtain (3.35) from (3.37). Theorem 3.8 is proved. 17

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00056___19f191ff89340f950d55fbcf4c72ae16.pdf42 I. Some mathematical problems of the theory of elasticity

$4. Perforated Domains with a Periodic Structure. Extension Theorems

4.1. Some Classes of Perforated Domains

Let w be an unbounded domain of R" with a 1-periodic structure, i.e. w is invariant under the shifts by any z = (zl, ..., z,) E Z".

Here we also use the notation:

Q = { z : O < s j < l , j = l , ..., n } ,

p(A , B ) is the distance in R" 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 R" with a 1-periodic structure.

B2 The cell o f periodicity w n Q is a domain with a Lipschitz boundary.

B3 Theset Q\W and the intersection o f Q\w with the 6-neighbourhood (6 < i) of aQ consist of a finite number of Lipschitz domains separated from each

other and from the edges of the cube Q by a positive distance.

W ~ Q w

Fig.,

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00057___f6ae13976d060302c03abe93a31fbbfc.pdf$4. Perforated domains with a periodic structure 43

We shall consider two types of bounded perforated domains R" with a periodic structure characterized by a small parameter e.

A domain 0' of type I ha5 the form (see Figs. 1, 2 , 3):

RE = R ~ E w , (4.1)

where R is a bounded smooth domain o f R", w is a domain with a 1-periodic structure satisfying the Condition B; R' is assumed to have a Lipschitz boundary.

R

Fig.

Fig..

The boundary of a domain R" of type I can be represented as doc = I',uS,,

A domain R' of type ZZ has the form (see Figs. 4, 5a, 5b): where rc = dR n E W , S, = (ane) n R.

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00058___4b5abc00bb94da93bfb837e53f868bdb.pdf44 I. Some mathematical problems of the theory of elasticity

where R is a bounded smooth domain,

T, is the subset of Z" consisting of all z such t h a t

E is a small parameter.

Fig. 5a.

Ql

Fig. 5b.

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00059___8dd40f3c858e1f65e8959dd4e8a59712.pdf$4. Perforated domains with a periodic structure 45

We assume that R1, R;, R' (the sets of interior points o f !=ll, !=li, a') are

The boundary 8R' of a domain R' of type I I is the union o f dfl and the bounded Lipschitz domains.

surface S, c R of the cavities, S, = (dV) n R.

4.2. 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 0' we shall construct extensions t o R of vector valued functions defined in R' and prove some inequalities (uniform in E ) for these extensions.

Lemma 4.1.

Let G c V c R" and let each of the sets G, V, V\G be a non-empty bounded Lipschitz domain (see Fig. 6). Suppose t h a t y = (8G)nV is non-empty. Then for vector valued functions in H'(V\G) there is a linear extension operator

P : H'(D\G) + H ' ( V ) such that

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00060___a8a0f84533e3be1b66e3aca91ebffc30.pdf46 I. Some mathematical problems of the theory of elasticity

Proof. Let us first show that each w E H'(V\G) can be extended as a function 6 E H ' ( D ) satisfying the inequality

116llH'(.D, L CIIwIIH'(D\a) (4.9)

with a constant c independent o f w.

Indeed, consider the ball B c R" containing a neighbourhood o f the set V . According t o Proposition 2 of Theorem 1.2 the function w can be

extended from V\G to the entire ball B as a function w 1 E H'(B) . Taking the restriction of w 1 on D we get a function 6 which satisfies the inequality

Denote by W the weak solution o f the following boundary value problem

(4.9).

for the system of elasticity

(4.10) 1 C(W) = 0 in G , W = w on V D a G , a(IV)=O on a G n a V , where C is an arbitrary operator o f class E ( K ~ , ~ 2 ) with constant coefficients. Note t h a t the last boundary condition in (4.10) should be omitted if aGnaD = 0. By Theorem 3.8 W exists and satisfies the inequality

IIWIIH'(G) 5 llG1lH'(G)

Therefore due t o (4.9) we obtain

Set

W(Z)

W(Z) for z E G . for z E V\G ,

P(w) = (4.12)

It is easy to see that P(w) is a vector valued function in H'(D) . By virtue of (4.10) we have Pv = 17 for any 7 E R. Taking into account (4.11) and the Korn inequality (2.3) in D\G (see Theorem 2.4) we conclude that estimates

(4.5), (4.6) hold with constants c1, c2 depending only on G and V. Let us prove the estimate (4.8) for Pw. Suppose tha t (4.8) does not hold.

Then there is a sequence o f vector valued functions vN E H1(D\G) such that

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00061___458c7233b3f150561c9c0cc3c3f17137.pdf54. Perforated domains with a periodic structure 47

IIPVNIIH'(D) I c1 II"NIIH'(D\o) 7 (4.13)

Ile(PvN)IIL2(D) 2 Ile(vN)IIL2(D\G) I (4.14)

(4.15)

Without loss o f generality we can assume t h a t (vN,r])da: = 0 for any rigid

displacement q , since P(v t r ] ) = Pw + r] due t o (4.10), (4.12), and for any bounded domain wo and any v E H'(w0) we have Ie(v + q)IZdx =

but

1 Ile(vN)IILz(D\G) = .

J D\G

J wo / le(v)12ds. By (4.15) and the Second Korn inequality (2.14) in V\G (see

Corollary 2.6) we get wo

N C 112) llHqD\G) I c lle(vN)llL2(D\G) = .

Thus vN -+ 0 as N -+ m in H1(D\G), and therefore IIPvNIIH1p) -t 0 as N -+ co due t o (4.13). On theother hand, (4.14) implies t h a t Ile(PwN)llLz(D) 2 1. This contradiction establishes the inequality (4.8).

J P ( w + C)dx = 0. Because of the Poincare inequality (1.5) in D\G it D\G follows from (4.5) t h a t

To prove (4.7) we choose a constant vector C such t h a t

IIVP(w + C)IILZ(D) 5 I KO [IIPb t C)lILZ(D\G) + IIVP(w + C)llL2(D\C)] I 5 K 1 JIVP(w + C)IILzp\c) , K O , K 1 = const .

Therefore (4.7) is valid since VC = 0, PC = C. Lemma 4.1 is proved. 0

Theorem 4.2 (Extension o f functions in perforated domains of type 1 1 ) . Let 0' be a perforated domain of type II. Then for vector valued functions in

H'(R') there is a linear extension operator P, : H'(R') -+ H'(R) such tha t

p c q = q , b E R , (4.16)

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00062___54757c86a21257da902d3c1ec6afabd8.pdf48 I. Some mathematical problems of the theory of elasticity

(4.18)

for any v E H'(R'), where the constants cl, ..., c4 do not depend on E , v. Proof. Let v(z) E H1(Rc). Set V(() = v ( E [ ) and fix z E Tc, where T, is the index set in the definition o f a perforated domain RE o f type II (see (4.3)). Consider the function V([) in the Lipschitz domain w n ( z + Q ) . By Lemma 4.1 one can extend V(() as a vector valued function PIV E H ' ( z + 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 T, with constants KO, ..., K3 independent o f z .

If the distance between Q\G and aQ is positive (i.e. Q\G lies in the interior

of cube Q ) , then the function (PIV)(z) is the extension whose existence is

asserted by Theorem 4.2, and therefore we can take (Pcv)(x) = (PIV)(-), where V(() = w ( E [ ) .

However, if Q\a has a non-empty intersection with aQ (as in Fig. l), the function PIV(J) may not belong t o H'(&-'R), since its traces on the

X

E

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00063___d47e9c0fd790595556ab06ef043ad3a1.pdf$4. Perforated domains with a periodic structure 49

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.

n

For 1 = 0 , l set &Q = U {[ E aQ, [k = I } . Due t o Condition 63 on w the intersection of the 6-neighbourhood of aQ

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 2 = 0 and 1 = 1 denote by 7;) ...,&, those o f the domains just mentioned whose closure has a non-empty intersection with a,Q (see Fig. 7). Therefore each 7: lies in the 6-neighbourhood of aQ and is adjacent to a face o f

k = l

lying on the hyperplane [k = I for some k.

Fig. 7 -

Let the domain R1 and the set T, E Z" be 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 (7: -t z) n ~ ( E - ~ O ~ ) # 0 for some j = 1, ...) ml.

The extension PIV([) constructed above is such that PIV E H'(g) for any open g C e-'R which has no intersection with any of the domains 7: + z, z E T,, 7: + z , z E T,'. Let us change P1V in these domains so as t o obtain a function in H'(&-'R).

Fig.. The domains G1, ...) GN are shaded pale.

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00064___8541302cd2d1b12d2cd106655140c2dc.pdf50 I. Some mathematical problems of the theory of elasticity

Denote by GI, ..., GN all mutually non-intersecting domains having the form either $ + z , z E T, or 7: + z , z E T: (see Fig. 8). Obviously p(G,, Gt) > b1 for s # t . The number N tends t o infinity as E t 0, however, G1, ..., GN are the shifts o f a finite number o f bounded Lipschitz domains. Consider the extension PlV(t). We have constructed the sets GI, ..., GN in such a way t h a t the set bG1 U ... U dGN contains a l l those parts o f the faces of the cubes z + Q , z E Tc, 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 Gj the b1/2-neighbourhood o f Gj.

By virtue of Lemma 4.1 let us extend PIV to each of the sets Gj as a vector valued function P2V satisfying the following inequalities

IIp2vllH1(C,) I IIPlVllHl[6,\G,) >

IIp2v11L2(G,) + Ile(P2V)IIL2(G,) I I M2 ( I I Pl v I I LZ (8, \G, ) + I I 4 Pl v 1 I I L2 (6, \G, ) ) , } (4.22)

I Ilv,p2vllL2(c,) I M3 lIVEPlVIlL2(6,\G,) ? Ilec(~zV)IlL~(c,) 5 M4 IIedPlV)IILZ(C,\G,) 7 and such that P277 = q if 77 E R, where the constants Ml, ...,A& do not depend on V, j .

Set U ( ( ) = (PIV)(() for ( E (E- 'S~)\G~, U ( ( ) = ( P 2 V ) ( ( ) for ( E GO. Applying the estimates (4.21), (4.22) we finally conclude that U ( - ) can be taken as the extension (P,v)(z) satisfying the conditions (4.21). Theorem 4.2

2

&

is proved. 0

Theorem 4.3 (Extension of vector valued functions in perforated domains of

Let R' be a perforated domain o f type I and let Ro be a bounded domain such that fi c 00, p(dR0,R) > 1. Then for every sufficiently small E there exists a linear extension operator P, : H1(S1',rC) -+ HA(Ro) such t h a t

type 1 ) .

(4.23)

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00065___d530f28af548ae666575e6576090a8ad.pdf$4. Perforated domains with a periodic structure 51

(4.25)

where the constant C1, C,, C, do not depend on E , u. Moreover, (Peu)I, = 0 for any open g such t h a t g c Ro\52, if E is suffi-

ciently small.

Proof., Denote by T, the set of all z E 22" such that ~ ( z + Q n w ) n 52 # 0. Let @ be the interior o f ~ ( z + Q n LO), and let fil be the interior of

E ( Z + 0). For each u E H1(52c,I'c) we introduce the following vector u

zET~

u P E T ,

valued function

u ( z ) , 2 E 52' ,

0 , 2 E f i ; \ R ,

0 , 2 ERO\f i l . It is easy t o see t h a t U ( z ) E H'(fif). According t o Theorem 4.2 one can extend V(z) to the domain 520. Denote this extension by P,U, and set P,u = Feu. Obviously the conditions (4.23)-(4.25) are satisfied. The last statement of the theorem holds since P,u = 0 in flO\fil. Theorem 4.3 is proved. 17

4.5'. The Korn Inequalities in Perforated Domains

In this section we prove the Korn inequalities (with constants independent

of E ) for perforated domains 52' 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 1 1 ) .

Let R" be a perforated domain o f type II. Then for any vector valued function u E H'(52') the inequality

'The proof is based on the extension of a function u from H 1 ( Q c , I ' t ) 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.

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00066___badcc084e25a85f96153d5c37229a87f.pdf52 I. Some mathematical problems of the theory of elasticity

holds with a constant C independent o f u, E . Moreover, if one of the following conditions is satisfied

(4.27)

(4.28)

(4.29)

where the constant C, does not depend on u , E.

Proof. The estimate (4.26) immediately follows from the Korn inequality (2.3) in R (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 t h a t u(z ) satisfies (4.27). Then

for any rigid displacement q E R. Let P,u E H'(R) be the extension of u constructed in Theorem 4.2. Denote by qo the orthogonal projection of P,u on R with respect to the scalar product in H'(R). Then

(Pc. - qo,C)Hl(n) = 0 , vc E R . (4.31)

Due to the Corollary 2.6 we have

since lle(Pcu - qo)ll,p(n) = Ile(PCu)11pp). By virtue o f (4.30) and Theorem

4.2 the last inequality yields

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00067___8bbd9796a5930d4463789bd8dcbd0f67.pdf54. Perforated domains with a periodic structure 53

lluIl&(n.) 5 IIU - r1112tz(n.) y VVEER. (4.32)

Choosing 7 = qo such that (4.27) holds for u - qo, we obtain by (4.29) for u - qo, that

1121 - rlOll&(*.) I c,z lle(u)lI22(n') . Therefore,

IIuIIi2(n*) 5 c6 l le(u)IIiz(n*)

by virtue o f (4.32). This inequality together with (4.26) implies (4.29) for vec-

tor valued functions u ( z ) satisfying (4.28). Theorem 4.4 is proved. 0

Let us now prove the Korn inequality in a perforated domain 0' o f type I for vector valued functions in H'(R') vanishing on rc. 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,rc) the inequality

\lvllH1(ne) 5 l l e (v ) \ \L2(ne) (4.33)

is valid, where C is a constant independent of E and v.

Proof. Let v E H1(OC, I',) and denote by P,v E H i ( n o ) the extension of 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 no. Therefore by (4.25) we have

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00068___3dd016f44565342f54af49bd034254b3.pdf54 I. Some mathematical problems of the theory of elasticity

Directly from Theorem 4.2 and Proposition 3 of Theorem 1.2 we obtain

Lemma 4.6. Let nc be a perforated domain of type II. Then

(4.34)

for any w E H1(Rc), where C is a constant independent o f E , w.

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00069___fc0641439949b8573d02d4aff96e177b.pdf$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 t h e system depend on a parameter

E , the constants in these estimates may depend on E . In this section we show

t h a t for perforated domains R' 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 to the class E ( n l , n z ) with tcl, K Z

independent o f E .


Let R" be a perforated domain of type I (see (4.1)), dR' = S, U rc, where S, is the surface o f the cavities, S, = R n dR', rC = dR n 80".

Consider the following boundary value problem

where fj E L2(Rc)), j = 0, ..., n, ip E H1(R'), L is an elasticity operator of type (3.1) belonging t o the class E ( q , ~ 2 ) .

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.

Let RE be a perforated domain o f type I and let the coefficient matrices o f the operator L belong t o the class E ( K ~ , ~ 2 ) with constants nl , I C ~ > 0 independent o f 6. Then there exists a weak solution u(x) of problem (5.1), which is unique and satisfies the inequality

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00070___6d1a9d7ce5342c6063c8f5c1bb8b39e9.pdf56 I. Some mathematical problems of 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) follow immediately from Theorem 3.8 with S1 = 0, S2 = S,, y = re. As stated in Theorem 3.8, the constant C in (5.2) depends only on tcl, t c Z , and the constant in the Korn inequality (4.33) for vector valued functions in H1(RE, re). According to Theorem 4.5 the last constant can be chosen independent of E,

and therefore (5.2) holds with a constant C which is also independent o f E. Theorem 5.1 is proved. 0

Remark 5.2.

Every vector valued function f o E L2(Rc) defines a continuous linear functional I ( w ) on H1(RC,I',) by the formula I(v) = ( f o , w ) L y n c ) . Denote by 1 1 f o l l l the norm o f this functional in the dual space (H'(R', re))*. Then

llfOll* = = { I(fo,v)LZ(nc)l, 2, E H1(f l e , r~ ) , IlvllH'(W) = I}. (5.3)

v

Obviously 1 1 f O 1 l * 5 1 1 folJLz(ne) . It follows from the proof of Theorem 3.3 t h a t 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 of type II consider the second boundary value problem o f elasticity

(5.5)

a? L ( u ) = p + - in R' , o(u) = cp + vJ on a R ,

d X i

u(u) = vif' on S, ,

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00071___300677690d8943746ba4cea8b563c731.pdf35. Estimates for solutions of boundary value problems of elasticity 57

where

fj E L2(R') , j = 0, ..., 12, cp E L2(dR) , (5.6)

dR' = dR u S, , (5.7) S, = (doc) n R .

In contrast t o Theorem 3.5 the next theorem establishes estimates uniform

in E for the solutions of problem (5.5).

Theorem 5.3. Let Rc be a perforated domain o f type 11, and

for any rigid displacement 7 E R. Suppose tha t the coefficient matrices of the operator L belong to the class E(lc1, l i 2 ) with I E ~ , K ~ > 0 independent of E. Then problem (5.5) has a unique solution u(z) such that

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 61, lc2 and the constant in the Second Korn inequality (4.29), which does not depend on E . Therefore (5.9) holds with a

constant independent of E . Theorem 5.3 is proved. 0

In order to 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:

(5.10) 1 a au afi ax h axk axi - ( ~ h k ( x ) -) - p(z)u = f o + - in RC , o(u) = v,f' on S, , o(u) = cp + uif' on dR ,

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00072___31c1919b659fef5859ed43f099347f24.pdf58 I. Some mathematical problems of the theory of elasticity

where fj E L2(Rc), j = 0, ..., n, cp E L2(8R), the matrices Ah'($) belong t o the class E(/c1,/c2), p ( z ) is a bounded measurable function in R' such tha t

0 < co I p ( z ) L c1 , ~ , c 1 = const . (5.11)

We say that u(x) is a weak solution of problem (5.10) if u(x) E H'(R") and the integral identity

(5.12)

holds for any w E H'(R'). Denote by u ( u , w ) t he bilinear form in the left-hand side o f (5.12). This

form satisfies a l l conditions o f Theorem 1.3 for H = H1(R2') with constants c1, c2 independent o f E. This fact is due to 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 R' be a perforated domain o f type II, and le t the family of matrices A h k ( t ) , h,lc = 1, ... ,T I , belong t o the class E(nl,tcZ). Suppose that conditions (5.11) are satisfied and the constants Q, cl, nl, n2 do not depend on E. Then problem

(5.10) has a unique solution u ( z ) , and this solution satisfies the estimate

(5.13)

where C is a constant independent o f E.

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00073___09d8a40fa3ef88db68e85797d1354b26.pdf$6. Periodic solutions of boundary value problems 59

$6. Periodic Solutions of Boundary Value Problems for the System of Elasticity

To study homogenization problems for the system o f 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

(6.1)

in w , a 8 W dF" - ( A h k ( x ) -) = F'(x) + - ax h 82 k axm U ( W ) = v,Fm on dw ,

1 w d x = O , I w is 1-periodic in x , Qnw where the vector valued functions F j ( z ) are 1-periodic in z, F j E L2(w n Q ) , j = 0, ..., n, the family of matrices Ahk(x) belongs t o the class E ( K ~ , ~ 2 ) and their elements at / (z ) are 1-periodic in x.


w E @(w) such that w d z = 0 , and the integral identity I Qk

holds for any v E k@(w).

Theorem 6.1.

Let Fo dx = 0 . Then problem (6.1) has a unique solution, and th is solu- J

Qnw tion satisfies the estimate

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00074___f66001acfa1119e14082ec7705b4634d.pdf60 I. Some mathematical problems of the theory of elasticity

where the constant C depends only on K ~ , ~ 2 , 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 i@i(w) such t h a t v d x = 0; the Korn inequality is .I furnished by Theorem 2.8.

Q h w

In what follows we shall often use the fact that solutions o f problem (6.1)

are piece-wise smooth, provided t h a t the coefficients of the system (6.1) and the functions Fj, 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 t h a t there are mutually non-intersecting open sets Go, ..., G, with a 1-periodic structure and such t h a t Gj C w, j = 0 ,1 , ..., m; Gjn& = 0, j = 1, ..., m; Go = w\(G1 u ... u Gm); GI, ..., G, have a smooth boundary.

We say t h a t a function 'p which is 1-periodic in x belongs t o class 6 ('p is called piece-wise smooth in w and smooth in a neighbourhood o f dw) if cp has

bounded derivatives of any order in Gj, j = 0,1, ..., m.

Theorem 6.2.

Le t w(x) E l$'i(w) be a weak solution o f problem (6.1), and suppose tha t the elements of A h k ( z ) , F j ( x ) belong t o class 6. Then w also belongs t o 6, i.e. w is piece-wise smooth in w and smooth in a neighbourhood of dw.

Proof. The smoothness of w in a neighbourhood o f dw follows from the general results on t h e smoothness of solutions of the elasticity system near the

boundary (see [17]).

Let zo E dGj, xo # dw, and consider the set Gj n {x : 1 1 - xoI < 6) = q,"(zo). It is shown in [17], Section 13, Part I , that for sufFiciently small 6 the function w has bounded derivatives of any order in q:(xo) . The smoothness

of 20 a t the interior points of w, which do not belong t o dGj, is also proved 0 in (17). Therefore w E d.

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00075___73555498d891a8fd1b22cff85ec3435e.pdf6. Periodic solutions of boundary value problems 61

6.2. Solutions of the Elasticity System Periodic in Some of the Variables

Let the coefficient matrices A h k ( z ) of the differential operator C belong to the class E(x1, Q), and suppose that their elements are 1-periodic in ? =

In this section w is an unbounded domain with a 1-periodic structure, which

satisfies the Condition B of $4 (see Fig. l), the domains w(a, b ) and ;(a, b) are defined by (1.6).

( 5 1 , *.., 2,-1).

Set

d t = w n (2, = t } . (6.4) Let gt be a non-empty open set belonging t o dt and invariant with respect

t o the shifts by any vector z = (21, ..., z,-1,0) E Z". Set d t = d t n { z : o < z j < i , j = i ,..., n - 1 1 ,

& = g t n { z : o < x j < l , j = l ,..., n - 1 ) . (6.5)

Fig..

Consider the following boundary value problem

in w(u ,b ) , f3F" L ( w ) = F y z ) -t - f3zm

w d x = O , J w is 1-periodic in P , o(a.b)

where $ b , Fj are vector valued functions 1-periodic in 2 , Fj E L2 ( ; (a , b ) ) , j = o,...,n; '$0 E L2(ga), 'd)b E L2(gb), 0 5 U < b < 00, U, = -1 on ga, v, = 1 on gb. The domain &(a, b) is assumed to have a Lipschitz boundary.

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00076___37a76a1109e65684a6cf868f675cd64f.pdf62 I. Some mathematical problems of the theory of elasticity


w E &'(w(a ,b) ) such t h a t for any w E Z?(w(a,b)) the following integral

identity is valid:

Theorem 6.3.

Let

Then there exists a weak solution w of problem (6.6), which is unique, and w

satisfies the est ima t e

where C is a constant depending only on w , a, b, tcl, Q .

This theorem is proved in a similar way t o Theorem 3.5. In this case we

take as H the subspace of &'(w(a, b ) ) formed by a l l vector valued functions

w such t h a t J vdx = 0. Then the Second Korn inequality follows from Theorem 2.9. To estimate the right-hand side of (6.7) we should use the ineq ua I ity

&(ah)

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00077___e1438861531efdbaf092f98d243314dc.pdf$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:

in w(a ,b ) , a F m L(w) = F0 t - dxm

w = a(?) on 19, ,

w is 1-periodic in P ,

(6.11)

where @(P), $ b ( ? ) , Fj(x), j = 0, ..., n, are 1-periodic in 2 , F i E L2(; (a , b ) ) ,

A vector valued function w is called a weak solution of problem (6.11), if $b E L 2 ( i b ) , @ E H1"(da).

w E -fi'(w(a,b)). w = @ on da, and the integral identity

is satisfied for any v E I? (w(a, 6 ) ) n H 1 ( b ( a , b), da).

Theorem 6.4.

There exists a weak solution w ( x ) of problem (6.11), which is unique and

satisfies the inequality

where C is a constant depending only on w, n2, a, b.

Mathematical_Problems_in_Elasticity_and_Homogenization/0444884416/files/00078___66755531822b71e015d1cc48e65ec61b.pdf64 I. Some mathematical problems of the theory of elasticity

Proof. By virtue o f Theorem 2.7 the Korn inequality (2.19) holds for any 2) E &(w(a, b ) ) n H1(&(a, b) , ia) (i.e. v = 0 on da). Moreover, it follows from Proposition 3 of Theorem 1.2 and the Korn inequality, t h a t

(6.14)

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)

0 and the validity o f the estimate (6.13).

6.3. Elasticity Problems with Periodic Boundary Conditions in a

Perforated Layer

In this section R' denotes the perforated layer

W = { O : o < x , < ~ ) ~ E w ,

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, E - ~ is a positive integer. Set

In R' consider the following boundary value problem:

(6.15) I u ( ~ , o ) = W ( 5 ) on ro , u ( i , d ) = P ( 2 ) on r d , .(u) = V, f m on (aW)\(r, u rd) , ~ ( z ) is 1-periodic in i . I

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 , a', ip2 are also 1-periodic in

Mat

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