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Chapter II Homogenization of the System of Linear Elasticity. Composites and Perforated Materials
119
CHAPTER I I HOMOGENIZATION OF THE SYSTEM OF LINEAR
ELASTICITY. COMPOSITES AND PERFORATED MATERIALS
This chapter deals with homogenization problems in the mechanics of strongly non-homogeneous media. Most of t h e 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
I I we give estimates for the closeness between the displace-
ment vector, t h e 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 a t 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
R" we consider the following boundary value problem
$4,Ch. I.
II. Homogenization of the system of linear elasticity
120
where
A h k ( t )are ( n x n)-matrices of class E ( K ~ , KI C~ ~) ,, K=~ const > 0
t . It is also assumed t h a t the surfaces across which they or their derivatives may loose continuity do not intersect dw i.e. the functions u:! belong t o the class 6' defined in $6.1, Ch. I. Existence and uniqueness of solutions o f problem (1.1) for f' E L2(n"), a' E H'(S2") 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
whose elements uihjk(() are functions 1-periodic in u : / ( ( ) are piece-wise smooth in w and
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
k'9
( p , q = 1, ...,n ) are given by the formula
and matrices N * ( t ) are solutions of the following boundary value problems for the system of elasticity
a ( N 9 ) = -&Akq on dw
Nq(1) is 1-periodic in
t,
)
J Qnw
Q=
~ ~ ( t )=d0 t
{ t , O < ( j < 1 , j = 1,...,n } .
)
I
(1.4)
31. Mixed problem in a perforated domain Existence of the matrices
Nq
121
follows directly from Theorem 6.1, Ch. I.
According t o Theorem 6.2, Ch. I, the elements o f the matrices piecewise smooth functions in w belonging t o the class
6.
Nq
are
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],
[llo]). 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
if; of the
matrices
Ak' satisfy the conditions
ik! =
= ii/
1.j
jt
klqihvih
kj
5 hikj'vikvjl 5 k 2 v i h v i h
for any symmetric matrix 7 = other words the operator
Proof.
(1.5)
?
{vih},
(1.6)
where
itl, k 2 are positive constants. In
L belongs t o t h e class E ( k 1 , k 2 ) .
In the special case of w = R",i.e.
R'
= R , the relations (1.5), (1.6) 2
can be obtained from Theorem 9.2, Ch. I, since t h e matrices N,9(x) 5 a N q ( - ) and
ak'satisfy the Condition N which can be easily verified on account of (1.3), &
(1.4). In t h e general case when w may not coincide with R",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 ( be a column vector with components
(Cl,..., (").
C1,..., Cn.
Denote by (* the line
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 = ( i u i j , j = 1, ..., n.
It is easy t o see t h a t the second equality in'(1.5) follows directly from (1.3) and the properties of the elements of the matrices APq(t),since
II. Homogenization of the system of linear elasticity
122
where NAi are the elements of the matrices
Nq.
Let us establish the first equality in (1.5), which is equivalent t o
(AP'J)'
=
A9P.
It follows from t h e integral identity for solutions o f problem (1.4) t h a t for any matrix
M ( J )E
k i ( w ) we have
Making use of the relations ( A k j ( t ) ) * = Aj'(t), matrices A,
B , we obtain from (1.8) t h a t
Setting M =
(A")'
=
(AB)' = B'A' for
NP'
in (1.9) and taking into account (1.3) and the relation
= Ajp we find
J (EAjP((r) + AqP(t))dt= (mes Q n w)(Ap'J)* . Qnw
&j
It follows t h a t the coefhcient matrices of the homogenized system can be written in the form
= (mesQnw)-'
/ Qnw
d
-(NP'+tpE)Ak3(t) atk
d -(Nq+tqE)dt.
at3
(1.10)
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
= (Aqp)'.
In order t o prove the inequalities (1.6) let us note t h a t i $ / v , h 7 j k where 7' is a coIumn with components q l k r . . . , y n k , and
= vh*Ahkvk
vh* = ( ~ ~ h , . . . ~ v ~ h ) .
For any symmetric matrix 7 with constant elements q ; h we obtain due to (1.10) that
(1.11) Let w = (NI
+ &E)qI be a
vector valued function with components
w1, ..., wn. It then follows from (1.11)t h a t
(1.12) Suppose that for a symmetric matrix 17 we have
iLy:v;pv3q
= 0. It then
follows from (1.12) and the estimate (3.13), Ch. I, t h a t Ile(w)llLz(Qnw)= 0. Therefore w is a rigid displacement (see the proof o f Theorem 2.5, Ch. I). On the other hand w(<) = (NI
+ tqE)vI. Therefore due t o the periodicity of
NI(1) the vector valued function NqvI must be constant, and the matrix v must be a skew-symmetrical one. It follows that q = 0. Thus iy!v;pqjI > 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.
0
1.2. The Main Estimates and Their Applications Let us take as an approximation t o the solution of problem (1.1) t h e fol-
lowing vector-valued function
G(x) = u"x)
duo + & N P x( - ) axp ' &
(1.13)
II. Homogenization of the system of linear elasticity
124
where NP(E) are the matrices defined by (1.4) and u o ( x )is t h e solution of the problem
E(u0) = f O in 0
, uo = ipo
on
.
(1.14)
Theorem 1.2. Suppose that u'(x) is a weak solution of problem (1.1) in ip' E
W ,f' E L2(R'),
H'(R'), fo E H1(R), ipo E H3(R)and u o ( x )is a weak solution of the
homogenized problem (1.14). Then
IIu'(x) - uo(x)
Proof. Applying the operator L, t o uc - ii
I
we obtain the following equalities
which hold in the sense o f distributions
Since the matrices
Lc(U"- i i ) =
N" satisfy the equations (1.4), it follows that
31. Mixed problem in a perforated domain
125
Define the matrices N P q ( t ) ( p , q = 1, ...,n ) as weak solutions of the boundary value problems
a
- A p j ( < )-N q ( t ) , - A P q ( t+ ) ipq in w ,
alj
b ( N P q )= - V k A k p ( t ) N q ( t ) on dw NPq
is l-periodic in
t,
/
>
,
N p q ( ( ) d t= 0
(1.17)
.
Qnw
The existence of N p q ( ( ) follows from Theorem 6.1, Ch. I and the equalities (1.3). Thus we deduce from (1.16), (1.17) that
Lr(U'
a Fk , + axk
- G ) = f'- f0 + &Fo
E
(1.18)
where
Let us consider now the boundary conditions on S, for u' - G . We have
126
11. Homogenization of the system of linear elasticity
6,(U"
- C) = 6 , ( u e ) - O"(C) =
By virtue of the boundary conditions on aw for
N q
and
NPq
it follows t h a t
(1.20)
On the outer part o f the boundary o f 0' we have U"
- fi =
@'-
@O
auo = 1c, __
- E N S
(1.21)
c .
ax,
Let us show t h a t
where c is a constant independent o f valued function qc E
We define \I1,(x) t h a t cp.(x)
IVcpl 5
E.
To this end it suffices t o find a vector
H1(02')such t h a t qC+ EN'
as follows. Let cpe be a scalar function in C"(Q) such
= 1 if p(qa0)
5 E,
cp.(~) = 0 if p(x,a0)
C2E-l.
Set 8UO
QJX) = -cp.EN8 - ; ax,
I(,
=
{X
It is easy t o see t h a t
E H1(W,re),
: p ( ~80) , 5 QC
E
2 ~ n) 0'
H ' ( 0 ' ) and
2 2 ~ 0, 5 cpc(x) 5 1,
127
51. Mixed problem in a perforated domain
Therefore taking into account the properties of ve and the fact t h a t the matrices N " ( [ )and d N 8 ( ( ) / a ( j have bounded elements, we obtain t h e inequality
5
~ ~ ~ ~ ~ ~ H C3(11UoIIH1(Kr) 1 ( O e ) -k E l l u o l l H 2 ( K c ) )
By virtue o f Lemma
.
(1.24)
1.5,Ch. I, we get
IC4E112 l l ~ O 1 l H ~ ( R )
IIUOIIHl(K.,
9
is a constant independent of
where
c4
yields
(1.23). Therefore estimate (1.22)is valid. (1.18), (1.20),(1.21), (1.22) we conclude t h a t
E.
This inequality together with
On the basis o f
(1.24)
uc - ii is
a weak solution o f the following mixed boundary value problem studied in
Ch. I:
LC(uc- ii) = f'- fo u c - C = +L
Here
+c
on
a Fk + eF0 + e axk
re; u . ( u c - C )
satisfies the inequality
in Re
55,
,
=evkFk on S,
.
(1.22)and
n
IlFkllLz(n*) k=O
5 c5 l l U o l l H 3 ( n ) .
where the constant c5 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 IIU'
5
- CllH'(n*) 5 % ( E l l Z IIUOIIH3(R)
This inequality implies of elliptic systems (see
+ Ilf'
-
fOll* t
- @ollHl,2(re))
(1.15)since due t o the a priori estimates for solutions [l]) we have:
5 c(llfOllHl(n)+ ll@ollH"2(an)) .
11~011H3(n)
.
128
11. Homogenization of the system of linear elasticity
Theorem 1.2 is proved.
0
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
(1.25) (1.26)
The integrals E c ( u e ) ,E0(uo)represent the energy contained in R' and
n R'
R' respectively.
Theorem 1.3 (On the Convergence o f the Energy). Suppose that all conditions of Theorem 1.2 are satisfied. Then
(1.28)
(1.29)
129
$1. Mixed problem in a perforated domain
(1.30) where
Since the elements of the matrices
6.2, Ch. I) we get
dN8 A'j, - are bounded (see Theorem X i
Let us introduce the matrices
(1.32)
11. Homogenization of the system of linear elasticity
130
In the rest o f the proof it is assumed t h a t the matrices A ' j ( ( ) , N s ( ( ) ,
d N 5 / d ( i are defined in R" and are equal t o zero in
Thus we obtain
Q\w.
from (1.10)
J
H"(()d( = 0
.
(1.33)
Q Note t h a t due t o our assumptions we can replace the domain o f integration
RE n R' in (1.30) by 0'. Therefore after a suitable transformation, (1.30) becomes
(1.34) From this equality and (1.32) we conclude t h a t
Ee(uc)- (mes Q n u)Eo(uo)=
J
R'
duo*
x
axt + P,
8210
-Hst( -) -da:
ax,
E
(1.35) Note t h a t due t o Theorem 6.2, Ch.
I,
the elements o f the matrices H s t are
bounded functions. Denote by J, the set of all vectors z E Z" such t h a t by J: denote the set of all z E
Z"such that E ( Q
E(Q
+ z) c R' and
+ z ) n dR' # 0. Then
E,(u') - (mes Q n w)Eo(uO)=
It is clear t h a t t h e 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 E.
dR' and 6 is o f order
Therefore using Lemma 1.5, Ch. I, we deduce t h a t
Consider now
K 2 , i.e. the second sum in the right-hand side of (1.36).
Denote by 0'' the set formed by the cubes
E(Z
+ Q ) , when z takes values in
J,. Set
The vectors y t ( x ) are constant on each o f the sets E ( Z
+ &). We have
(1.38)
It is easy t o see that
(1.39) Taking into account the Poincark inequality in
E(Z
+ Q ) we obtain (1.40)
Let us estimate the sum
K2.
Due t o (1.33) the last integral in (1.38) vanishes
since yt are constant on each o f the sets
E(Z
+ Q ) for z E J,.
It follows that
11. Homogenization of the system of linear elasticity
132
Therefore taking into account (1.39), (1.40) we obtain 1K21
I C8E l l ~ o I l $ ( * , r )
(1.41)
*
The inequality
IIu0 IIHW I C9(llf011L2(*)
+ 11~ollH3/l(an))
and (1.31), (1.36), (1.37), (1.41) imply estimate (1.27).
Theorem 1.3 is
proved.
0
Let us consider now the convergence of the stress tensors, i.e. o f matrices whose columns are
o,"(x)
x du'
Apk(-) E
ax,,
p = 1, ..., 12
,
(1.42)
where u' is the solution of problem (1.1).
The stress tensor corresponding t o the homogenized problem (1.14) has the form duo
O;(X) E APk6
axk
,
p=l,
..',1 2 .
(1.43)
In the homogenization theory the matrices with columns o,", 00' are also referred t o as weak gradients or flows. In the next theorem it is assumed t h a t
and t h a t
Theorem 1.4. Suppose t h a t the conditions of Theorem 1.3 are satisfied. Then
133
$1. Mixed problem in a perforated domain
+ IlfO - Yll.
lPO- @cllH1/2(re)]
-t
where c is a constant independent o f
E,
(1.44)
1
and the matrices GP'J(<) are defined
by the formulas
< E Q nw , for < E Q\w .
GPs(<)= A P 8 ( (+ ) Api aN" - (mes Q n w)&'
for
ati
GPs(()= -(mes Q n w ) k 8 Moreover, if
f' = fo, W
= cPo then
u,"(x)--+ (mes Q n w)uOp(x) weakly in L2(R) as
Proof.
(1.45)
E
+0
.
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
Therefore taking into account (1.43) we get
o,"(x)- (mes Q n W ) U ; ( X ) =
This equality and (1.29) imply (1.44). The weak convergence of u,"(x)t o (mes Qnw)uOp(Z) for follows from (1.44) and the fact that convergence
I.
auo
J
Theorem 1.4 is proved..
QC
= cPo
GP"(<)d(= 0 which implies the weak
Q
G p " ( - )- -t 0 in L2(R)as E
ax,
f" = fo,
--f
0 by virtue o f Lemma 1.6, Ch. 0
11.Homogenization of the system of 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 t h e Neumann problem in a perforated domain
R' 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 t h e conormal derivative of a rapidly oscillating corrector
x duo E axs
which has the form E N ' ( - ) - outside a neighbourhood of do. Therefore in order t o clarify the main ideas used in the proof of 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 t h a t 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
a ax;
consider the following Neumann problem
x auc
L c ( u e )= - (aI3(-1 "
R
E
-)axj
= f in R
where ( u l , ...,un) is the unit outward normal t o are smooth functions in
where
K ~ , = K const ~
R",1-periodic in [,
> 0.
The functions
f
,
dR. It is assumed that aij([)
and such that
and 'p are sufficiently smooth and
satisfy the condition of solvability of problem (2.1)
J n
fdx =
J
an
'pdS.
(2.2)
We define the functions N p ( t ) ,p = 1, ..., n , as solutions of the problems
135
$2. Boundary value problem with Neumann conditions
N P ( J ) is
1-periodic in
t,
J
N P ( t ) d t= 0 .
Q Set
hhk =
J
[uh"((s)
Q
Nk + uhi 1-dati
dt ,
Q = {X : 0 < x3 < 1 , j = 1,...,n} . As a n approximation to the solution of problem (2.1) we take the function 6, = u'(x)
+ E N ' ( - ) -, dUO(X)
2
(2.4)
ax,
where uo is the solution of the homogenized Neumann problem
duo = 6'3.. duo u; = cp
auA
on dR
ax
(2.5)
.
In analogy with (1.16) simple calculations show that CE(uC - 6,) = -EU'~N'
a3uo
axiax,ax
Let us define the functions
Then
a
- 6 ) = E -(ukh
CE(UC
axk
+
N i s ( ( ) as solutions of the
problems
d3U0
ax,axiaxj
-
11. Homogenization of the system of linear elasticity
136 where
(2.11)
Lemma
2.1.
Functions &"t) defined by
1 n
cu'a(t)cif = o
a Qis ( f ) = 0 at;
J
s;
( 2 . 1 1 ) satisfy the relations
,
,
i , s = 1, ...,n ; s = l,...,n ;
aJ3(<)dij =0,
(there is no summation over j ) .
tlt E R', s , j = 1 ,...,n ,
(2.12) (2.13) (2.14)
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 Qilt2 the set
Multiplying (2.13) by (, - tl and integrating over
8
J
0=
*i a ( < ) ( t i - t l ) d t
we get for each
j
=
Q:, t 2
J
=
c P ( ( ) ( t z - tl)d& -
J
a3'(()d( .
Qil
3i2
t2
Setting tl = t z - 1 and taking into account (2.12) we obtain (2.14) for t = tz. Lemma 2.1 is proved. 0 Lemma 2.2. Let a"(() be functions in H ' ( Q ) 1-periodic in ( and satisfying the conditions (2.12)-(2.14). Then for any v E H ' ( R ) the following inequalities are valid
(2.15) where c is a constant independent of
E,
v.
Proof. Denote by 1; the set of all indices z
+
p ( ~ ( z Q ) ,80)2
E.
Set
01
=
u
(EZ
+
-t Q) Q ) . It is easy to see that
E Znsuch that
E(Z
ZEl:
o
=
J n\n1
-
J anl
Therefore
d dXi
X
- (aik(--))vdx =
J
a"u;vdS -
an
aiku;vdS -
J
R\RI
dV
aik- dx
dXi
,
k
= 1, ..., n
c R,
11,Homogenization of the system of linear elasticity
138
J
aik(;)u;vdS 2 =
an =
J
aiku,vdS
+J
an
&
aik
n\nl
dx
.
(2.16)
We clearly have
(2.17) Let us estimate t h e first integral in the right-hand side of (2.16).
It is easy t o see t h a t
an1 consists o f ( n- 1)-dimensional faces o f the cubes
+ Q) for some z E If. Denote by o:,..., ojI ) the ( n - 1)-dimensional faces of the cubes E(Z + Q ) for z E If such that ojk is parallel t o the hyperplane xj = 0 and lies on an1,j = 1, ..., n. We thus have
E(Z
j=l
a=l
Denote by q; the cube E(Z
+ &) whose surface contains the set a;. It is
obvious t h a t among the cubes q;, j = 1, ..., n; s = 1, ...,13, there cannot be more than 2n identical ones. Due t o the condition (2.14) for any n; we have
J
atkv,dS =
0:
since u, = 0 for
J
aJkv3 dS = o
,
(2.18)
0:
i#j
(there is no summation over j ) . Set
~ ( x =) (mesq;)-’
J
vdx
,
9:
X E $ ,
j = 1 , ...,n ; s = 1 , ..., l J .
Thus the function ~ ( x is) constant on each surface a;. Taking into account (2.18) and the fact that not more than 2n cubes q; can have a non-empty intersection we obtain
139
$2. Boundary value problem with Neumann conditions
(2.19) Here we have also used the inequality
J
1v - r]l2dx 5 c3.5
0:
J
1vvl2da:
(2.20)
9:
which can be proved if we pass to the variables [ = E
- ~ X and
apply Proposi-
tions 3 and 4 of Theorem 1.2, Ch. I , in the domain R = &-I$. The relations (2.16), (2.17), (2.19) imply (2.15). Lemma 2.2 is proved. Therefore due to (2.7)-(2.10) we have (2.21)
where Fl, F t , F2 are bounded uniformly with respect t o Setting w = u ' -6 obtain from (2.21)
E.
+ vc where vC is a constant such that nJ
w dx = 0 , we
+ F2wdS1] . auo
Applying Lemma 2.2 to v = -20 and using the Poincare inequality (1.5), Ch. I, we find that
8x9
II. Homogenization of the system of linear elasticity
140
Thus we have actually proved Theorem 2.3.
f, cp are smooth functions satisfying the solvability conditions for problems (2.1), (2.5). Then there is a constant q'such that Suppose t h a t u', uo are solutions o f problems (2.1), (2.5) respectively, and
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
R' denotes a perforated domain of type II defined dRc is a union o f dR and the surface of the cavities
In the rest o f this section in $4, Ch. I. The boundary
s, c R . In
R'
we consider the boundary value problem o f Neumann type for t h e
system o f linear elasticity:
d x du' Lc(uE= ) - (Ahk(--) -) = ~ ( x in) dxh E d2k a,(u') = O on S, = dR'\dR , a c ( u c )= $f
on dR
w,
1
(2.22)
J
.
It is assumed t h a t the elements of the coefficient matrices Ahk satisfy the same conditions as the coefficients of the system ( l . l ) , f' E L2(Rc), @ E L2(dR) and satisfy the conditions of solvability for problem (2.22), i.e.
J (f',rl)dx = J ( V , r l ) d S vv E
n*
an
3
'$2. Boundary value problem with Neumann conditions where
R is the space of
141
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 (2.23)
(mes Q n u)+(uo) = ?,bo where 6(uo)
"hahk
&Lo
-, the matrices Ahkare defined by the formulas (1.3), axk
',1 E L2(dR), fo E L2(R), satisfy the solvability conditions (mes Q n u)-1
J (GO,
V ) ~ S=
an
J
(fo, q ) d x
VV E
R.
R
It is important t o note that the factor mes Q nu appears in t h e Neumann a R . This factor is equal t o 1 if R' coincides with R (see $2.1, formula (2.5)). To characterize the closeness between functions fo, ',1 and j ' , 1,' we conditions on
introduce the following notation.
L2(dR) the scalar products (f,v ) p ( n . ) , ($, v),p(an) define continuous linear functionals on H'(R'), and therefore f and 1c, can be considered as elements of the dual space H1(R')*. Let us denote the norms of the respective functionals as IlfllH1., ll11,llH1*,i.e. For any vector valued functions
f
E L2(Rc), 11, E
{ (f,v)Lz(ne),v E H1(R'),
Ilf llH1* =
IIOIIH1(Re)
= 1)
7
U
II$IIH1*
=
SUP
{(11,,")L2(an),v E
H'(R'),II~IIHl(n.) = I} .
v
G(x) = uo + &cp(X)N$(-)- . & ax, x
duo
(2.24)
Here uo is the solution o f problem (2.23), N 8 ( t ) are solutions of problems (1.4), cp(x) is a truncating function which satisfies the following conditions:
142
Homogenization of the system of linear elasticity
x E R1 such that p(x,dR,) L where cl, c are constants independent of
E;
R1 is defined
,
(2.25)
C ~ E
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 uc is considered in t h e per-
Rc but neighbourhood o f dR.
forated domain
the matrices
X
N " ( - ) are in general not defined in E
a
The main result of this section is Theorem 2.4. Suppose t h a t
f'
E
Then the solutions
L2(R'), fo E H'(R), @ t i E , uo
E L2(dR), 6 ' E H3l2(dR).
of problems (2.22), (2.23) respectively satisfy t h e
following inequality
L
t (2.26) 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
143
$2. Boundary value problem with Neumann conditions
Lemma 2.5. ) the following conditions The matrices a i d ( (satisfy
J
ai"(()d(= 0 ,
i , s = 1, ...,n , (2.27)
Qnw
-aa
ati
is
in w , s = l , ...,n ,
(()=O 1.
u,aid = u,A'" on
J
dw ,
s = 1, ..., n
,
(2.28)
a j d ( ( ) d i j= (mes j?j - mes Q n w ) P
,
(2.29)
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 wnQ:,
Ajqd( t2
/
(j
- t l ) E , where
E
n w , (tl < t z ) ,we obtain
is the unit matrix, and integrating over
( ( j - tl)ukAkqdS.
(2.30)
a(Q:, t2 n w )
Each integral in (2.30) over d(Q;,,,
n w)
can be represented as a sum of
integrals over the sets
and the integrands are 1-periodic in (2.30) that
ST, T #
j,
= 1,
..., n , it follows from
II. Homogenization of the system of linear elasticity
144
(there is no summation over j ) . Setting tl = tz - 1 in this equality and taking into account (1.3) we find that
It follows from the definition of
ajs t h a t
= (mes S,3)A3"- (mes Q n u)ka = A
.
A
.
= (mes 2; - mes Q n u)&" . Lemma 2.5 is proved.
0
Remark 2.6.
If the domain Re is not perforated, i.e. w = R",Rc = R , then aj"(()d& = 0 , since mess; = mesQ n u = 1 (see Lemma 2.1).
/
A
.
9; Lemma 2.7. Let ol,...,oZn be (n- 1)-dimensional faces of the cube EQ = { z : 0
~ ,=j 1, ..., n}. Then each u E H1(eQ) satisfies the inequality
< zj <
145
$2. Boundary value problem with Neumann conditions
(2.31) where c is a constant independent of
Set o1 = {q = 0) n EQ, o2 = { x 2 = 0} n E Q , S1 = &-lo1, - E -1 6 2 . Consider the points = (O,yZ,y3, ...,yn), ?j2 = (y2,0,y3, ..., yn)
Proof.
s
2
i, j, E.
6'
on thefacesS1, SzofcubeQ. Thesegmentg(t,y2,y3, ...,yn) = t$'-k(l-t)i2 for t E [0,1] belongs t o Q. It is easy t o see t h a t for any v E H ' ( & ) we have v2(g(l,
~ 2 ,
Y,)
- v2 (g(0,
~ 2 ,
Yn))
=
Integrating this equality with respect t o y2, ..., yn from 0 t o 1 we obtain the estimates
J
v2dS-
J
v2ds
X
= - , C = const &
J
&+-I) v2
J Ivl(Vpldl, Q
sz
s1
,
do - &-(,-I)
01
v2 do 01
5CPE
J
IvI IV,vl dx
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
dR1 which is the boundary of the domain Ol given (4.3),Ch. I. The domain R1 depends on E and its boundary 6'01 consists of the (TI - 1)-dimensional faces of the cubes E ( Z -k Q), z E T,. Denote by C T ~..., , o) the faces o f the cubes E ( Z -k Q ) for z E T, parallel t o the hyperplanes xj = 0, j = 1, ...,n , and laying on 801. Then
functions defined on the set by formula
II. Homogenization of the system of linear elasticity
146 The cube
E(Z
+ Q ) , z E T,, on whose boundary lies the set ojs is denoted by
q;. It is easy t o see t h a t among g;, j = 1, ...,n , s = 1, ..., Zj, the number of
the identical cubes is not greater than 2n. Lemma 2.8. Let u E H'(R). Then
Ill'llH1(n)
ll'llLz(ani)
where c is a constant independent of
Proof. According (n -
(2.33)
7
E.
dR1 consists of the sets ofi, and each ofi is an 1)-dimensional face o f the cube q i . The boundary dR is a smooth t o (2.32)
surface, therefore each cube q; possesses a face os,jsuch t h a t os,j is parallel
0 and oSj is the orthogonal projection along t h e axis OX^(^,^) of a surface S8,j C dfl which is given by the equation
to the hyperplane
2,
xrn(j,*)=
= $m(?rn)
and
CIE
where constants cl,
7
j.m
E
osj
I 15 - yI I C ~ Efor c2,
~ 5
~
I $M
7
mm
=~m ( j ~,3)
~
~
E os,j, y E S8,j ,
M do not depend on
E,
s, j .
Denote by Qs,j the set formed by the segments orthogonal t o o,j and connecting the points o f a,,j and Sd,,. Then using a suitable diffeomorphism mapping
QS,j to
EQ and taking into account Lemma 2.7 we find t h a t
2
IIuIILz(u,,,)
5 c ( ~ ~ u ~ ~ ~ z ( S+s ~, ,~) u ~ ~ ~ ~ ( Q s , , ) ) *
Therefore by Lemma 2.7 we get 2
+
5
.
~ ~ u ~ \ L z ( u'1 ~( )~ ~ u ~ ~ ~ z ( S , ~, ,~)u l ~ ~ l ( Q , , , ) )
Summing these equalities with respect t o s, j we obtain estimate (2.33), since due t o the smoothness o f dR there is an integer and such that each
Ic independent of
E
can have a non-empty intersection only with a finite
number of Q{,$which is not greater than
Ic.
Lemma 2.8 is proved.
Lemma 2.9. Let the matrices y h k ( z )E
Lm(dfl1)be such that
0
$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), functions uo E
147
y = const. Then for any vector valued
H3(R),w E H'(R) the inequality
holds with a constant c independent o f
Proof. Consider a function
E.
r ( x ) defined almost everywhere on
dRl by the
for m u Ia
Obviously r ( x ) is constant on each
0;". Therefore
setting
and taking into account (2.34) and the Poincarh inequality in a;" we obtain
J
anl
lrI2dS =
CCJ n
11
j=1 s=1
a:
lrI2ds =
II. Homogenization of the system of linear elasticity
148
It follows from Lemma 2.8 that
J
lr12dSI czEZrZlu011;3(n)
(2.36)
1
anl where c2 is a constant independent of It is easy to see that
E.
(2.37) Due to (2.36) and Lemma 2.8 we have
L
.
C3EY 11~011H3(n) l14IH’(n)
(2.38)
Let us estimate the second integral in the right-hand side of (2.37). Define the vector valued function ~ ( x on ) do1 by the formula T(x) = (rnesqm)-’
J
1 w dx for x E om
.
s!n
Therefore
J
Iw-qf’dS=
anl
5
55 1
Iw-qI2dS
5
j = 1 s=l
g :
CCJ
n 11 C ~ E j=1 a = l
9:
I ~ w l ~ d5xC
J
~ E
IVw12dx.
(2.39)
n
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 ZJ = w. By the definition of r we have
149
$2. Boundary value problem with Neurnann conditions
J
(yhkvh auo - r ) d S = 0
.
4 Therefore since q(x) is constant on each c i we find by virtue of (2.39), (2.36) and Lemma 2.9 that
Ic77+’/2
lluollH3(n)llWllHl(n,
*
This inequality together with (2.37), (2.38) yields (2.35). Lemma 2.9 is proved.
0
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 of Theorem 2.4. Let us apply the operator
ii is given by (2.24). Then
C, to the vector valued function
uc - 6 , where
11. Homogenization of the system of linear elasticity
150
Let us define the matrices Nhk(E)as weak solutions of the following bound-
a r y value problem
$2. Boundary value problem with Neurnann conditions
151
152
II. Homogenization of the system of linear elasticity
153
$2. Boundary value problem with Neumann conditions
du" duo Q,(u"- G) = (1 - v ) u ~ A ' ~ - (1 - cp)Aiju; "
ax + 43; + ~ $ 7on
Set w = u" - 4
+ q E , where 7'
dXj
+
dR' .
(2.43)
is a rigid displacement such! that
ER .
( w , ~ ) H I ( ~= . ) 0 for any
Due to the boundary conditions for uo, uEand the fact that cp = 0 in R\Rl
it follows from (2.41), (2.43) that
+J
J
( u ; A l j ( l -duo ( ~ ) ~ , w ) d ~((mesQnw)-'1Clo-1Cl',w)dS. + an
s.
(2.44) 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 Rl,p(x,aRl) 2 of
E.
ax
:
x E
C ~ E and } IsVql 5 c , where c , c1 are constants independent Therefore by Lemma 1.5, Ch. I, we obtain
5 cz IlwllH1(n*)&"2lluollH3(n)> where c2 is constant and does not depend on
(2.45) E.
It follows from (2.42) that
5
C3E l l w l l H 1 ( n e , ) lluollH3(n)
(2.46)
7
where c3 is a constant independent of E. Taking into account (2.27), (2.28) and setting
cy
= mes Q n w we get
11.Homogenization of the system of linear elasticity
154
+J
(a-1+0
- +‘“,w)dS =
an
(2.47) It should be noted here that in the integral over (aRl)\S, the normal v is exterior t o to
dR1, whereas in the integral over dR1n S,,
the normal v is exterior
0‘. T h e last two integrals on the right-hand side of (2.47) can be estimated
by C@
11~011H3(n) llwllHl(n.,
similarly to (2.45). Let us introduce the matrices
phk(t)setting
ah’([) in w
,
in P \ w . Then p h k ( ):
=
a (--) in
It follows from (2.47), (2.48) that
(2.48)
$2. Boundary value problem with Neumann conditions
155 (2.49)
+J
(2.50)
(+o-v,~)~s.
an
The integral identity for uo yields t h a t
Therefore by virtue of Lemma 1.5, Ch. I, mesQnw
an
= (mes &\w)
/
anl
dUO
(uhAhk -,w)dS
a xh
+
53
,
(2.51)
We thus obtain from (2.50), (2.51) t h a t
(2.53)
yhk = (mes &\w)Ahk - phk
in Lemma 2.9. It is easy to verify that conditions (2.34) are satisfied for y h k . Indeed due to (2.48) and (2.29) we have
156
J
11. Homogenization of the system of linear elasticity
-
yhkdS = tn-l(mes &\w)Ahk
J
= En-'(mes &\w)Ahk
-
J
J
phkdS-
q\s*
ohm
phkdS =
urnsc
ahkdS - (mes .; nI .)Ahk " =
a,"\&
J
= En-'(mes &\w)Ahk - cn-' c-1
= En-l(mes &\w)Ahk
a h k ( ( ) d S- (mes
n sC)Ahk =
(u,m\S,)
- (mes Q n w ) a h k -
- c"-'(mesE-'(a~\Sc)
~ ) =A en-' ~ ~(mes &\w - mes E-'(ur\sc) + + mes Q n w - mes c-l(u;~"n s,))Ahk= o , - (mes u r
ns
since
mes&\w t m e s Q n w = 1 , mesc-'(ur\Sc)
+ rnesc'(@
n Sc)= 1 .
W e conclude from (2.52), (2.53) and Lemma 2.9 t h a t lJ2l
I c [Ell2
IbOIIHJ(*)IIWIIHl(n.)
+ E l l 2 IlfOllLz(n)IIWIIHl(n.)
+ I I 4 O - @IIH'* IlWIIH'(n.,]
t
(2.54)
*
It follows from (2.44), (2.45), (2.46), (2.49), (2.54) that ~ ~ e ( W ) ~ ~ ~5z (c1P )[&'I2 11uo11H3(n) IIWllH1(R*)
+
+ E l l 2 Ilf011L2(n)llWllHl(n.) + llfO - f C l l H' * l l ~ I l H 1 p + )
+ I I 4 O - PIIH1. IIWllH'(n.)]
(2.55)
*
From the well-known results on the smoothness of solutions of elliptic boundary value problems we have lluollH3(n)I c2(llf011Hl(n)
+ l14°11H3/1(an)) ,
since dR is a smooth surface and fo E H'(R),
(2.56)
4 ' E H312(dR).
Therefore by virtue of Theorem 4.4, Ch. I , the inequalities (2.55), (2.56) yield (2.26). Theorem 2.4 is proved. 0
157
$2. Boundary value problem with Neumann conditions
2.5. Estimates for Energy Integrals and Stress Tensors
1.3 and 1.4 on the convergence
Slightly modifying the proof o f Theorems
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 instead of
(2.26)
(1.15).
2.1Q (On the convergence of energy integrals). 2.4 are satisfied and EE(u'),Eo(uo) are defined by (1.25), (1.26). Then
Theorem
Suppose t h a t all conditions of Theorem
IEE(uE) - (mes Q n u)Eo(uo))I
5
+ 11$011~3/2(an))+ l l f o
[E1'z(llfoll$l(n)
+ (1I@O11L2(an)+ IlfollL2(n)) (1l.f'
- f"llH1* +
- fCIIH'*
where C is a constant independent of
+
E ; u E , uo
- +'IIH'*
- @'IlH'*)]
7
+ (2.57)
are solutions of problems
(2.22),(2.23) respectively. The proof of this theorem in the main repeats t h a t of Theorem 1.3. However, slight modifications should be made. In particular we consider the solutions uo and u' such that
J
(ue,q)dZ =
ne
J
n*
(u0,q)ds = 0
,
vq E
R.
(2.58)
This choice of uc and uo is possible since solutions of problems
(2.22),
(2.23) are defined t o within a rigid displacement. In this case one can take qE = 0 in (2.26),and use the estimates 0 llH'(n)
11u011H3(R)
5 C ( ~ ~ f o ~ ~+L ~2~( @~ o) ~ ~ L 2 ( a n ) ) 9
5 c(llfolIH'(n)
+
11$o11H3/2(aS2))
9
which are well known from the theory o f elliptic equations (see Similarly t o Theorem
1.4 we establish
[l]).
II. Homogenization of the system of 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
o!(x),o:(z) be defined by the formulas (1.42), (1.43). Then
where c i s a constant independent o f E , t h e matrices GPq are defined by (1.45). Moreover
oP(x) 4 (mes Q n w)oOp(z) weakly in L2(R) as
E +0
2.6. Some Generalizations For the homogenization o f eigenvalues and eigenfunctions related to 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 ,Cc(uc)- pe(x)u' = f' in o,(uc) = $f
on dR
,
Re ,
o,(u') = 0 on S,
,
(2.60)
and also the corresponding homogenized problem
L(uo)- po(x)uo = f o in (mes Q n u ) & ( u o )= $O where operators L ,
P, E L"(R'),
PO
,
on dR
,
E are the same as in problems (2.22),
(2.61) (2.23), the functions
E L"(R) are such t h a t (2.62)
52. Boundary value problem with Neumann conditions and constants
Q,
cl, c2, c3 do not depend on
159
E.
In Theorem 2.4 we established the closeness of solutions of problems (2.60), (2.61) when pc
= 0, po = 0.
If we introduce a parameter characterizing the
closeness o f pe 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 ( ( , ~ )is 1-periodic in 5 and satisfies the Lipschitz condition with respect t o z E
R
uniformly in <, i.e.
p ( t , z) E QR. x 0) in terms of
Lemma
1.6, Ch. I.
Let po(z) = ( p ( . , z ) ) ,where ( p ( . , z ) ) is defined by (1.23), Ch. I, and is
equal t o the mean value o f p ( [ , z )with respect t o
t.
It follows from Lemma 1.6, Ch. I, t h a t for any vector valued functions u,o
E H'(R') we have
Ic-5 l l U l l H l ( W ) IIUIIHl(n*)
3
c = const .
(2.63)
Indeed, set g(<,z)= ( p ( < , z ) - p o ( z ) ) x w ( t ) in Lemma 1.6, Ch. I, where
xw(<)is the characteristic function of the domain w with a 1-periodic structure. It is easy t o see t h a t g(<,x)E L(Rnx !=I), J g(t,z)dt = 0. Consider the extensions P,u, P,v of u , v t o the domain R Q
which were constructed in Theorem 4.2, Ch. I. Then
Note t h a t the set fl\R1 belongs t o a 6-neighbourhood o f dR and 6 is of order
E.
Therefore applying Lemma 1.6, Ch. I, t o estimate the first term in
II. Homogenization of the system of linear elasticity
160
the right-hand side of this inequality, and Lemma 1.5, Ch. I, to estimate the second term, we obtain qc
5 C1E IIPeUIIHl(C2) IIPeVIIHl(fi)
*
This estimate together with (4.17), Ch. I, yields (2.63). 2 Therefore the functions p ( - , s ) and po(z) are close in the sense of the & inequality (2.63). In a more general situation we shall characterize the closeness of pE and PO by the norm
IllPlll =
SUP U,v
{
P(U,VldZI
?
1
IIuIIHl(n*)= 1 1 ~ 1 1 H l ( n ~=) 1
7
(2.64) where the supremum is taken over all vector valued functions u , w in H'(R'). Relation (2.64) implies that for any u , w E H ' ( 0 " ) we have (2.65)
It is easy t o see that estimate (2.63) implies Lemma 2.12.
2
Let p'(z) = p ( - , s ) , &
IllPr
PO
= ( p ( . , s ) ) , p ( t , z ) E i ( I R " x fi). Then
- Poll1 < = 7
where c is a constant independent of
(2.66) E.
Theorem 2.13. Suppose that f' E LZ(R'), fo E H'(R), $f E LZ(aR),go E H312(dR),
E C'(n) and u e , uo are the solutions of problems (2.60), (2.61) respectively. Then PO
(2.67)
$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 to the proof of Theorem 2.4. An approximate solution of problem (2.60) is sought in the form
ii = U O
duo + ~ c p N "xE ( -) , ax,
where uo is the solution of problem (2.61), N" are the same as in Theorem 2.4. Applying the operator Cc - p e l to uc - fi we obtain Cc(UC -
where .Fl,
a) - p c ( U = - a )
=
q ,e, e, are defined by the formulas (2.42)
c = (pc
-
PO)^'
+ E P ~ Q N: . ax, aUo
and (2.69)
For oc(uc- 6 ) the formula (2.43) remains valid. Setting 20 = 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 fl\fl, for the solution uo of problem (2.61). Further changes in the proof of Theorem 2.4 are obvious.
$3. Asymptotic expansions for solutions of boundary value problems 163 $3. Asymptotic Expansions for Solutions of Boundary Value Problems
of Elasticity in a Perforated Layer
3.1. Setting of the Problem Consider a domain
RE o f the form
where w is an unbounded domain with a 1-periodic structure satisfying t h e Condition
B
of $4.1, Ch.
I,
E
> 0 is a
small parameter, and
E - ~ is
an integer
number. Set
If w In
# R",then RE is a
perforated layer.
0' we consider the following boundary value problem
u c ( i , o )=
W ( i ) on
uE is 1-periodic in
Here Ah'([)
ro,
u c ( i , d )= i p 2 ( i ) on
,
rd
1
I
I
i = (q, ..., xn-l) .
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 of problem (3.1) under suitable assumptions on
f , Q1,
a2are 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"),
@3
E Cm(Rn-'),
f , @j
II. Homogenization of the system of linear elasticity
164
Our aim is t o find an asymptotic expansion for the solution uc 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
Here we reproduce the results obtained in [87].
k > 0 the solution
u' of problem (3.1) can be represented
in the form k U E ( 2 )=
+
81
PA(? ,&)Y:(2,&)
&'
E
m=O
I=O
E k p k ( & , 2)
where PA((,&)are n x n matrices such that
PL(E,E) = PL,(t) and PA,, are 1-periodic in (,
d
+ PA,(t)+PAz(i,tn- -) &
7
PAl(()and PAz(i,En - f ) define boundary lay-
ers, the components of the vectors Y;(x,E)are polynomials in
.E
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 {z : 0
< x , < d}, the remainder IIpk(E,x)IIHi(fie)
with a constant
5 Mk
Mk, independent
p k ( ~ , z satisfies ) the inequality
k = 1,2,*** 7
1
of
E.
3.8. Formal Construction of the Asymptotic Expansion We seek the solution o f problem (3.1) in the form uc =
2 c €'
I=O
(+I
Na(t)DD"va(x) ,
t = €-lz.
(3.2)
In contrast t o Chapter I, here for the sake o f convenience we use the following notation
a, takes the values 1, ..., n; N a ( ( )are matrices whose elements are 1-periodic in
t ;v , ( x ) = (vf, ...,vi) 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
a (v(x,€-%)) = av(x7 t ) + &-1 %x, ~
dXi
Xi
we obtain the formal equality
Here we used the following notation
at;
0,
[ =e-lx,
11. Homogenization of the system of linear elasticity
166 for ( a )=
1,
5 c N,(t,;)D% - d
where
for ( a )> 0 and
E
~($1,
E
rd ,
$3. Asymptotic expansions for solutions of boundary value problems 167
for ( a ) = 0. Let us represent
Na(<) in the form
NOI(t) = N,o(E)4where
mo + mo
>
N : ( [ ) are matrix valued functions 1-periodic in t , N i , Nz define bound-
ary layers near the hyperplanes Set
2,
= 0 and
2,
= d respectively.
N: = E , N t = N t = 0, where E is the unit ( n x n)-matrix. Denote
a
a
T,"(5)= (AkawN:2...011+ ( t )A) " * W - ~:2...01,(o + %k atj +
AOIlW
(oN:3...*l 7
z:(E)= ~ k A k a 1 ( E ) N ~ 2 . . . a t P( t=) 0 , 1 , 2 7
7
where N z = 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
< E aw , N : ( E ) is 1-periodic in t , J N , O ( ~ ) ~ E= o ,
a(N:) = - Z z ( t ) on
(a)= 1,2, ...
(3.5)
Qnw
.
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
168
J
i,
N : ( t ) is l-periodic in
where h i , h: are ( n x n)-matrices with constant elements chosen in such a way t h a t the inequalities
s=o,1,2
s =
,...)
d d
-, - - 1, ... , &
(3.9)
&
hold with constants Cp, C;,
K:,
K:
independent o f
E.
(3.6), (3.7) and existence o f the h i can be proved by induction on the basis o f Theorem
Existence o f the solutions for problems constant matrices h;,
8.4, Ch. I. Note t h a t because o f the boundary condition in (3.7) on the hyperplane
&,
d = - , the matrices N," and h i depend on &
from the l-periodicity in
N : ( t ) on
E
t
E.
If d is a multiple of E it follows
of the matrices A h k ( J )t h a t the dependence of
is determined by the relation
d N:(t) = fi:ci,tn - );
9
53. Asymptotic expansions for solutions of boundary value problems 169 where
fi:([)
are solutions of the corresponding sequence of problems of type
(3.7) in w(--oo, 0) with the boundary conditions
N:(t,O)= -NZ(t,O) + kz . Obviously the matrices Having thus defined
fiz,i t do not depend on E .
NZ, p = 0,1,2,, let us substitute v, in (3.1). We get
the formal equalities
nc, c c , a*(?)= c c +N:(t c c ( h i + N i ( ; , ;))D% W
f ( z ) 2i'
zE
ho,D"vc(z)
2-2
(+I
I=O
(3.10)
00
(hL
€I (+I
I=O
&
3
00
@(3)
1=0
,O))D%, d
(3.11)
(a)=/
It is easy t o verify t h a t here
hO, = hO,l = 0
, hi = hi = E
(3.12)
(c3W)\(I'o U r d ) are satisfied due t o the boundary conditions in (3.5),(3.6), (3.7) for the matrices N,, NA, N:. Note t h a t by virtue of (3.4)the constant matrices h:,,, are defined by the
and the boundary conditions on
formulas
Comparing these equalities with
(1.3)we conclude that h$ = aij,i , j =
1, ...,n, i.e. h:j are the coefficient matrices of the homogenized elasticity system. Let us seek vc in the form of a series
c 00
v,(z) =
€jVj(X)
.
(3.13)
j=O
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 of linear elasticity
170
0 0 0 0
Therefore by virtue of (3.10) we find that (3.14) Consider the first equality in (3.11). Due to (3.13) it is obvious that
=
c c c
h;Z)"Vk-,
Ek
k=O
.
m=O ( a ) = m
By virtue of (3.9) we have ?
I I N ; ( ; ,o ) I I ~ ~ , Z ( ~ , ) L ce-
, y,c = const > o .
Therefore the first equality in (3.11) yields
a'(?)2
c c c k
00
Ek
k=O
m=O ( a ) = m
In the same way we find that
hiDaVk-m
I
(3.15) =o
I "
(3.16) k=O
m=O ( a ) = m
Equating the terms of the same order with respect to E in (3.14), (3.15), (3.16) we get the following recurrent sequence of problems for y(x):
$3. Asymptotic expansions for solutions of boundary value problems 171
& ( x ) is 1-periodic in
P,
j = 0,1,2,
I
... .
Here
j = 1 , 2 , ... . Existence of
Vj follows from Theorem 6.5, Ch. I, when w = IR" and the
coefficients of the elasticity system are independent of
E.
3.3. Justajication of the Asymptotic Expansion. Estimates for the Remainder In the previous section we constructed a formal asymptotic expansion for t h e function u' which is the solution of problem (3.1). This asymptotic ex-
pansion has the form (3.2) where
No = N:
+ NA + N:,
N,, NA, N: are
solutions of problems (3.5), (3.6), (3.7) respectively, u, 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
(3.19) where L.
(3.20) j=O
In t h e next theorem we give an estimate for t h e remainder term of the asymptotic expansion for t h e solution u' of problem (3.1). Theorem 3.1. Let u' 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
172
IIdk)- U=llHl(fi.) I M k E k + 1 ,
(3.21)
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, p = 0,1,2.
Lemma 3.2.
The solutions N,P, p = 0,1,2, o f problems (3.5), (3.6), (3.7) satisfy the inequalities
J
(lvm;)12 + IN,(;)I - ) dx < - M," ,
A*
2
ll~;(;)llHl,z(~d) X llm;)llH1,2(Fo)
where ME,
cj, -yj
,
Ise-? , I cze-
(3.22) (3.23)
21
(3.24)
7
are positive constants independent of
Proof. Let us establish (3.22) 0,1,2,
p = 0,1,2
E.
for p = 0. By induction with respect t o (a)=
... we obtain from Theorem 6.1, Ch. I, t h a t
J
( I V ~ X ( O I+~I N , o ( ~ ) I ~ I ) ~C, E
Qnw
Changing the variables of
2
=
E<
and taking into account the 1-periodicity
N,"(t) and the fact t h a t the domain
h. contains
not more than (d
+ 1 ) ~ - "cells ~ ( +z Q nu), z E Z",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 t h a t
J t(O,
where
9)
( I V ~ N ; ( O+ I ~1 ~ ; ( 5 ) 1 ~ ) I d t ~a
M , is a constant independent of
E.
Passing t o the variables
in this inequality we obtain (3.22) for p = 1 since and the domain z =
(2,O).
he contains exactly ~ - " + l
domains
x =
E(
N:(E) are 1-periodic in [ E(Z
+ h(0, i)), z E Z", E
$3. Asymptotic expansions for solutions ooff boundary value problems 173 Estimate (3.22) for p = 2 is proved in the same 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 1 / 2 ( t ~ H'I2(f'd). )H112(i!d). , Lemma 3.2 is proved. of the norms in H H1I2(t0), Proof of Theorem 3.1. Let us show that the vector valued function u ( ~given ) by (3.19) is the solution of the problem . ~ , ( u ( ' ) )= f
+
+
E ~ + ~ ~ OE () Z ,E ~ + '
d k ) ( ; , 0 ) = a'(?)+ ~ ( ~ ) ( ; ,= d )a'(?)
a
-gm(z,E ) in R' ,
\
axm e k + + ' d 1 ( i , e on ) ro,
+ E ~ + ' ~ , ( $ , E ) on
C = ( U ( ~ )= ) V , E ~ + ~ ~ ~ ( Z ,on E )
u ( ~ ) ( xis) 1-periodic in f
,
r d
>
(3.25)
(anc)\(rou r d ) ,
,
1
where n
Mo, MI, Mz
(3.26)
are constants independent of
E.
Then estimate (3.21) would follow directly from Theorem 6.5, Ch. I. Consider first the boundary conditions for ti@)("). We have
C
k+l
~ ( ~ ) ( f ,=d )
E'
f=O
C s=o
ht
(a)=f
k
=
C
cc S
E'
t=O
(+t
Ck
+ 7,' =
E'V"V,
j=O
x,=d
I +
hX'DD*V,-,
x,=d
II. Homogenization the system system of of linear linear elasticity elasticity Homogenization of the
174
where where
(3.
^.
and hz are assumed t o be zeros if the length of the index larger than k 1. Due to the conditions (3.18) we have
Q
is negative or is
+ 7: .
(3.29)
+
2k+l
u ( k ) ( ? , d )=
+C
k
C C
hZ,D"V,
Taking into account the smoothness of V, in the layer { z : 0 5 z, 1. d } we conclude from (3.23), (3.28), (3.29) that u ( ~ ) ( ? , = ~ 2 ( ? ) + & k f l 2 9 2 ( ? , € ) and the second inequality (3.27) is satisfied. In the same way we prove that u ( ~ ) ( ? , O ) = @ I ( ? + ~ ' + + ' d , ( ? , . z )and that the first inequality (3.27) is satisfied. Let us now calculate o,(u(~)) on d q r o u rd). Setting E = .z-'z, due to (3.3) we have
(3.30)
$3. Asymptotic expansions for solutions of boundary value problems 175 Here we used the equality B,(() = 0 for
t E & - l (bR‘\(ro 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
+
A*lW ( t ) N a , . . . a k + z ] D a v ( k ) ( x )
C
+ &k+l
AWWN a 3 . . . a k + 3 ( E ) D a V ( k ) ( x )
7
(a)=k+3
t = &-12
’
(3.31)
Since N,“, N:, 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 kA2
(3.32)
+
Ek+l
c
(a)=k+3
A a1w ( ~ ) N o s . . . o * + 3 ( ~ ) D ,v(k) (2) .
Let us transform the expression (3.32) setting ( a ) 2 k 3. We have
+
i:= 0 for
C,(u(”) =
c c s=o
k+2
(+I
j=O S
c k
k+2
k0
(3.33)
&
hO,DD*
(a)=s-j
j=O
&j<
+ 7,“
ft:
=
= h: for ( a ) 5
Ic + 2,
II. Homogenization of the system of linear elasticity
176
s=k+3
j=o
s=k+3
j=o (.)=s-j
(a)=.-j
Therefore it follows from (3.17), (3.18), (3.33) t h a t
(3.34) where
(3.36) Estimate (3.26) holds due to the inequalities (3.22), Lemma 3.2 and the smoothness of V,. 0 satisfies (3.25). Theorem 3.1 is proved. It is obvious t h a t Remark 3.3. It follows from the estimate (3.21) and the equalities (3.19), (3.20) that u'(x) =
=
u(k)(x)
+&k+lQ(X,€)
=
c c N$) c
k+l
k
€'
I=O
(+I
&
j=o
. € T V j ( X )
+&k+'q(x,€)
,
(3.37)
53. Asymptotic expansions for soJutions of boundary value problems 177 where JIq(z,E)JIHl(fie) u"(2) =
5M
with a constant M independent of
c c c k
k
&*
t=O
j=o (a)=t-j
N,(qD"1/, &
+E"+'q1(s,E)
E.
Therefore
,
where llq1(x,E)llL2(fi.)I Ml, and constant
Ml does not depend on E , N , = 0 for negative (a).In particular we obtain for k = 0 llu'(5) - VollL2(fiC)
I CE
7
where C is a constant independent of
E.
It also follows from (3.37) for k = 0 t h a t
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 of order
E
for the remainder
term, whereas without the boundary layers we can only get an estimate of order
E
~
as/ in ~ Theorem 1.2 with
= a', fo = f" (see estimate (1.15)).
II. Homogenization of the system of linear elasticity
178
$4. Asymptotic Expansions for Solutions of the Dirichlet Problem for the Elasticity System in a Perforated Domain Here we consider asymptotic expansions in
E
for solutions o f the Dirichlet
problem for the elasticity system in a perforated domain
R' with
a periodic
structure. The displacement vector is assumed t o vanish on t h e surface o f t h e cavities
S,.
Similar asymptotic expansions for solutions o f t h e Dirichlet problem for t h e equation AuE = f in a perforated domain
RE were
obtained in [52], where
the estimates for the remainder term were proved in the case
f E C,"(R).
In order t o justify t h e asymptotic expansion, when f ( x ) is sufkiently smooth and may be non-vanishing in a neighbourhood of
dR, we construct boundary
layers which exponentially decay in x with the increase o f the distance from z to
aa.
4.1. Setting of the Problem. Auxiliary Results Consider a perforated domain domain o f
R" with
by the vectors z E class C' and
RE = R n E W , where
w is an unbounded
a 1-periodic structure, i.e. w is invariant under the shifts
Z".It is
assumed here t h a t Q\w contains a surface of
R is a smooth bounded domain.
Note that in this section we do not impose any restrictions on the smoothness o f w. In R' we shall study the following Dirichlet problem
U'
= 0 on
dR',
f E C"(R) ,
where Ah'(<)are ( n x n)-matrices o f class E ( n l , n2) and their elements a$(<) are 1-periodic in (. The aim o f this section is t o justify the asymptotic expansion
=c M
uE(x)
1=0
<=
&-I2
,
&It2
c
(+I
N c r ( & , ( ) D a f ( ,x )
$4. Asymptotic expansions for solutions of the Dirichlet problem
179
for solutions of problem (4.1). Here a,ID" are t h e same as in $3, N o ( [ ) are matrices of the form N , = NL
+ NZ, where the elements of N t are functions 5
defined in w and 1-periodic in (, the elementscof N ~ ( E -) , decay exponentially
Re with on f. in
the increase of t h e distance from
E
z t o dR, N:, N:
do not depend
Let us now prove some auxiliary propositions t o be used below for the justification o f the asymptotic expansion (4.2). Lemma 4.1. For any vector valued function w II4IL2(n.)
E HA(R') the following inequalities are valid
I ME lle(4llLqn*)
IIV'IUIIL2(n.)
5
(4.3)
I
Ile(w)IlLqn.)
where M is a constant independent of
(4.4)
I
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 H;(R) such that 6 = 20 in R', 6= 0 in R\Rc. Let us prove (4.3).
Obviously we can assume t h a t
R"\Re. Denote by T' the set o f all + Q ) n R # 0 and consider the function W ( [ )=
20
is defined in
R"
and vanishes in
z E Z" such t h a t
E(Z
~ ( E J ) .
Taking into
account the properties of aw and the fact t h a t 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
J (Z+Q)nw
IW(t)124 I c
J
I V W 2 4.
(4.5)
(Z+Q)nw
Summing up these inequalities with respect t o z E
T"
variables x = E [ we obtain (4.3). Lemma 4.1 is proved. Lemma 4.2. Let U ( z ) E H1(Rc)be a weak solsution of the problem
0'
I
and passing t o the 0
II. Homogenization of the system of linear elasticity
180
r
7
(4.7) 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
J R C
b2J r
5
I V W I ~ ~~z~
n*
+J
~ f ~ l ~ d a : (fi,f')dz
1fol2da:+ & z
Q
J
n*
n.
+J
J
7
I V Q I ,~ ~ ~
R'
(f',fi)dX
+ €2 J
lviplzdz]
,
nz
K3 do not depend on E. Therefore estimates (4.7) are valid, since w = U - i p . Lemma 4.2 is proved.
where the constants Kz, 0
The next theorem shows in particular that the solutions of problem (4.6) have the form of a boundary layer in the vicinity of
dR, provided t h a t @ = 0
54. Asymptotic expansions for solutions of the Dirichlet problem on
181
( d P ) n R and f'(z), i = 0, ..., n , rapidly decay in Re with the growth of
the distance from x t o 30.
C'(n) such t h a t T = 0 in a neighbourhood of 8 0 , r 2 0 in R, (Vr1 5 M = const. It is assumed that E is so small t h a t there is a subdomain 0' c R whose closure consists of the cubes EQ+EZ with z belonging t o a set T, c Z", and dR' lies in the neighbourhood of 80 where T = 0. Consider a scalar function r(x) E
n'
Theorem 4.3. Let U(x) be a weak solution of problem (4.6) with @(x)= 0 on
(aW)\dO
(i.e. Q, E H'(R",S,). Then E-2
J
Iu12exp(fi)dx+ &
nennl
where
J
67 IVU12exp(-)dx &
+
n*nW
K , 6 are positive constants independent of E .
Proof. Set
v = (epT- 1)U in the integral identity (3.15), Ch. I, for U(x),
where p = const
Since
T
> 0 is a
parameter t o be chosen later. We have
= 0 outside R',
we find by virtue o f (3.13), Ch. I, t h a t
II. Homogenization of the system of linear elasticity
182
(4.10) Due t o the Korn inequality for vector valued functions w E H1(Qnw, a w n
Q ) (i.e.
20
= 0 on (dw) n Q ) we have
J
( I ~ I+ ' I
I
v C ~ I ~ ) M ~ ~
Qnw
J
lec(w)l2dt .
(4.11)
Qnw
This inequality follows from Theorem 2.7, Ch. I, if we extend
w as zero t o
Q\w and note that Q\w contains a surface o f class C'. Passing t o the variables
t = E - ~ X in (4.11)
we obtain for any w, = E(Q
w)
+ E Z C R' n R', z E T', the following inequalities
J
IUl2eXP(P~)dz5
n
WZ
(4.12)
54. Asymptotic expansions for solutions of the Dirichlet problem
U
Therefore since p = -we find from (4.13) that 2 d G E
183
II. Homogenization of the system of linear elasticity
184
+J
Ie(U)l2dz .
(4.15)
n* Choosing u sufficiently small and independent o f E and taking into account U
-we
obtain from (4.15), (4.14), (4.13) the estimate (4.9). 2 Theorem 4.3 is proved. 0
that p =
,m& .- - -
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 a w , w is 1-periodic in [ ,
(4.16)
where A h k ( [ )are matrices o f class E(rcl, Q ) , w = (w1, ...,wn)*, 3j E L2(Qn u), 3 3 are 1-periodic in l , j = 0,1, ..., n. A weak solution of problem (4.16) is defined as a vector valued function
w E$ ( w ) = k i ( w ) n H1(Q nu, Q n 8u)which satisfies the integral identity (6.2), Ch. I, for any v €$ ( w ) . Theorem 4.4. There exists a weak solution of problem (4.16) which is unique and satisfies the estimate
c n
5
~~wI!H1(Qnw)
j=O
IIF’llLz(QW)
*
185
54. Asymptotic expansions for solutions of 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
c
c
00
.ii"(.) g
(+l
I=O
X
Na(c,t)DDolf(x) , [=-
(4.17)
E '
in the equations (4.1). Formal calculations similar t o those o f 53.2 yield
L"(.ii.(Z))
2
g c 6'
I=O
(,)=I
H L 1 ( t ) D 0 ( f (gX )f(x)
,
(4.18)
where
t A""(()
a
- Naz...a,(t)t A"1"2(t)N03...a.(() ,
atj
Let us seek
3
L2
.
N , ( t ) in the form N , = N,O(t) t NA([), where N , ( t ) are
matrices whose elements are l-periodic in ( functions belonging t o the elements of from
2
to
X
~
(w), and
N : ( - ) decay exponentially with the increase o f the distance
dR.
G
We introduce the notation
T,OEI, T,'=O,
11. Homogenization of the system of linear elasticity
186
where I is the unit matrix. Define the matrices NE(J) as weak solutions of the problems
N,"(()= O on bw
,
N,O(t) is l-periodic in
t .J
The matrices N t ( J ) are defined as weak solutions o f the problems (4.21)
J
NA(() = -NE(() on a(&-lfl").
Existence of N,, NA can be easily proved by induction with respect t o
I
on the basis o f Theorems 4.3, 4.4, Ch. I. Lemma 4.5.
2
The matrices N i ( -) satisfy the following inequalities &
llv,NillL~(n.) + ((N:llLZ(n.) I c, (4 2 j=o,1,
'5
7
(4.22)
01
where C, are constants independent o f
Proof. Relations (4.19),
E.
X
X
&
&
(4.20), (4.21) show that N , ( - ) = A':(-)
are solutions of the following boundary value problems:
&(No) = &-'I
in R'
,
No = 0 on aR"
,
(4.23)
Let us use induction with respect t o 1. For 1 = 0 it follows from (4.23) due to (4.7) with
+ N ~ ( E-)2& ,
= 0 that
§4. Asymptotic expansions for solutions of the Dirichlet problem
where C, is a constant independent o f
187
E.
Let 1 = 1. By virtue of (4.7), (4.24) we get
+ E-l IlLqn.) I M z ( E IIVz~ollL2(n.)+
IlV,Na, IPa1
IlLW)
I Ml (IlvzNollL2(n.,
IINoIlLZ(nc))
IINollL2(nc))
These inequalities and (4.26) imply that for lc IIL2(n') I 'al...akE-'
llvzNal...ak
I\Nal...aklIL2(n')
I
cC71...a,
5
k
.
1 we have
;
(4.27)
'
Suppose now that the inequalities (4.27) hold for t h a t they also hold for
,
kI 1 - 1.
Let us show
= 1.
It follows from (4.25), (4.7) t h a t
I c 1 [ IlVr~a2...a1 Ilv(n.) + E-l IINaz...allIL2(n.)
llvz~al...al11;2(n.)
+ E-l IPa1...a1
IILqn.)
+
IINa3...at11~2(ne)]
I cz [E IlV~a2...a1 IIL2(n*)+
+ II~a2...alllLZ(n.) + IINa3...aIIIL2(n')]. kI 1 - 1 we obtain
Therefore due t o (4.27) for
(4.27) for
k = 1.
The elements of the matrices N,"(() are 1-periodic in (. Therefore estimates (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.
0
Lemma 4.6. The elements o f matrices N:(a, subdomain
Ro such
that
2
-) are of boundary layer type, i.e. for any
E noC R the following inequalities are valid
(4.28) where C,, 7 are positive constants independent of
k f . Consider a domain
a'
such that
E.
0' c R , noc R'
and the distance
II. Homogenization of the system of linear elasticity
188 between of
E,
and
Ro and dR' is larger than
0'
consists of the cubes
The parameter
E
K
> 0,
where
h:
is a constant independent
+ z ) , z E T , for some subset T c Z".
E(&
is assumed so small t h a t
R'
with the above properties exists.
C'(n), r f 1 in Qo, r = 0 outside the --neighbourhood o f Qo, IVr1 5 C K - ' , C = const. Let us construct a scalar function r(x) such that r E K
2
Using the induction with respect t o s = 0,1,2, for a = (a1, ...,a,) satisfy the inequalities
X
... , let us prove t h a t N : ( -E)
where C,, 6 are positive constants independent of
E.
Let us first show t h a t (4.29) holds for the matrix N,' which is a solution of the problem 2
, NJ(-)&
L c ( N J )= O in
X
= -N,O(-) on a R c . E
X
N : ( - ) = 0 on dR'\dR, we can apply the estimate (4.9) of Theorem & 4.3 t o N i . We get Since
E-2
J
INA12exp(fi)dx &
n*nw
2 (I
J
+J
IVzNJ12exp(-)dx 87 E
R'
5
IV,N,'I2dx.
n* This inequality together with (4.22) implies (4.29) for
N t . Fix a positive
NA with a = (a1, ..., a l ) , 1 2 s - 1. Let us show that (4.29) holds for NA , with 1 = s, where NAl,.,ae
integer s and suppose that (4.29) is valid for all
is a solution of the problem
a
d
Lc(NAl,,,a,)= &-'Aa1' -N:2,,,aa
-(Ak"'N~Z...018) -
dXj
k
- & - ~ A ~ Ia3...aS ~ Z Nin* a" , X
N:l...,8($ = -NO
X
on do' 2
.
Taking into account the fact that N , ( - ) = 0 on &
Hlc\Xl and using the
estimate (4.9) of Theorem 4.3 applied t o NAl,,,,,,we obtain
189
$4.Asymptotic expansions for solutions of the Dirichlet problem
J
E-2
R"V
I K1
+
[l
+ J
67
l~;l...,,12
exp(-)d. E
n*nw
lvzN:l...a8 I2d. t E2E-2
E-4&2
J
67
lvz~:l...,a12 exp(-)dz E
J
67
n*nw
+
6T
lvzN:z...ae I2 e x p ( 7 ) d .
n.nw
IN:,...,fexP(p.
I
+ E-2
J
IN;z...a.12 exP(-)dz ST &
nwni
1
.
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 NL1...,#. The estimates (4.28) follow from (4.29), since
7
f 1 in
Ro. Lemma 4.6 is
proved.
0
Theorem 4.7. Let u'(z) be a weak solution of problem (4.1) with
c c 8
u:(.)
=
(+l
d
? I : ( = . )
l=O
where
N, = Nz
&1+2
+ NA, N,,
I
c N,(paf(4 c N,O(p"f(4 7
1=0
f E C"+'((S2).Set
1
(,)=I
(4.30)
NA are weak solutions of problems (4.20), (4.21)
respectively. Then
113.).
- uC(4IIH'(n.) 5 COEa+2Ilfllc.t~(n),
(4.31)
IIv:(z)
- Uc(z)IIH1(n*nn0)I C~E"+'Ilfllc+z(n) ,
(4.32)
Ro is a subdomain of R such that not depend on E ; C1 may depend on RO.
where
h f . Let us apply the operator
C, t o u:
noc R, the constants Co, C1 do - u'. Assuming
N , = 0 in (4.17)
for (a)2 s we obtain in the same way as (4.18) that
L,(uZ - u C )= Ed+1
2
AOlk
ol,...,astl=i
a
@k
,I
...O . + l
Naz-'O'tlD
f+
11. Homogenization of the system of linear elasticity
190
c n
+
a], ...,a,+2=1
7p...a,+2 f in 0' , A"'"' N"3...as+2
u: - u ' = 0 on dR'
I
Note that, because of (4.22), the matrices
x
d x N a ( - ) , - Na(-)are E
atj
E
Lz(Rc) norms of the elements of the
bounded by a constant independent of
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) it suffices to observe that
No = N:
the inequalities (4.28). Theorem 4.7 is proved.
+ N:
and
N i satisfy 0
191
55. Some generalizations for the case of perforated domains $5. Asymptotic Expansions for Solutions of t h e 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 :
A2u'(x) = f(z) in R'
auc
,
,
u ' = - = 0 on dR' dU
(5.1)
and obtain a complete asymptotic expansion for solutions of this problem.
R' is a perforated domain of type I with a periodic structure described in 54.1, f(x) is a sufficiently smooth function in R; u is the outward normal. Here
We seek the asymptotic expansion for the solution o f problem (5.1) in the form
u: =
c N,(&,[)DDLIf(x) ,
c
(+I
k 0
[ = &-lX
,
(5.2)
where DD", 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
Hi(R') the following inequality is satisfied
E - 2 )I'UIILZ(*.)
where
E2(v) = (-
+
d2V
dxidzj
E - l Ilvv(lL2(n*)I
Ml llEz(")IlL2(n.)
aZv '", -)dx;dxj
Proof. Obviously it is sufficient
>
(5.3)
MI is a constant independent of E .
C,"(R'). Set v = 0 in Rn\Rcand denote by T' t h e set o f all z E iz" such t h a t E ( Z + &) n R # 8. Consider the function W ( [ )= v(e[). Since W = 0 in W \ w , the Friedrichs inequality for each o f the sets w, = z Q yields t o prove (5.3) for v E
+
II. Homogenization of the system of linear elasticity
192
Summing these inequalities with respect t o z E
T' and passing t o the
variables x = E<, we obtain (5.3). Lemma 5.1 is proved. Let @ E
H2(Rc),f j E L2(Rc), j = O,1,2, ...,n.
We say t h a t U(x) is a weak solution o f the problem
A'U(x) = f o
af' +axi
in Re ,
au-- a@ U=@, au
on
du
(5.4)
anE,
if W = U - @ belongs to H;(R") and satisfies the integral identity
+J
(fov
- f' 5 ) d x dXi
n' for any
v E H,2(Rc).
Denote by @(w) the completion with respect t o t h e norm IIvII@(snw)of the functions and
v(t) such that v E
C2(G), w = 0 in a neighbourhood o f dw
v(E) is 1-periodic in t. Here w is an unbounded domain with a 1-periodic
structure, the same as in s4.1. We say t h a t w is a weak solution o f the problem
Aiw = F o ( t )
aw av
w=-=O
dFj +in atj on d w ,
w(<) is 1-periodic in
t,
w
,
I
J
where Fj E L2(wr l Q),F j ( < ) are 1-periodic in and satisfies the integral identity
t ,j
= 0, ...,n, if w E @ ( w )
$5. Some generalizations for the case of perforated domains
for any
193
v E &(u).
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 ( 2 ) of problem (5.4) sdtisfies the following inequalities
II. Homogenization of the system of linear elasticity
194
J
It follows t h a t
Since W = U - 0, this inequality implies (5.8). Due t o
(5.3) we have IIWIIL2(nq 5
M 1 E Z IIEz(W)IlL2(n.) >
IIVWIIL2(n.) I MlE IIEz(W)IlL2(n.) . From these inequalities and (5.11) we obtain (5.9), (5.10), since Lemma 5.2 is proved. Let r(z) be a function of class Cz((sl)such t h a t
7
W = U - CP. 0
= 0 in a neighbourhood
a R , 7 2 0 in R. Consider a subdomain R' defined just before Theorem 4.3 and assume t h a t 7 = 0 outside 0'.
of
Theorem 5.3.
acp
U ( z )be a weak solution of problem (5.4),CP = - = 0 on aR'\aR (i.e. av cp E HZ(Rc, aRc\an)).Then Let
(5.12)
195
55. Some generalizations for the case of perforated domains where K O
> 0, 6 > 0
can depend on
are constants independent of
E.
(Note that
KO
and
S
52' and Il~(~)llczcfi,.)
Proof. For any v(x) E H,2(RC)the function U ( x )satisfies the integral identity
Set v = (ep'
-p
J
- 1)U, where p > 0 is a
d2U
n*
a2r
U -eprdx-p2
axiaxj
parameter t o be chosen later. W e have
J
n c
a2u
a7 a7
-U -
axjazj axi ax, e p T d x t
(5.13) Since
T
= 0 outside of a', we obtain by virtue of the Holder inequality
(5.13) that
and
II. Homogenization of the system of linear elasticity
196
In the same way as in the proof of Theorem 4.3 to obtain (4.14), we find that
J
IU12epTdx5
J
IVU12ep7dx,
~ 1 . 2
nCnW
(5.15)
Wnnl
where p = u/I
E,
u E (0,l) is a constant
Since U ( x ) can be approximated in the norm of H2(R")by functions vanishing in a neighbourhood of ane\Xl it follows that inequality similar to (5.15) holds for the first derivatives of U ( x ) , i.e.
J
J
IVU12eFTdx5 K2e2
nwnl
1 ~ 2 ( ~ ) 1 2 e o. ~ d x
(5.16)
nwnl
Estimates (5.15), (5.16) yield
J
IU12ep7dx5 K3c4
ncnnl
J
where K2, K 3 are constants independent of From (5.14), (5.16), (5.17) we obtain
J
IE2(U)12epTdxI c e p e
n*
(5.17)
,
n*nW E.
+
J
IEz(U)12epTdx
ncnnl
+ c,(p t p2)c2 J
(
+ C,e2 J
n*nnl
n*nw
+
IE~(U)I~~'T~~
(J
lfo12ep7dx)1'2
nennl
IE2(U)12ep7dx
$5. Some generalizations for the case of perforated domains
197 (5.18)
n* where p = u / K e . If we choose u sufficiently small but independent of
c , we
get from (5.18) the following inequality
J
IE2(U)12eMTdx I ~
nc
+ Mzc2
1
ljol2epTdx
€
4
n*nnl
+
f ' f ' e p T d x M3
WnW
where
J
~
+
1
IE2(U)12dx ,
(5.19)
na
Ml, M2, M3 are constants independent of c .
Estimate (5.12) follows from (5.19), (5.16), (5.17). Theorem 5.3 is proved. 0
5.2. Justification of the Asymptotic Expansion for Solutions of the Dirichlet Problem for the Biharmonic Equation
f E C8+4((s2) in (5.1). Let us seek an asymptotic expansion for the solution o f (5.1) in the form (5.2) where N,(E, [) = N:([) + N:((E, t), X N,(t) are l-periodic in (, NA(e, -) are functions o f boundary layer type in E R", which decay exponentially with the increase of the distance from x to dR. Suppose t h a t
It is easy t o verify t h a t
Therefore
+
2€-3
a
- A, N,D"
at;
af + 4€-2 a IN, D" a2f -
axi
atj ati
+
axiaxj
198
11. Homogenization of the system of linear elasticity
+ E - ~ A ~ N , Vf~+A~, - l 2P atj
Aaf + N,V"A: f .
ax
(5.20)
From (5.2), (5.20) we obtain
(5.21)
where S, is the Kronecker symbol. Let us define the functions N , ( E , ~as ) weak solutions of the following boundary value problems
NO
AiNo = 1 in E - ~ Q ' , No = -= 0 on
au
a(~-'n')
(5.22)
199
$5. Some generalizations for the case of perforated domains
a
Nal...a, = - Nol ...a, = 0 on av
a(&-'Q").
On the basis of Theorem 1.3, Ch. I, we can easily prove by induction t h a t
N a ( [ )exist. Let us show t h a t periodic in
t
N , ( E , ~= ) N:
+ NA, where
N : ( [ ) are functions 1-
and belonging t o & ( u ) ; N:(E, f ) are of boundary layer type in
0". Set
j=O,l,
m24,
Define the functions N : ( [ ) as solutions o f the following boundary value problems w
,
8N,o N , = = z -- 0 on dw
,
A i N : ( t ) = T,O([) in
N , ( [ ) is 1-periodic in
Obviously Theorem 1.3, Ch. I, guarantees the existence of In the domain Dirichlet problems
&-*nC define
the functions
.
I
(5.23)
N,([).
N: as weak solutions o f the
II. Homogenization of the system
200
Obviously
of linear elasticity
N , = N," -k NA.
where the constants
M, do
not depend on
E;
j = 0,1.
This lemma is proved by induction in the same way as Lemma 4.5. Lemma 5.5.
X
The functions N ~ ( E -) , are of boundary layer type, i.e. for any subdomain such that !=lo c IIN:(E,
where
R
&
no
the following inequalities hold X
c,
,)lla(nonn=) I exP(-7&-')
(5.26)
7
C,, 7 are positive constants independent o f E, (C, and 7 may depend
no). Proof. The estimate (5.26) on
is obtained in the same way as (4.28) in Lemma
4.5. Let us indicate the main steps of the proof. Consider a subdomain some z E
Z", and
R' c R
which consists of the cubes
E(Z
+ Q ) for
let ~ ( x E) C2(O)possess the same properties as in the
proof o f Lemma 4.5. 5
The function N;l.,,,m(~,-) &
. IS
a weak solution o f the problem
55. Some generalizations for the case of perforated domains
Due t o
J
E-4
aN:
-= 0 on aW\aQ,we can apply av (5.12) for U = N:l,,.am we get
Since N o =
n*nn'
J
67
~~:~...,,~~exp(-)dx E +&-'
where ICz is a constant independent of
nennl
Theorem 5.3 t o
201
U = N:.
67 I V N ~ , . . . ~ , I ~E ~ X P+( - ) ~ ~
E.
From these inequalities and (5.25) we obtain by induction with respect t o
... that
m = 0,1,2, E-4
J
n*nn!
where the constant 7
+
67
~ ~ : , . . . , , , , ~ ~ e x p ( F ) dEz -
K,,,,,,,
~
J
nCnW
67 I V ~ N : , . . . , , I ~ ~ X P+( ~ ) ~ ~
does not depend on
= 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 of solutions of problem (5.1)).
Let
~ ' ( 5be )
).(:it
a weak solution o f problem (5.1) and let
=
2 I=O
c Ncr(€,
"Vf(a:)
(CI)=I
&
,
f E CSf4((s2),
0
202
11. Homogenization of the system of linear elasticity
where N a ( e , [ ) = N:
+ NA, N:,
Ni
are weak solutions of problems (5.23),
(5.24). Then
IIu'(x) - U:(X)IIH~(W) L C 1 ~ s +Ilfllc*+4(n) 3 7
(5.27)
llu'(x) - C(X)IIH~(~OW*) I C 2 ~ s +Ilfllcw(n) 3 ,
(5.28)
C1, C2 are constant independent of E , Ro is a subdomain of R such that fi0 c R, the constant C2 may depend on no. where
Proof.
It is easy t o see that by virtue of (5.21), (5.22) u:
-u ' is
a weak
solution of the problem
C
A'(uZ - u c )= &'+l -
&J+1
c
a1,...,a,+1=1
&
Y
1
d
~ E ~ A ~ N ~ ~ .Z)a1 . . ...~a.+l, + ~ ax,,
011 ,...,as+ 1=1
rn=s+1
a
4~~-[(A2Nal...o.+l)Z)a'..'a'+l
...,am=l
a4,
f+
f] -
$5. Some generalizations for the case of perforated domains 5+4
+ E5+1&2 C
m=a+l
n
C a s ,...,
Since N , = A ':
l&(.:
IINa5...om
o,=l
I
IILz(~*) IlfIIcs+4(ii) .
+ N:, we obtain by virtue o f (5.25) - UC)llL.l(W) I M E 5 + ' ( E 2
5 MI
203
IlfIlc.+,(n, ,
+ E3)
that
IlfIlc.+4(n)
I
M l , =~ const .
This estimate and (5.9), (5.10) imply (5.27). Inequalities (5.28) follow from (5.27), (5.26). Theorem 5.6 is proved.
0
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 st ruct u res. Suppose t h a t a subdomain
R' c R
is such that
a' c R and a' =
u d,
s=l
&,
B: are bounded domains of R" such that B: n Bj" = 0 for i # j . Suppose also t h a t ,:'I s = 1, ...,de, are closed sets r: c @ and for each 2, E C1(&) such t h a t v = 0 in a neighbourhood o f r:, the Friedrichs inequality where
holds with a constant C* independent o f Let
T(Z)
and s.
be a function in C 2 ( a )such that
5 M',
))T))~z(Q
E
where
M'
T
= 0 in Cn\s2',
is a constant independent of
E,
s.
Theorem 5.7. Let
U ( Z )be a weak solution of the boundary value problem
au d@ U = @ , - -- - on 30' , dv
au
T
2 0 in R,
II. Homogenization of the system of linear elasticity
204
dip
HZ(R'), ip = - = 0 on R' n dR', f j 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'. aip Suppose t h a t f j = 0, j = 0, ..., n in R'nR', ip = - = 0 on R'ndR' and av the domain Ro is such that noc R', p(dRO,80') 2 K > 0 with x independent of E . Then the solution U ( z )satisfies the inquality
where ip E
II UllHz(no"n') I
c
L where
[
II@IIH2(W)
+2
l 6
l l . P I l L ~ ( n ~ \ * ~ )e-G
j=O
,
(5.30)
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 ~ ( xsuch
~ ( x= ) 1 in Ro, ~ ( 2 = ) 0 outside the ~/2-neighbourhood o f Ro, the C2(n) norm of 7(2) is bounded by a constant independent o f E.
that
Consider now the system of elasticity. Suppose that the sets
rz,s = 1, ...,dc, are such that for each TJ E C'(&),
= 0 in a neighbourhood of E - l )l"llLz(E:)
l?:
the following inequality is valid
+ IIv"IILz(E:) 5 c; IIe(")llL2(B:)
where C; is a constant independent o f
a',
5 M;,
where
E.
let
T(Z)
7
(5.31)
E C'(O), ~ ( x= ) 0 outside
M; is a constant independent of E .
Theorem 5.8. let
U ( x ) be a weak solution o f the boundary value problem for the elasticity
system
U = ip on dR'
,
L2(R'), j = 0,..., n , ip E H'(R'), ip = 0 on R ' n 80' and matrices A h k ( x , e )belong t o the class E(nl,n2) with tcl,lc2 > 0 independent
where
fj
E
$5. Some generalizations for the case of perforated domains of
E.
Then for U ( z ) the estimate (4.9) holds with constants I(
depending only on C; in (5.31), Suppose t h a t the domain
fj
= 0, j
E.
> 0, S > 0
KZ.
n R', = 0 on 0' n doZ'and p(8Ro,dR') 2 ~d> 0, where IE is a constant
= 0, ...,n , in R'
Ro c R' is such t h a t
independent of
M;, n l ,
205
Then the solution U ( z ) satisfies the inequality
where C is a constant depending only on C;,
M:,
~ 1 K ,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 ~ ( z such ) that of
7
Ro, 1(7(lCl(n)is bounded
= 1 on
Ro, 7 = 0 outside the ~/2-neighbourhood
uniformly in
E.
11. Homogenization of the system of linear elasticity
206
$6. Homogenization of the System of Elasticity with Almost-Periodic CoefFicients In this section we consider homogenization o f solutions o f the Dirichlet problem for the system of elasticity with rapidly oscillating almost-periodic coefficients.
6.1. Spaces of Almost-Periodic Functions Denote by Trig IR" the space o f real valued trigonometric polynomials. Thus Trig R" consists o f all functions which can be represented in the form of finite sums
4Y)= y,E E
cc C€exP
{i(Y,E))
R", (y,E)
= yit;,
7
ct = c-( = const
.
(6.1)
The completion of TrigR" in the norm sup Iu(y)I is called the Bohr R"
space of almost-periodic functions and is denoted by A P ( R " ) (see [50],[51]).
The space of all finite sums having the form (6.1) and such t h a t
Q
= 0 is
0
denoted by Trig R". Let II,E
L;oc(EP).We say that M {$} is the mean value of II,,if
II,(&-'z) + M { $ } weakly in L 2 ( G ) as
E --+
0
for any bounded domain G c R". It is well known that for any function g E L:,,(R"), which is T-periodic in y, the mean value exists and is equal t o
J
1 M{g}= 5
g(y)dy
I
[OYTI"
[O,T]"={y
:
O l y j S T , j = 1 , ...,n } .
Thus each function belonging t o TrigR" possesses a finite mean value, and therefore we can introduce in TrigR" the scalar product defined by the for rnu la
$6. Homogenization of the system of elasticity
207
The completion of T r i g R with respect t o the norm corresponding t o t h e scalar product (6.2) is denoted by B 2 ( R " )and is called the Besicovitch space of almost-periodic functions. We keep the symbol
M { $ g } for the scalar product of the elements II,and
g in B2(R").
As before we say that a matrix (or vector) valued function belongs t o one
, P ( R " ) , if i t s components belong t o the of the spaces Trig R",B 2 ( R " ) A corresponding space. In this case the mean value is a matrix (or vector) whose components are the mean values of 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.
As usual e ( u ) denotes the symmetric matrix with elements e U ( u ) =
auj 1 (k+ -), 2 8%
dYl
where u is a vector valued function u(y) =
(u1
,...,u,).
Lemma 6.1. Suppose that f , g E Trig R", and u =
(u1,
...,u,)
E Trig IR". Then
F I E~ Trig R" such that F/h = Fhl, 1, h = 1, ...,n , there is a vector valued function w E Trig R" such t h a t Moreover for any functions
Proof. Note that
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 = (u1, due t o (6.6) we find that
...,u , ) , uj = E
4e’(yx).Then
This implies (6.4). Let us show now the existence of the solution of equations (6.5). Suppose that
$6. Homogenization of the system of elasticity
Obviously for each
# 0 the coefficients w:
209
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 ( # 0, c y = 0 ,
I , h = l,...,n. Then multiplying the equations (6.7) by W: and summing up with respect t o I from 1t o n we obtain h 2 1-1 - 5 IwcthI - 5 W(W(
=0 .
Therefore w: = 0. Let us replace
by -( in (6.7) and write the complex conjugate equation.
One clearly has w: = wkt, since ckh = C(.!'
Lemma 6.1 is proved.
0
Consider the Hilbert space of ( n x n)-matrices whose elements belong t o
Bz(LT)and denote by W
the closure in this space of the set
S = { e ( u ) : u = ( u l ,...,u,) E Trig R"}. Elements o f W will be denoted by e, Z, etc. The norm o f an element e E W is given by
M {e,je1j)'/2 = M { ( e ,e))'" . It should be noted that not every element e E W can be represented as e = e ( u ) with u E B2(1Rn).Nevertheless for every e E W there is a sequence o f vector valued functions { u 6 } with components in TrigR" and such t h a t
M {Ie - e(u6)12}+ o as 6
--.)
0.
6.2. System of Elasticity with Almost-Periodic Coeflcients. Almost- Solut ions Consider the system o f linear elasticity
11. Homogenization of the system of linear elasticity
210
a
621
aYh
auk
- ( A h k ( y )-)
dfj
= -, &j
E ( I E ~ , II E E I~ , K) .,~ = const. > 0, whose elements belong t o A P ( R " ) ,u = (ulr...,un),f j = ( fij, ..., fnj) are column vectors, f j r = f,j E A P ( R " ) . 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 ( R " )o f system (6.8) has yet been
where A h k ( y )are matrices of class
found. However we can construct the so-called almost-solutions u6 o f (6.8) with components in Trig R".This fact was established in [149]. Following [149] 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:
d
- ( u y ( y ) e j k ( u ) )= ayh
afij ~
&j
, I = 1, ..., n .
In the rest o f this paragraph we shall denote by v h t h e column vnh)*
o f the matrix r) with elements
vih.
(vlh,
...,
Then system (6.9) becomes
(6.10) where e k ( u) =
...,e n k ( U ) ) * .
(elk(u),
If the coefficients u k k ( y ) and the functions f , 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
{ ( M e( u) ,e ( 4 ) - (f,+,>} for any
w E +j(R"),where f is a matrix with elements
( M v)ih
hk aij v j k
Let the coefkients u:/
(6.11)
=0 flh
and
. be almost-periodic functions of class A P ( R " ) .
Then in analogy with (6.10), (6.11) we consider the system (6.12)
§6. Homogenization of the system of elasticity
21 1
and define a weak solution of (6.12) as the element E E W , 2 = { E i j } , which satisfies the integral identity
M ( M 2 , e ) - ( f , e ) }= 0
(6.13)
for any e E W .
It follows from Lemma 3.1, Ch. I, that the bilinear form M {(ME, e)} is continuous on W x W , i.e. (6.14) fot any 2,e
E W , since for a ( y ) E A P ( R " ) , f E B2(R")we have af E
I ( ~ ~ I I B ~ ( R ~5)
B ~ ( R and ")
SUP
R"
la1 I l f l l ~ 2 ( ~ n ) .
Moreover the condition (3.8), Ch. I, yields the inequality
for any e E W . By virtue of (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 Us(y) = (U,",..., U,") E Trig R" which approximate solutions of the system (6.9) in the sense of 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 of system (6.12). Then there exist sequences of vector valued functions U s E Trig R" and matrices gs E A P ( R " ) with columns g: = ( 9 3 , ...lg$), gfj = g:, j , l = 1, . . . l n , such that (6.16)
lim M (12 - e(U')12} + o
6-0
as 6 -+ 0, 6
> 0, and the integral identity
(6.17)
II. Homogenization of the system of linear elasticity
212
holds for any +(y) = (&, ...,$,,) E
Cr(Rn).
Proof. By
the definition of t h e space W we see that there is a sequence U 6 E TrigR" which satisfies the condition (6.17). Therefore due to (6.13), (6.14) we have
for any e E
W , where y(6) t 0 as 6 + 0.
Set = M e ( U 6 )- f ,
Since the elements a;,, of matrices Q6 belong to A P ( R " ) ,we can represent @ in the form Q6
where
= F6
+ G6 ,
(6.20)
a6,F 6 , G6 are symmetric matrices with
elements
ath,Fph, Gfih, F6 E
Trig Rn, G6 E A P ( R " ) , and lim M { lG6I2}= 0 6-0
.
(6.21)
Since
it follows from (6.19), (6.21) that
for any e E W , where n ( 6 ) --t 0 as 6 + 0.
According t o Lemma 6.1 there is a vector valued function w6 E Trig R" such that
a
aFp,
8Yh
ayh
-e,h(w6) = -
,
1 = 1,...,n
.
(6.23)
213
$6. Homogenization of the system of elasticity
Multiplying each o f these equations by wf and summing with respect to
I
from 1to n, we find by virtue of (6.3) and Lemma 6.1 that
(6.24) Therefore
M {le(w6)I2) I M {lF612} .
(6.25)
It follows from (6.20), (6.17), (6.21) that M { lF612} are bounded by a constant independent of 6. Therefore due t o (6.24), (6.25) we obtain
M {le(ws)12} -+ O as 6 -+ o .
(6.26)
Obviously by virtue of (6.20) we have
where g6 = e(w6) butions.
+ G6;and the equations (6.27)
hold in the sense of distri-
The convergence (6.16) is due to (6.26), (6.21), and the integral identity (6.18) follows from (6.27) and the conditions (3.2), Ch. I, for ubk. Lemma 6.2 is proved. 0 The vector valued functions U 6 are called almost-solutions of system (6.9) with almost-periodic coefficients. Let us now establish some other properties of the almost-solutions U s , which are essential for the study of G-convergence of elasticity operators with almost-periodic coefficients. Lemma 6.3. Suppose that f j , Ahk E A P ( R " ) ,2: is a weak solution of system (6.9), 2: E W , and U s (6 -+ 0) is a sequence of almost-solutions of system (6.9). Then for any sequence E -+ 0 there exists a subsequence € 6 -+ 0 as 6 + 0, such that E6(u6(;)
+ cs) -+ o
weakly in ~ ' ( 0, )
(6.28)
II. Homogenization of the system of linear elasticity
214 where
cg
is a constant vector,
rsP(E)+ M { A P k & - f p } weakly in L 2 ( R ) ,
p = 1, ...,n , (6.29)
a
X
-r 6 h ( E g ) +
axh
o
in the norm of H-'(R)
,
(6.30)
as 6 -+ 0, where
(6.31)
0 c R" is a bounded Lipschitz domain.
Proof. Taking
into account the inequality (6.4) o f Lemma 6.1, the fact t h a t
U s E Trig R",and the convergence (6.17) we obtain (6.32) where
I<
is a constant independent of 6.
Denote by G6'(x) the matrices whose elements are
G~:(x)
a
x
-U f ( - ) E TrigR" .
ax,
E
Note that the matrices G6icare not necessarily symmetrical. By the definition of mean value we have
lim
E'O
J
1 ~ 6 i ~ ( z ) 1 2 d= z
n
= lim E-10
J(
au6(C)
dU6( 2)
n
dYl
'
dYl
)dx =
(6.33) Similarly
lg6(z)12dx= (mesR)M{lg612},
lim E'O
n
&
where g6 are the matrices from Lemma 6.2.
It is obvious that
(6.34)
215
56. Homogenization of the system of elasticity
M e(U6)(f) + M { M ,(Us)} weakly in L2(R) as
E
-i
0. (6.35)
Moreover G6lC(x)-i 0 weakly in L2(R) as
aus x 0 since-(-)ETrigRnande'(~l€) 8Yl
E
E -+
,
0
(6.36)
+OweaklyinL2(R)as~-+OforJ#0.
Let V = {ql,$, ...} be a countable dense set in the Hilbert space of all matrices with elements in L*(R). For each 6 by virtue of (6.33)-(6.36)
for m
I 6-l,
we can find
€6
such that
m = 1 , 2 , ...; qm E V .
It follows from (6.32), (6.39) that the norms IIG6vC611Lz(n,are bounded by a constant independent of 6, and inequalities (6.40) imply that for any qm E V we have
J
(~69~6,qm)dx -+
o
as 6 -i o
.
n
Therefore G6lc6(x) -i 0 weakly in L2(R) as 6 -i 0 Set
where the constants
Cg
are chosen such that
(6.42)
II. Homogenization of the system of linear elasticity
216
J
X
V6(-)dX
n
E6
(6.43)
=0.
Then due to the Poincari inequality we have
where c is a constant independent of 6. Since the right-hand side of (6.44) is bounded in 6, it follows from (6.42), (6.44) that E6'V6'(?) + V weakly €6,
in H'(R) and strongly in L2(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'(R) c L2(R). By virtue of (6.43), (6.42) lished.
V
= 0. Thus the convergence (6.28)
is estab-
Since the elements u p of matrices Ahkare bounded, it follows from (6.17), (6.37) that the norms 1IM e ( L i 6 ) ( E ) / I L Z ( are n ) bounded by a constant independent of 6, and
lim M { M e ( V 6 ) }= M { M 2 ) .
6-0
Therefore we conclude from (6.41) that
M e ( U 6 )E6( z )
M { M 2 ) weakly in L2(R) as 6 -+ 0
To complete the proof of (6.29) it is sufficient to observe that
Therefore
X
I'6p(-)
are
€6
given by (6.31). Let us prove (6.30). For any $(x) = (&, (6.18) we obtain
.
..., &)
E C,"(R) due t o (6.31),
217
$6. Homogenization of the system of elasticity
This inequality together with (6.38), (6.16) implies (6.30). Lemma 6.3 is 0 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
a
- (Ahk(-)
&(ti')
axh
E
-)aau= xk
= f ( x ) in R , u E HA(R) , (6.45)
f E H-'(R), matrices Ahk(y)belong t o the class E(n1, Q ) , n l , n2 = const > 0, and their elements ahk(y) are almost-periodic functions of class
where
AP(R").
If matrices A h k ( 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
(a?(y) -k a:j"ejk(N:))dy 7
=
(6.46)
6 where
N," = (N:8, ..., dN!8
1 dNZs
- (2
ayj
+ -), dYk
is the s-th column of the matrix N q , ejk(N:) =
and the columns
N,' are 1-periodic solutions o f the system
d
a
ayh
ayj
- ( a F ( y ) e j k ( N i ) )= - -a!," Setting A:q = (a;:, ..., a::),
&I
=
,
(A;:, ...,i::),
(6.47) in vector form
=
J 6
1 = 1, ..., n
( A F ( y ) t Ahkek(N,'))dy,
.
(6.47)
we can rewrite (6.46),
II. Homogenization of the system of linear elasticity
218
belong t o AP(R"). It was shown above that for fixed q, s
Now let a(:y)
we can find weak solutions e"' E W (27' is a matrix with elements
a):
o f the
system (6.48) which is similar t o (6.12) with fj
= -(a{:, ...,a;:) = -A!'
.
(6.49)
Set
FI;:
= M {a:,"
and denote by
ahqthe
+ a:j"Z;;} , A!q
= M {Atq
+ AhkZY} ,
(6.50)
matrices with t h e elements ;I:,".
Theorem 6.4. Suppose that APQ(y) are matrices of class E(nl,riz),
6 1 ,= ~ const ~
> 0,
and
their elements aQ'(y) are almost-periodic functions belonging t o AP(R"). Then the sequence of operators
(
CC,(u) a A h"(') axh
&
au -)dxk
(& + 0)
is strongly G-convergent t o the elasticity operator
L
whose coefficients are
given by (6.50).
Proof. Let
us show t h a t there is a sequence 6 -+ 0 and matrices X
l , . . . , n , such t h a t matrices Ahk(-), I,'as 6 + 0, where
N:, q
ahkare matrices whose elements are defined by (6.50). &6
=
ahksatisfy the Condition N of $9, Ch.
virtue o f Theorem 9.2, Ch. I, this means t h a t
C,,
3 L as 6 + 0.
By
Due t o
the uniqueness of the strong G-limit (see Theorem 9.3, Ch. I) it follows that
C,
S L a s & -+o.
Fix q, s and consider the almost-solutions
U& = (U&,, ...,U:ns) of system
(6.48) constructed in Lemma 6.2. Set
N ~ s (=~(N6qa,-..rN69,~)* ) = &6('%(&)
X
i-'%)
7
(6.51)
$6. Homogenization of the system of elasticity
219
where c:# are constant vectors satisfying the condition (6.28) with U s = U,P,. Denote by N:(x) the matrices whose columns have the form (6.51). Let us verify that the matrices
(2
N i , APQ - , h ' q satisfy the Condition N as 6 -+ 0.
Indeed, the Condition N 1 follows from (6.51) and (6.28).
Consider the
Conditions N2, N3. Due t o (6.29)-(6.31)
A:'(:)
we have
+APX(:)
d ~ N i ~ (-+ xM{APkEg,"+Az4} )
(6.52)
weakly in LZ(R),
in the norm o f H-l(R), as 6 ---t 0. These relations show that Conditions N2 and N3 are satisfied, since due t o
(6.50) t h e expression in the right-hand side o f (6.52) is equal t o 6.4 is proved.
,@.
Theorem 0
II. Homogenization of the system
220
o f linear elasticity
57. Homogenization of Stratified Structures
7.1. Fonulas for the Coeficients of the Homogenized Equations. Estimates of Solutions Consider a sequence
{L,}
o f differential operators o f t h e linear elasticity
system
a
ax;(@(p(X),xi, ...,2,)
&(u)
dU
-)axj , > 0 independent
belonging t o class E ( I c nZ) ~ , with constant 53, Ch. I). Here
E
is a small parameter,
E
of E ,
2
(see
E (0,l); the elements of matrices
AF(t,y) are bounded (uniformly in E) measurablefunctions of t E R', y E R" with bounded (uniformly in
function in Cz(!=l) such t h a t 0
...,yn; p(x) is a scalar 1, (Vpl 2 const > 0; R is a bounded
first derivatives in
E)
5 p(x) 5
y1,
smooth domain. Let us also consider the following system o f linear elasticity
E(u)
d dU =( k j ( q 9 ( X ) , X 1 , ...,2") -) dX; 8x3
=f
,
whose coefficient matrices belong t o E(k1,Zz) and k l , stants which may be different from
~ 1 n2; ,
k2
are positive con-
the elements of the matrices
a ' j ( t , y ) are bounded measurable functions o f t E R',y E R",possessing bounded first derivatives in
y1,
...,yn.
In this section we consider the following Dirichlet problems
L,(u') = f
in
R , u" = GJ
E ( u ) = j in R , u
f E H-'(R) ,
=
on
dR ,
on ~ 80,
(7.3) (7.4)
a) E H q d n ) .
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 p ( x ) (see [go]). Here we obtain estimates for the difference between the displacements u" and u,the corresponding stress tensors and energies. We establish explicit
§7. Homogenization of stratified structures
221
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
2.
k
as
E +
0, and give
The corresponding spectral problems are studied in $2, Ch. Ill. Let t h e matrices N,'(t,y),
~ ( Y) t = ,
t
J [cpr(y)ipk(y)~:'(T, Y)]-lcpp(Y) 0
M;At 7 Y)
t
=J { 0
-
M $ ( t , y ) be defined by the formulas
AY(T,Y))
cpj (Y
)A!
+ A;(T,Y)
(7, Y) [cpl(Y )cpk(Y
-a
( a p u ( T , Y)
-~
( 7y))dT ,
,
)A%, Y )] -l YP(Y 1('WTY) -
' y ~ ~ ~ ,) } d ~ (7.5)
where (cpl(y) of
,...,cp,(y)) = (3 ,..., a ')= Vcp, B-' 8Yl
8Yn
is the inverse matrix
B . It will be proved in Lemma 7.5 t h a t the matrix [(pl(y)(pk(y)A,k'(~,y)]-'
exists and t h a t its elements are bounded functions (uniformly in
E).
To characterize the closeness between solutions of problems (7.3), (7.4) we introduce a parameter 6, setting
Theorem 7.1. Let u", u be the solutions of problems (7.3), (7.4) respectively, and u E Then the following estimates hold
H'(0).
II. Homogenization of the system of linear elasticity
222
where
..
depend on
auc
.
A:' -, 9'
rf E
axj
au , z = 1,...,n , the constants q,,c1 do not = A'3 ,
axj
Proof. Define v,(x) as the solution o f the problem
au axj
,Cc(vc) = 0 in R , v, = N;((P(x),x) - on dR
,
v, E H'(R).
(7.9)
Then it is easy to calculate that
Therefore
(7.10) T h e right-hand side of this equation is understood as a n element o f Let us show using the definition of
a
- - - Mi',
at
+ aYia(X,E)
,
S,, N,", MG
H-'(R).
that
(7.13)
0 7.
223
Homogenization of stratified structures
IP;.(x,E)I 5
] a , ( x , ~ )5 I c&, stants c2, cg, c4 do not depend on E. Indeed, we obviously have where
c&,
Multiplying these equations by
(pk
Iai..(x,~)\
I c46,,
and the con-
and summing them up with respect t o
1-
aMfs Ic, we obtain (7.11) due t o the inequality 5 c6,. aY1 Setting k = i in (7.14) we find by virtue of (7.5) that
a
-kf;",(cp(x),z)= (~;'~jA~[cplcpkA,k']-'cp,(A~" - AY) t ax;
This equality implies (7.12). According to the formulas (7.6), (7.5) we have
Let us estimate the H-'(R)-norm of the right-hand side of (7.10). For any column vector $ = ($1,
...,$,)
E
(?,-(a),due to (7.13),
au -)dx a$
-,
ax, ax;
(7.11) we obtain
-t
224
II. Homogenization of the system of linear elasticity
Therefore, taking into account (7.12) and the definition of 6, we find that
where c5 is a constant independent of E . Let us estimate the second term in the right-hand side of (7.10) in the norm of H-'(R). Using the definition of 6, we get
It thus follows from (7.10), (7.14), (7.15) that
$7. Homogenization of stratified structures where
c7
is a constant independent o f
E
225
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 (7.17) where
c8
is a constant independent o f
We now estimate the norm wc
where
$E
= V& - d c
,
E.
(1?&IIHl(n). Set
au
19' = *cN3f((P(x),x) dzj 7
= 1 in the 6,-neighbourhood of dR,
neighbourhood of dR, $' E C-(S=l), 0 follows from Theorem 3.1, Ch.
I,
5 $' 5
= 0 outside the 26,1, 6,1V$cl
I const.
It
that
and therefore
I
l l ~ c l l & ~ ( * ) ClOC IIUll&qn)
+ c11 l l V ~ l I i ~ ( w ) 9
where w1 is the 26,-neighbourhood of dR. By virtue of Lemma 1.3, Ch. I,
IIvuII~z(wl) L C126'
IIUll&l(n).
Hence
5
~ ~ ~ c ~ ~ H 1c13s~/2 ( S I ) IIuII@(n)
.
(7.18)
Estimates (7.17), (7.18) imply (7.7). Let us now prove (7.8). It follows from (7.7) that du'-
ax,
du
dN'
axj
axj ax,
-du+ G ,
--+2
(7.19)
II. Homogenization of the system of linear elasticity
226
and thus the estimate (7.8) is valid. Theorem 7.1 is proved.
0
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 of Theorem 7.1 we have lluc
1
- u - N' a
ax,
H*(W
< Q 6 f / 2 ( l l f l l L 2 ( S l ) + II@IIH3/2(aSl)) 7 (7.20)
(7.21) where Q, c1 are constants independent of E . Estimates (7.20), (7.21) follow from (7.7), (7.8) due t o the inequality 11u11H2(n)
5 c2 (IlfllL2(Sl) + ll@llH3/2(aR))
7
(7.22)
which is known from the theory o f elliptic boundary value problems in smooth domains (see [l]). 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 u" and u by the formulas
Theorem 7.3. Let u", u be the solutions o f problems (7.3), (7.4) respectively, u E H 2 ( 0 ) . Then
IJ%(~")
-
5 c1(G)6?(11UtI&2(n)
+ IIuIIHz(Sl) ~ ~ u c ~ ~ H 1 ( S l ) ) (7.23)
§ 7. Homogenization of stratified structures where q ( G ) is a constant independent of
227
E.
Proof. For the sake o f simplicity we prove this theorem assuming the elements of the matrices
k? 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) t h a t
Taking into account
(7.19), (7.11) we find
II. Homogenization of the system of linear elasticity
228
(7.25)
Therefore it follows from (7.25) that
aMf8
I
au
c4 [ l l u l l & 2 ( C l ) 6 .
From (7.19) we obtain
auc
+ IIVU11~2(8G)GC + 'l"IIU1lh(n)]
'
(7.26)
§ 7. Homogenization of stratified structures
5
c 5 ' ~ ( ~ ~ u ~ ~llu\lH'(n) ~2(n)
+
~
~
229
"
~
+ ~ ~ V U ~ ~ & ( a +G )bfl ) ~
~
2
(
~
)
3
(7.27)
:c66f/* 11u11$2(n). where lpil I Since by the imbedding theorem we have IlVullp(ac)5 ~ l l ~ l l ~ for 2 ( any u E H2(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.
[7
Corollary 7.4. If the coefficients of system (7.3) are smooth, it follows from (7.22), (7.23) that
IE&(u")- EG(U)\ 5 c2(G)S:/2 (Ilfllil(n)
+ ll'll&D(8n))
a
Note that the matrix [cpkcp,A:']-' was used in (7.5) to define N i , M;'. Let us show that this matrix exists and its elements are bounded functions (uniformly in
E).
Lemma 7.5. , tcl, Let A ' j ( z ) , i , j = 1, ...,n , be matrices o f class E ( K K~ ~, ) where positive constants independent o f 2. Let 'p E C'(fi), ( V y ( 2 const.
are
> 0,
Vcp = ( 9 1 , ..., cpn). Then there exist two constants such that for any E R"
t
K3
K ~ K, ~ depending ,
It?I ([cpl(.)(P,(.)A"'(.)l-l~,t)
only on
I 6 4 1tI2,
K',
n2 and
K 3 , K4
'p,
>0 . (7.28)
hf Set. q i h = ( P i t h 4-cp& in (3.3), Ch. 1. Then aY"lihqj1
(Cpith
= 4'pp,(PpayTtitj
,
+ ( P h t i ) ( ( P i . t h+ ( o h t i ) = qih%h
=
~
)
230
II. Homogenization of the system of linear elasticity
Set
K ( s )= ( p , ( x ) ( p p ( x ) A p q ( x )Then . by (3.3), Ch. I, for any [ E R" we get
where the constants cl,
MI depend only on
nl,
x 2 , cp. It follows that
K-l
exists. Setting [ = K-lC we obtain
13
These inequalities imply (7.28). Lemma 7.5 is proved.
7.2. Necessary and Suflcient 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 the sequence {C,}
to the operator
2 in terms o f convergence o f certain combi-
nations of the coefficients o f Ce, and t o obtain for the coefficients o f
E explicit
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 Coip the space of bounded measurable functions g ( t , y ) , ( t ,y) E [0, 11 x 0, equipped with the norm
t varies over a set of full measure. By Clip we denote the space o f functions
CobP,j = 1, ..., n.
g ( t , y) such t h a t g ( t , y ) , 89 E ayj
§ 7. Homogenization of stratified structures
231
Lemma 7.6. Consider a family o f functions $,(t,y) bounded in
E
whose norms in Cotp are uniformly
E ( 0 , l ) . Then there exists a subsequence E’ + 0 and a function
@ E Covflsuch that &(t,y) for any y
E
-+
@(t,y) weakly in L2(0,1) as
E’ +
0
a.
Proof. Let V
be a dense countable set in L2(0,1). For a fixed
w 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’ + 0 such that
j
$et(t,Y)w(t)dt + q,(y)
uniformly in y
,
(7.29)
0
where @,,(y) is a function of y
E i=2, Since V is a countable set, one can use
the diagonal process to construct a subsequence L‘
--t
0 such t h a t (7.29) holds
w E V. Now let w be an arbitrary function in L2(0,1) and j -+ 00, vj E V .
for any
Let us show that there exists Q,(y) S,(y)
-+
Q,,,(y)
wj
4
w in L 2 ( 0 ,1) as
such that
uniformly in
a
as j
-+ 00
.
Indeed, it is easy t o see t h a t
Choosing
EO
sufficiently small in order that for
E’
< EO
we have
11. Homogenization of the system of linear elasticity
232
II
1
qUk(y) 1
1
J $Ei(t,Y)vkdt + Q ~ , ( Y ) - J $ c i ( t , y ) v j d t 5 6/2 , 0
0
~,(y) + qW(y) uniformly in y E
0
as j
+ 00
.
Choosing a sufficiently large j in the inequality
we find t h a t
1
+ct(t, y)w(t)dt
+ U,(y)
as
E’ +
0
0
uniformly in y
E
a.
Obviously Qw(y) is a bounded linear functional on w 6 Lz(O,l) for any
y E
a. Therefore
1
Qw(Y) =
J
Q(t,y)w(t)dt
7
0
where
@ ( ty) , E L2(0,1) for
any y E
a.
Thus
J + a l ( t , y ) w ( t ) d t + ] qt,y)w(t)dt 1
0
as
+0
0
for any w(t) E L2(0,1). The function @ ( t , y ) satisfies the inequalities
-c(y’-y”(P
5 @(t,y’)
-@(t,y”)
5 cly”y“(4,
(7.30)
$7. Homogenization of stratified structures
233
f weakly in L2(0,1) as E --t 0 and m 5 fs 5 M , then m 5 f 5 M for almost all t E ( 0 , l ) . Therefore correcting, if necessary, CP on the set o f measure zero we get iP E Co~@ due t o (7.30). Lemma 7.6 is owing t o the fact t h a t if fb
-+
proved.
0
Corollary 7.7. Let {&(t,y)},
E
E (0,1), be a family of functions, whose norms in C'vfl are
bounded uniformly in
E.
Then there exists a subsequence E' -+ 0 such t h a t
weakly in L'(0, 1) for any y E
0, j
= 1, ..., n , where
~
E C'@.
Proof. It follows from lemma 7.6 t h a t there is a subsequence E'
j = 1, ...,n weakly in L2(0,1) for all y E Obviously for any g E
= lim a'+O
JJ 1
0 where $,(pj
,
E Co3p.
CF(f2) we have
"' I (t,Y v(t)g(y)dydt = j
o n
Therefore p j ( t , y ) =
0 such t h a t
-+
-in the a yj
sense of distributions. Since q j , G E
Coiflthe last equality holds in the classical sense for almost all t . Lemma 7.8. Suppose t h a t the functions
&(t, y) are bounded in Coifluniformly with respect
E E (0, I ) , and t h a t &(t,y) y E 0. Then
to
---t
0 weakly in L2(0, 1) as
E -+
0 for every
11. Homogenization of the system of linear elasticity
234
t
J
Y)
~ c ( t ,
$e(T,
Y)dT
-+
0
0
in the norm of Co([O,11 x Moreover, if
0 ) as
E
--t
811,e . = 1, ..,, n , are also bounded ,
layj
I
in
Q c ( ~ ( x ) , z-+ ) 0 weakly in H'(R) as for any cp(x) E
.
0
E,
E
then
-+
0
C'(0).
Proof. The family
{Qe(t,y)}, E E ( O , l ) , is equicontinuous and uniformly
bounded in [O, 11xa. Therefore due t o the Arzela lemma there exists a function
$ ( t , y ) such that E' -+
-+
11, in
the norm of
Co(([O,l]x 0 ) for
a subsequence
0. Since
/ t
@,, =
$Q(T,
y ) d -~+ 11,(t,y) for all t , y E [0,1] x
0, and &(t,y)
0
-+
0
weakly in L2(0,1) as
E -+
0 for any fixed y E
Let us prove t h a t Qc(cp(z), z)
-+
0 , it follows that 11, = 0.
0 weakly in H'(R) as E -+ 0. Indeed, we
have already proved that Q , ( ( P ( z ) , L ) -+ 0 in the norm o f
L"(R) as E
a Moreover the derivatives -Q,(cp(z),z) are bounded uniformly in E .
axi
due t o the compactness of a ball o f L2(fl) there is a subsequence E'
a that - Q,~((P(z), x) ax;
7.8 is proved.
-+
-+
+
0.
Thus
0 such
~ ( xweakly ) in LZ(R),and therefore x = 0. Lemma 0
(7.31)
$7. Homogenization of stratified structures
235
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 t h a t the elements of the matrices AY(t,y), z,j = 1, ...,n , have norms in
C ' ~ puniformly bounded in
G-convergent t o t h e operator
Then t h e sequence
E.
t as E --+
{L,}is strongly
0 if and only if the following conditions
are satisfied
B,"(t,y)+h'(t,y), B?(t,y) + S"(t,y) weakly in L2(0,1) as
E
s=O,1,
,
. .
Z,J
-+ 0 for any y E
...,n
= 1,...,TI ,
E
(7.32)
0.
Proof. Assume first t h a t the conditions (7.32) in this case 6, + 0 as
,
are satisfied. Let us show t h a t
+ 0, where 6, is defined by formula (7.6). Indeed,
one can easily check that
'p,B,p" = 0 , cppBps= 0 , cppB,P= E 'ppAfld = (B,O)-'B:,
A! = -B; Therefore
'ppAps=
+ (B;)*(B;)-'B;,
,
(bo)-'bs , = -bs + (&)*(@)-1&s
.
II. Homogenization of the system of linear elasticity
236
- ( ~ i ( ~ , y ) ) * ( ~ o ( ~ , y ) ) - ' ~ s (-~ B'"(.r,y)] ,y) dT
.
(7.33)
Denote the integrands in the above formulas for N,'(t,y), M,"(t,y) by n:(t, y), rnt(t,y) respectively. By virtue of (7.32) we have
n:(t,y), rnL(t,y) as
E -+
o for
-+
(7.34)
0 weakly in L'(0, 1)
any y E fi.
According t o Corollary 7.7 it follows that
a
-a
t , Y)
+
0
7
a
-4
ayj a yj weakly in L2(0,1) as E
as
E
-+
-+
4 Y)
-+
0
0 for any y E S2
(7.35)
.
o for any y E s2.
Lemma 7.8 and (7.34), (7.35) imply that the matrices N,'(t, y), M,"(t, y),
a ??/ j
a
-N,'(t, y), -M,'(t, y ) converge t o zero in the norm o f E
-+
Co([O,11 x
j
a) as
0. Therefore due t o (7.6) we have
as ~ 4 0 .
&-+O
(7.36)
Moreover, it follows from Lemma 7.8 that (7.37) Taking into account (7.11), (7.36), (7.37) we find that
a
- M ; ( v ( z ) , z ) -+ 0 weakly in
at
'Pk
a
IV'Pl2
ask
L ' ( 0 ) as
E +0
,
(7.38)
since - -M,",((p(z),z)-+ 0 weakly in L2(R). Let us prove the strong G-convergence of L, t o C as E -+0. Set f = L ( u ) E H-'(R), iP = 0, u E C,"(R) in Theorem 7.1. Then estimates (7.7), (7.8) are valid. By virtue of (7.36), (7.37), (7.38) we have uc -+
u weakly in Hi(R), -yf
-+
3' weakly in
L2(R) .
87. Homogenization of stratified structures Now let us show t h a t the set {E(v),v E the convergence
L,
3 2
237
CF(R)} is dense in H-'(fi).
Then
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 =
v E H;(R), and for any f = e(w)E H-l(R), w E
(?,-(a), we have
k(v),
This means t h a t llg - fllH-l(n)
L c 1121 - wIIH;(n) .
w E Cr(R) close t o v in Hi(R) we get a functional f = E(w)close t o g in H-'(R). Therefore choosing
Let us now prove that the conditions (7.32) are necessary for the strong G-convergence of
L, t o 2. Suppose t h a t L,
3 k as
E
.+ 0.
Due to our assumptions about the coefficients of system (7.1) and Lemma
7.5, the elements of matrices & ( t , y), s = 0,1, ...,n , Bfj(t,y ) , i , j = 1, ...,n , belong t o C'vp and have norms in C'@ uniformly bounded in E. Therefore by virtue of Corollary
7.3 there exist matrices B o ( t , y ) , & ( t , y ) , @ ( t , y ) ,
s,i , j = 1, ...,n , with elements in C'@and such that for a sequence
E' +
0
we have
weakly in
L2(0,1) for any y E
a.
Set jiis
= (P)*(BO)-'fjs- B i s
,
2,s = l , . . . , n
.
(7.40)
Define the matrices f i J ( t , y ) , f i G ( t , y ) by the formulas (7.5) with A i j ( 7 , y ) replaced by . 3 ' j ( ~ , y )and define
& by (7.6)
with
N;, MG replaced by &,
kG.
The same argument t h a t we used a t the beginning o f the proof of this theorem shows t h a t
II. Homogenization of the system of linear elasticity
238
fi:'(cp(x),z) + 0 weakly in H'(R)
,
(7.41)
a
- AIi(p(z), z) + 0 weakly in L2(R) at as
E'
+ 0.
Let
ii E C,"(R). Denote by u' solutions of the following problems
Set
Similar t o the proof of Theorem 7.1 we obtain the inequalities
Therefore by virtue o f (7.41) we have u"
+ ii weakly in H'(R)
, y;, -+ Ti weakly in L*(R)
(7.42)
as E' + 0.
Denote by uo the solution of the problem
By the definition o f the strong G-convergence of we have uo = 4,
L, t o
2 and due t o
ahk= Ahk- almost everywhere in R. axk a3
a3
axk
an arbitrary vector valued function from C,"(R), it follows that almost everywhere in
-..
Since d is
Ahk = ahk
R.
Thus we have shown t h a t from any subsequence another subsequence
(7.42)
E' +
. .
E"
+ 0 we can extract
0 such t h a t relations (7.39) hold for
Bs = Bs,
B'J = B'J,s = 0, ...,n , z , = ~ 1, ...,n , where B s , B'J are expressed in terms I
.
.
of the coefficients of the G-limit operator
A
A
_
.
e by the formulas (7.31).
Since
3 7.
Homogenization of stratified structures
239
{ E " } is an arbitrary subsequence, it follows t h a t (7.32) is valid for Theorem 7.9 is proved.
E -+
0. 0
In the proof of Theorem 7.9 we have actually established Theorem 7.10.
C'@ are Suppose t h a t there exist matrices &(t, y), h'j(t,y),
Let the elements o f the matrices AY(t,y) be such that their norms in uniformly bounded in
E.
s = 0, ...,n , i , j = 1,
system (7.1).
...,n , such
efficient matrices AY(cp(x), coefficients
a'j(t,
a's
that (7.32) holds for the coefficients of
Then the sequence o f operators Cc corresponding t o t h e co-
x) is strongly
G-convergent t o operator
2 whose
y) have the form
,
= ( ~ ) * @ 0 ) - 1 ~ 5- B i s
i , s = l , ...)1 2 .
(7.43)
Let us consider some examples o f strongly G-convergent sequences
{LE}
which satisfy the conditions (7.32). Theorem 7.11. Suppose that the elements of the matrices A:j(z) o f class E(tcl,t c 2 ) have the form a ? , ( ~ - ' q ) , where a?,([) E A P ( R ' ) are almost-periodic functions of
E
E
R'.Then the sequence C, strongly
G-converges t o the operator
k whose
matrices of coefficients are given by the formulas
where (Aij) is by definition the matrix with elements m
Moreover estimates (7.7), (7.8) hold and 6, -+ 0 as
Proof. In the case under consideration x l . Set
E + 0.
we have A:f(t,y) = A'j(E-'t), p(x) =
II. Homogenization of the system of linear elasticity
240
Y;.(s)= (A"(s))-l
(A1j - A'j(s)) ,
+ A"( s
zij(s)= ~ i 1 ( ~ ) ( ~ 1 1 ( ~ )() ~- '1 -j A'+)) The elements o f matrices any almost-periodic f , g,
q, Z;j
SJ
- AS3
6..
.
are almost-periodic functions, since for
1
f 2 const > 0, the functions f g and - are also
f
almost-periodic.
It is easy t o see that
Obviously ( Z i j ) =
(5)=
0 and the elements o f N j ,
are uniformly
bounded and equicontinuous. Therefore 6,
M; converge t o zero as E
-+
-+ 0 in Theorem 7.1, since N j , 0 at any point z1E ( 0 , l ) .
L, t o 2 follows from the conditions (7.32), I -+ ( f ) weakly in L2(0,1) as E -+ 0 for f(-)
The strong G-convergence o f which hold due t o the fact that any almost-periodic
f. Theorem
E
0
7.11 is proved.
Let us consider some examples where the coefficients of We introduce a class A, consisting of functions Cf(Y)?
C depend on
z.
f (t, y) such t h a t for some
gf(t,Y) we have
j
f(s,y)ds - cf(Y)t = gf(t,y) .
(7.44)
0
The functions f ( t , y), -, af 8Yl
cj(y), dcf(y) , 8Yl
assumed t o be Holder continuous in y E
-
R
sf, 'gf, dYl
1 = 1,...,n , are also
uniformly in
t E [0,1], and such
that
d
Isr(t,Y)I + lay,gf(t,Y,I 5 4 1 where the constants Set
Q,
+ Itl)'-"
7
o do not depend on t , 0 E (0,1].
(7.45)
5 7.
Homogenization of stratified structures Obviously for any
f E A,
241
we have (f(.,y)) = cf(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 c = 1.
2. Consider a function f ( t ) of the form f ( t ) = M+cp(t), where A4 = const.,
f E A1, if N > 1; f E A, for any Q E ( O , l ) , if N = 1; f E A N , if 0 < N < 1. Ip(t)l I C(1+ Itl)-N, N
3. The sum c3
$1
+
$2,
= min(ol,ez), c
I /6 &
> 0.
where 1 , q
$1
We can easily check t h a t
E A,,
+2
E
A,
belongs t o
A,
with
E (0,1].
+
f ( ~ - ' ~ , y ) dI ~c~E,(T 1) , a = 0 , l ; I = 1, ..., n
,
(7.46)
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 y ~ and , dcf(y)/dy, = 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
242
where 0
< a < 6 < 1 and
X[a,b] is the characteristic function o f the interval
[a, 61.
L*(O,1) by linear combinations o f characteristic func1 aaf tions and taking into account t h a t f , a f / a y , are bounded, we get (; 0 Approximating v E
,y)v(s)ds + 0 as E
--t
For a given matrix
0. Lemma 7.12 is proved.
0
B(t,y) with elements B;j(t,y ) l e t ( B ( . , y ) )be the ma-
trix with elements (Bij(-,y)). Theorem 7.13. Let the elements of the matrices Aij have the form
t
= (p(z), y = ( 2 1 ,
and define for
...,,
n)
,
i, s = 1, ...,n , Ak' = {a;"j'}the following
matrices
1 Bf"(t,y )
=
G
t B y - , y) E
E
t (pj(Y)A'J(E,Y)[Q'(Y)Qk(Y)Ak'(E ' '
t
t t 7Y)]-1vp(Y)APs(;,Y) -AiS(; , Y ) .
I
(7.48) Suppose t h a t the elements o f
2
B'(T,Y),B " ( T , ~B'"(.r,y) ), belong t o A,
for some u E (0,1 . Then the sequence of operators t,corresponding t o the matrices A'j Q X) ,z) is strongly G-convergent t o the operator k whose
(T
coefficient matrices are
at"(,) = (B'(.,z))'(BO(.,,))-l(B'(.,2)) - (B'"(.,z)) .
(7.49)
243
$7. Homogenization of stratified structures Moreover, the number C E ~ where ,
6, used in Theorem 7.1satisfies the inequality 6, 5
the constant c does not depend on
E.
Proof. According t o Lemma 7.12 we have t B"(-,Y)4 (B"(.,Y)) , E
t
q 1 y ) --+ (B'j(.,y))
,
= 0,1,...,n
. .
2,J
,
= 1,".,12
a.
Lz(O,1) as E -+ 0 for any y E Therefore due t o Theorem 7.9 one can take
weakly in
&(t,y) = (B".,y)) s
=0,1,
Thus by virtue of by
,
S'j(t,y) = ( B q , y ) )
,
...,n , 2., J . = 1,...,n .
(7.43)the coefficients of the G-limit operator
E
are given
(7.49). Let us now show t h a t 6,
5 CE". It is easy t o see t h a t
(PP(Y).iP"(Y) = (BO(*, y))-'(BY, Y)) (Pj(Y)'wY)
Therefore, from
= (B'(.,y))*(BO(.,y))-'
1
*
(7.48)we see that the matrices Njc(t,y), Mfs(t,y) defined by
(7.5) can be written in the form t
N?(t,Y) =
J [BO(Z
- BY5 , Y ) ]
E 1Y) (B0(.,y))-'(B"*1Y))
0
7
7
&
E
d7
1
I
Denoting the integrands in (7.50) by nj(- , y ) , rnij(- ,y), respectively, we see t h a t the elements of n j ( t l y ) , rn;j(t,y) belong t o
A, and (nj(.,y)) =
(mij(.,y)) = 0. Thus by the definition of 6, and proved.
(7.46)we get 6, 5
CEO.
Theorem 7.13 is 0
244
II. Homogenization of the system
of linear elasticity
Corollary 7.14.
If p(z) = z1in Theorem 7.13, then the coefficients of the G-limit system are given by the formulas
245
58. Estimates for the rate of G-convergence
58. Estimates for the Rate o f G-Convergence o f Higher Order Elliptic Operators
8.1, G-Convergence of Higher Order Elliptic Operators (the n-dimensional case)
R c R" consider
In a smooth bounded domain
a differential operator of
the form C(U)
C
=
l45m
( - l ) a P ( a a p ( x ) V p u ),
xER ,
(8.1)
IOKm
where aap(x) are bounded measurable functions in a1
+ ... + an,u(x)is a scalar function in H,"(R). We say that a differential operator C :
R , a,/? E Z;,( a (=
H,"(R) + H-"(R) o f the form
(8.1) belongs t o the class E(Xo,X1, &), if its coefficients satisfy the following conditions
ZL E CF(R), where Ao, X l , A, are positive constants independent o f u. It follows from the last inequality (see (1341, [9])t h a t for any t E R" and
for any any
x E R we have
c
I4=IPI=m
where ~0 = const.
aap(x)S"tP
> 0, 5"
2 co ItlZrn 7
=
Thus every operator C of class E(X0, XI, A,) is elliptic. Now following [148] we give the definition for the strong G-convergence of a sequence of higher order elliptic operators.
We say t h a t a sequence of operators {Ck} of class E(X0,X I , A,) is strongly G-convergent t o the operator
k
o f class E ( i 0 ,
xl,i z ) , if for any X 2 i, (i,=
11. Homogenization of the system of 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
( L + X)u = f ,
t 00
t o the solution u of the problem
u E H,m(R);
(8.3)
and moreover, if the sequence of functions
converges in L2(R) weakly as
k
t 00
t o the functions
Here { a k p ( z ) }and { h a p ( x ) } are the matrices of coefficients of operators
Ck
and
E
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 ra(uk,&) t o I',(u,L)
in
L2(R) as k
4 00.
It is shown in [148] that the strong G-convergence o f Ck t o
E
as
is equivalent t o the following conditions, the so-called Condition N: There exists a sequence of functions { N , k ( z ) }such that
N1
N3
Ny"E H"(R), Ny" + 0 weakly in H"(R), IyI < m
C
lal=rn
Do(& - hop) + 0 in the norm of H-"(R) ,
;
k
t 00
247
$8. Estimates for the rate of G-convergence
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," we arrive a t a stronger condition (the so-called Condition N') which not only implies the
weak convergence o f uk t o u in
H,"(R) as k
-+ 00,
but enables us t o estimate
the difference between uk and u. We say that a sequence o f operators {&} E E(Xo,X1,X2) trices of coefficients { a $ ( z ) } ,
Icy[,
IpI 5
with the rna-
m , satisfies the Condition N' in 0,
if there exists an operator E E E ( i 0 , i 1 , A*) with the matrix of coefficients { Z l a ~ ( x )and } a family of functions N," E Hm(R),IyI 5 m , such t h a t
D a N t E L"(R)
N'1
for
Icy1
in the norm of L"(R),
5 m , IyI 5 m , D*Nt
0
la1 < m , )yI5 m ;
in the norm of H-">"(R), (for the definition of
-t
5 m as k
-t 00
H-"*"(R) see $9.2, Ch. I).
For the sake o f simplicity we assume t h a t the coefficients 2 1 , ~ are infinitely m o o th . Let us introduce the parameters which characterize the rate of convergence in the Conditions "1, N'2, "3. Set
II. Homogenization of the system of linear elasticity
248
Theorem 8.1. Let the Condition N' hold for the operators ck,
ji such t h a t for p
> ji
and s
k . Then there is a real constant
5 rn - 1 the solutions o f the
Dirichlet problems
satisfy the inequalities
(8.12)
Proof. For any operator L: E E(Xo,XI, A,) there is a real constant jl depending only on Xo, XI, Xz, and such t h a t if p > fi, then the solution w of the Dirichlet problem
(L:+ p ) w = f ,
f
E H-"(R),
w E H,"(R)
58. Estimates for the rate of G-convergence
249
satisfies the inequality
where c is a constant depending only on A,,
XI, Xz (see [9]). Let us choose
> p > 0 such that the solutions of the Dirichlet problems for .& + p , L + p satisfy (8.13) with a constant c the same for all k. p
operators
We shall use the following Leibnitz formulas (see [127]): (8.14)
(8.15)
where ( that
Pj
a
P
a1
)=(
Pl
)...(
an
pn
)=
each j = 1, ...,n.
5 aj for
a!
P!(a--P)!
, a ! = crl!
... an!, PI a means
Set
where u is the solution of problem (8.9) and N," are the functions entering the Condition N'. By virtue o f (8.14) we find (ck(U:),V) f
l4Im
IPKm
=
c 1
l49lPl
n
1
n
a$@u:D"vdx
atp@uVorvdx$
=
11. Homogenization of the system of linear elasticity
250
Denote by
50 the last integral. Then because o f (8.4) we have
where c1 is a constant independent of Ic. Transposing the indices y and
p
in the integral next t o the last one in
(8.16), we obtain
where 51, J z , 53 stand for the respective integrals on the left-hand side of the last equality. From (8.4) we have (8.19)
It follows from (8.6) and Lemma 9.1, Ch. I, that
$8. Estimates for the rate of G-convergence
251 (8.20)
Let us estimate 51. Using (8.15) we find
51 = J n
c
(i$-iafl)D~UDavdz =
bl=m IPlSm
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
(8.22)
for any v E H,"(fl), where
We obviously have U : - v k - Uk E H,"(R) and
II. Homogenization of the system of linear elasticity
252
: .1 5
- vk - ukllHm(f2) I c 3 [(ak
+ bf) + Y k ) IIuIlH2m(R) +
IlulIH2m+1(n)]
which implies (8.10), since for the solution u of problem (8.9) the following a
priori estimate is valid
One of the simplest examples, when the Condition N' holds, is provided
& with coefficients a $ ( e ) such that akp(s) + zl,p(z) in the norm of L"(R) as k + 00. In this case we can take N: G 0. Obviously the Conditions "1, "2, "3 by the sequence o f operators
are satisfied and
where c is a constant independent of
k,
$8. Estimates for the rate of G-convergence 6k =
max
253
.
Ila$ - &pIIw(n)
I46m IPISm
According t o Theorem 8.1 we have
Actually one can prove a stronger inequality in the case under consideration, namely:
- U I I H ' ( ~ )5 Kz6k I l f l l t 2 ( n ) s 5 m - 1, K1,K 2 = const . llUk
7
(8.25)
To obtain (8.25) we note t h a t in the proof of Theorem 8.1 the norm IlfII~l(n,estimates l l ~ l l ~ 2 ~ + in 1 ( (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 t h a t under our assumptions this integral can be estimated by 6k llUllH2"'(n) IIvIIHm(n)
*
Let us now consider a less trivial example, when the Condition N' issatisfied. Assume that the coefkients a$(z) of operators Lk depend only on z l , i.e. akp(z) = a$(zl).
Let the coefficients &p(zl) o f operator
\PI 5 m , o = (m,O, ...,0)
where 1 is such t h a t
Rc
L
be such t h a t for all la1
5 m,
we have
{z : 0
< x1 < I}.
Define the functions N j ( z , ) as solutions o f the equations
C
Jal=m
14=m such t h a t
Da(aL,D~~
pk= )
C
(al=m
~
~
(- a$) i ~
,
p
IPI I m , (8.27)
II. Homogenization of the system of linear elasticity
254
(8.28)
d" -N$(o)=o, dx;
s = 1 , ..., m - 1 .
(8.29)
It follows that the Condition N'3 is satisfied and /$) -+ 0 as k -+ m, -yk = 0 in (8.5), (8.7). By virtue of (8.26) the right-hand side of (8.28) tends to zero weakly in d" L 2 ( 0 ,I) as k -+ 00. Therefore -Ni(x1)-+ 0 in the norm of Co([O,11) as dx: k - + c o , s = O , l , ...,m - 1 . Owing to (8.28), (8.4) one has
and
0 as k
Thus the Condition N'1 is also valid. Let us consider the Condition N'2. We obtain due t o (8.28) that cyk -+
ii$
By
-+ 00.
- i,
= a!,,
2)"
Nj
+ akp - ii,p
=
virtue of (8.26) we have iikp(xl) - iop(zl)-+
0 weakly
in
L2(0,1).
Therefore
/
01
@$(zl) =
(h$(s) - i,p(s))ds -+ 0 in the norm of Co([O,Z]) .
0
d dXl
Since ii$(zl) - A,p(zl) = -@$(xl), we can assume in (8.6) that
255
58. Estimates for the rate of G-convergence
and ,@-) -+
0 as k + 00, c = const.
Now in order t o obtain an effective estimate for the closeness of
uk(2)
to
u ( z ) it is sufficient t o estimate IlvkllHm(n). We have
< Hm-'/Z(an)
5 C2a:l2 ~
~
~
5 C4a:l2 IlfllHl(f2)
~
~
IluII$zm(an) ~ ~ m - 5 l%":I2 (
~
IIuIIH2m+l(R) ~ )
5
Y
where cj are constants independent of Here we used the definition o f
(Yk,
Ic. the a prion' estimate (8.24), the in-
equality IIUIIL.(an)
5 c II4IH.-'(an)
IIUIIH.+f(an)
I
>0
Y
t >0
[9]),and the fact that Npk possess derivatives up t o the order m, which are bounded uniformly in k. Define the parameter 66, by 6 k = max {ak,,&'}. (see
Then, according t o Theorem 8.1, we obtain the inqualities lluk
- U(IHm-'(n) 5 C6:l2
IlfllHl(f2)
I
(8.31)
8.2. G-Convergence of Ordinary Differential Operators The results of the previous section are obviously valid for ordinary differential operators.
However in the latter case we can obtain more accurate
estimates. Here we prove some theorems in this direction.
II. Homogenization of the system of linear elasticity
256
Let 0 = ( 0 , l ) and let
Lk,
J? be ordinary differential operators of the form (8.32)
Theorem 8.2. Let U k , u be the solutions of the following Dirichlet problems
(Lk+ p ) u k where
c k ,
J?
=f,
( k + p)u = f
E H,"(o, 1)
(8.34)
are ordinary differential operators (8.32), (8.33). Then
where the constant c does not depend on
p=l,
216721
f , Ic,
...,m
and N i are the solutions of the equations
dmNk
iimp
akp
P=--dxm akm akm '
(8.38)
satisfying the boundary conditions d j Npk
X A l =0 ,
j = 0, ...,m - 1
(8.39)
58. Estimates for the rate of G-convergence
257
Proof. It follows from the above result for higher order elliptic equations, whose coefficients depend only on x l , that in order t o prove estimates (8.35), (8.36) we have only t o estimate the functions
z)k
which are solutions o f problems
(8.12), namely
For the functions
Moreover, if p
Npk we have
5 2m - 1 , it follows from
Sobolev's lemma t h a t
where c is a constant independent of u . Therefore due t o (8.24) we get (8.42)
Set
It follows from (8.41), (8.42) that
tdpk'l,b$!I 5 c2Ak
IlflILZ(o,i)
.
One can construct a continuous extension operator
(8.43)
P mapping any pair o f
numerical sets {a!"'),{ a ! ' ) } ,i = 0 , 1 , ..., m - 1 , into a smooth function cp(x) defined on [0,1] and such that
II. Homogenization of the system of linear elasticity
258
Obviously cp(x) can be defined by the formula (8.44) where e p ) ( x ) ,e { ' ) ( z ) are smooth functions which satisfy the conditions
Therefore
vk
is the solution of the Dirichlet problem
Lk(vk)
where
Vk
t P V k = 0 on (071) 7
vk
-Vk
E
H,"(o, 1) I
are the functions defined by (8.44) with a?) = a$!, all) = a$. By
vritue o f (8.43), (8.44) we get IIVkllH'"(0,l)
Set
wk
= 2)k
- (pk.
Lk(wk)
5 c3
{laf:A!l, IaI'pk'l} 5
maX
k 0 , ...,m-1
Then
w k
=Lk(9k)
9
c 4 A k IlfllL2(0,1)
.
(8.45)
is the solution o f the Dirichlet problem wk
E H,"(o, 1)
Using the inequalities (8.13), (8.45), we find that
5 c 5 l l v k I I H m ( o , l ) 5 '%Ak
IILk(qk)llH-m(O,l)
Hence
llWkllHrn
5 c 7 A k Ilfllp(o,l) and we finally
IIwkllHrn
It is clear t h a t
5 c8Ak 7k
=
'
,@)
IlfIlL2(0,1)
= 0;
IlflIL2(0,1)
.
obtain
*
&),(Yk
5
c&,
c = const, and therefore
estimates (8.10), (8.11) imply (8.35). Theorem 8.2 is proved.
0
Remark 8.3. Suppose that the coefficients
o f the operators
L k
have the form uk,(x) =
a,,(kx) where u p , ( ( ) 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
/
259
1
I rn - 1, (f)=
where p , q
estimates (8.35) become 1Iuk
f(l)d<. We also have Ak
0
- ullH*(0,1)5
C
~ ~ f ~ ~ L Z ( o , i )3
5
C
-
k
.
Moreover, the
= 0, ...,m - 1
where c is a constant indeDendent of k and f.
d d If Lk = - (ak((.) -), dx dx
-
reduced t o
Iluk - uoIILz(o,i) 5
d h = - (h(x) dx dx
c
-),
then the estimate (8.31) is
max XQ"
where C is a constant independent o f
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
du d d'u LI;(uk - U - N k -) = - (ak(X)Nk(X)7 ) dx dx dx Therefore
11. Homogenization of the system of linear elasticity
260 where So
5 uk(x) 5 M for L4w) = 0 ,
any
k
= 1,2,
R
R
- UIlL2(0,1)
2)k
is such t h a t
such that
on the coefficients of the operator
below. It is easy t o see t h a t
IIUk
the function
x E [O, 11 ,
Due t o (8.24) there exists a constant
The dependence of
... and
Ilell ax
1
2 will
be specified
< - ~ ~ f ~ ~ L z ( Thus ~ , l ) .we have
L~(OJ)
so
MR+1
5Ak 60
Ilflb(0,l)
+~~w~~~L2(o,1) I
where
In order t o estimate the norm
112)kllL2(0,1) we
apply the maximum principle,
which yields
since obviously
Thus
11%
- UIIL2(0,1)
MR+1
I (--
60
1
+ R + &)Ak
IlfllL2(0,*)
.
R > 0 in terms of the Soefficients of t h e d u dii du L.Squaring both sides of the equation i =f-- -
Now let us estimate the constant G-limit operator
and integrating it over [0,1], we obtain
dx2
dx dx
261
$8. Estimates for the rate of G-convergence
It follows t h a t
where
P > 0 is any constant such t h a t max x€[OJl
and therefore we can take
P2
R = - (1 + -) 60 6 : 2112
dii (z) 5 P.
Thus
112
W e finally obtain t h e inequality
where the constants M , R, So can b e easily calculated for t h e given coefficients o f t h e operators
&,
2.
CHAPTER I I HOMOGENIZATION OF THE SYSTEM OF LINEAR
ELASTICITY. COMPOSITES AND PERFORATED MATERIALS
This chapter deals with homogenization problems in the mechanics of strongly non-homogeneous media. Most of t h e 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
I I we give estimates for the closeness between the displace-
ment vector, t h e 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 a t 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
R" we consider the following boundary value problem
$4,Ch. I.
II. Homogenization of the system of linear elasticity
120
where
A h k ( t )are ( n x n)-matrices of class E ( K ~ , KI C~ ~) ,, K=~ const > 0
t . It is also assumed t h a t the surfaces across which they or their derivatives may loose continuity do not intersect dw i.e. the functions u:! belong t o the class 6' defined in $6.1, Ch. I. Existence and uniqueness of solutions o f problem (1.1) for f' E L2(n"), a' E H'(S2") 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
whose elements uihjk(() are functions 1-periodic in u : / ( ( ) are piece-wise smooth in w and
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
k'9
( p , q = 1, ...,n ) are given by the formula
and matrices N * ( t ) are solutions of the following boundary value problems for the system of elasticity
a ( N 9 ) = -&Akq on dw
Nq(1) is 1-periodic in
t,
)
J Qnw
Q=
~ ~ ( t )=d0 t
{ t , O < ( j < 1 , j = 1,...,n } .
)
I
(1.4)
31. Mixed problem in a perforated domain Existence of the matrices
Nq
121
follows directly from Theorem 6.1, Ch. I.
According t o Theorem 6.2, Ch. I, the elements o f the matrices piecewise smooth functions in w belonging t o the class
6.
Nq
are
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],
[llo]). 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
if; of the
matrices
Ak' satisfy the conditions
ik! =
= ii/
1.j
jt
klqihvih
kj
5 hikj'vikvjl 5 k 2 v i h v i h
for any symmetric matrix 7 = other words the operator
Proof.
(1.5)
?
{vih},
(1.6)
where
itl, k 2 are positive constants. In
L belongs t o t h e class E ( k 1 , k 2 ) .
In the special case of w = R",i.e.
R'
= R , the relations (1.5), (1.6) 2
can be obtained from Theorem 9.2, Ch. I, since t h e matrices N,9(x) 5 a N q ( - ) and
ak'satisfy the Condition N which can be easily verified on account of (1.3), &
(1.4). In t h e general case when w may not coincide with R",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 ( be a column vector with components
(Cl,..., (").
C1,..., Cn.
Denote by (* the line
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 = ( i u i j , j = 1, ..., n.
It is easy t o see t h a t the second equality in'(1.5) follows directly from (1.3) and the properties of the elements of the matrices APq(t),since
II. Homogenization of the system of linear elasticity
122
where NAi are the elements of the matrices
Nq.
Let us establish the first equality in (1.5), which is equivalent t o
(AP'J)'
=
A9P.
It follows from t h e integral identity for solutions o f problem (1.4) t h a t for any matrix
M ( J )E
k i ( w ) we have
Making use of the relations ( A k j ( t ) ) * = Aj'(t), matrices A,
B , we obtain from (1.8) t h a t
Setting M =
(A")'
=
(AB)' = B'A' for
NP'
in (1.9) and taking into account (1.3) and the relation
= Ajp we find
J (EAjP((r) + AqP(t))dt= (mes Q n w)(Ap'J)* . Qnw
&j
It follows t h a t the coefhcient matrices of the homogenized system can be written in the form
= (mesQnw)-'
/ Qnw
d
-(NP'+tpE)Ak3(t) atk
d -(Nq+tqE)dt.
at3
(1.10)
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
= (Aqp)'.
In order t o prove the inequalities (1.6) let us note t h a t i $ / v , h 7 j k where 7' is a coIumn with components q l k r . . . , y n k , and
= vh*Ahkvk
vh* = ( ~ ~ h , . . . ~ v ~ h ) .
For any symmetric matrix 7 with constant elements q ; h we obtain due to (1.10) that
(1.11) Let w = (NI
+ &E)qI be a
vector valued function with components
w1, ..., wn. It then follows from (1.11)t h a t
(1.12) Suppose that for a symmetric matrix 17 we have
iLy:v;pv3q
= 0. It then
follows from (1.12) and the estimate (3.13), Ch. I, t h a t Ile(w)llLz(Qnw)= 0. Therefore w is a rigid displacement (see the proof o f Theorem 2.5, Ch. I). On the other hand w(<) = (NI
+ tqE)vI. Therefore due t o the periodicity of
NI(1) the vector valued function NqvI must be constant, and the matrix v must be a skew-symmetrical one. It follows that q = 0. Thus iy!v;pqjI > 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.
0
1.2. The Main Estimates and Their Applications Let us take as an approximation t o the solution of problem (1.1) t h e fol-
lowing vector-valued function
G(x) = u"x)
duo + & N P x( - ) axp ' &
(1.13)
II. Homogenization of the system of linear elasticity
124
where NP(E) are the matrices defined by (1.4) and u o ( x )is t h e solution of the problem
E(u0) = f O in 0
, uo = ipo
on
.
(1.14)
Theorem 1.2. Suppose that u'(x) is a weak solution of problem (1.1) in ip' E
W ,f' E L2(R'),
H'(R'), fo E H1(R), ipo E H3(R)and u o ( x )is a weak solution of the
homogenized problem (1.14). Then
IIu'(x) - uo(x)
Proof. Applying the operator L, t o uc - ii
I
we obtain the following equalities
which hold in the sense o f distributions
Since the matrices
Lc(U"- i i ) =
N" satisfy the equations (1.4), it follows that
31. Mixed problem in a perforated domain
125
Define the matrices N P q ( t ) ( p , q = 1, ...,n ) as weak solutions of the boundary value problems
a
- A p j ( < )-N q ( t ) , - A P q ( t+ ) ipq in w ,
alj
b ( N P q )= - V k A k p ( t ) N q ( t ) on dw NPq
is l-periodic in
t,
/
>
,
N p q ( ( ) d t= 0
(1.17)
.
Qnw
The existence of N p q ( ( ) follows from Theorem 6.1, Ch. I and the equalities (1.3). Thus we deduce from (1.16), (1.17) that
Lr(U'
a Fk , + axk
- G ) = f'- f0 + &Fo
E
(1.18)
where
Let us consider now the boundary conditions on S, for u' - G . We have
126
11. Homogenization of the system of linear elasticity
6,(U"
- C) = 6 , ( u e ) - O"(C) =
By virtue of the boundary conditions on aw for
N q
and
NPq
it follows t h a t
(1.20)
On the outer part o f the boundary o f 0' we have U"
- fi =
@'-
@O
auo = 1c, __
- E N S
(1.21)
c .
ax,
Let us show t h a t
where c is a constant independent o f valued function qc E
We define \I1,(x) t h a t cp.(x)
IVcpl 5
E.
To this end it suffices t o find a vector
H1(02')such t h a t qC+ EN'
as follows. Let cpe be a scalar function in C"(Q) such
= 1 if p(qa0)
5 E,
cp.(~) = 0 if p(x,a0)
C2E-l.
Set 8UO
QJX) = -cp.EN8 - ; ax,
I(,
=
{X
It is easy t o see t h a t
E H1(W,re),
: p ( ~80) , 5 QC
E
2 ~ n) 0'
H ' ( 0 ' ) and
2 2 ~ 0, 5 cpc(x) 5 1,
127
51. Mixed problem in a perforated domain
Therefore taking into account the properties of ve and the fact t h a t the matrices N " ( [ )and d N 8 ( ( ) / a ( j have bounded elements, we obtain t h e inequality
5
~ ~ ~ ~ ~ ~ H C3(11UoIIH1(Kr) 1 ( O e ) -k E l l u o l l H 2 ( K c ) )
By virtue o f Lemma
.
(1.24)
1.5,Ch. I, we get
IC4E112 l l ~ O 1 l H ~ ( R )
IIUOIIHl(K.,
9
is a constant independent of
where
c4
yields
(1.23). Therefore estimate (1.22)is valid. (1.18), (1.20),(1.21), (1.22) we conclude t h a t
E.
This inequality together with
On the basis o f
(1.24)
uc - ii is
a weak solution o f the following mixed boundary value problem studied in
Ch. I:
LC(uc- ii) = f'- fo u c - C = +L
Here
+c
on
a Fk + eF0 + e axk
re; u . ( u c - C )
satisfies the inequality
in Re
55,
,
=evkFk on S,
.
(1.22)and
n
IlFkllLz(n*) k=O
5 c5 l l U o l l H 3 ( n ) .
where the constant c5 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 IIU'
5
- CllH'(n*) 5 % ( E l l Z IIUOIIH3(R)
This inequality implies of elliptic systems (see
+ Ilf'
-
fOll* t
- @ollHl,2(re))
(1.15)since due t o the a priori estimates for solutions [l]) we have:
5 c(llfOllHl(n)+ ll@ollH"2(an)) .
11~011H3(n)
.
128
11. Homogenization of the system of linear elasticity
Theorem 1.2 is proved.
0
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
(1.25) (1.26)
The integrals E c ( u e ) ,E0(uo)represent the energy contained in R' and
n R'
R' respectively.
Theorem 1.3 (On the Convergence o f the Energy). Suppose that all conditions of Theorem 1.2 are satisfied. Then
(1.28)
(1.29)
129
$1. Mixed problem in a perforated domain
(1.30) where
Since the elements of the matrices
6.2, Ch. I) we get
dN8 A'j, - are bounded (see Theorem X i
Let us introduce the matrices
(1.32)
11. Homogenization of the system of linear elasticity
130
In the rest o f the proof it is assumed t h a t the matrices A ' j ( ( ) , N s ( ( ) ,
d N 5 / d ( i are defined in R" and are equal t o zero in
Thus we obtain
Q\w.
from (1.10)
J
H"(()d( = 0
.
(1.33)
Q Note t h a t due t o our assumptions we can replace the domain o f integration
RE n R' in (1.30) by 0'. Therefore after a suitable transformation, (1.30) becomes
(1.34) From this equality and (1.32) we conclude t h a t
Ee(uc)- (mes Q n u)Eo(uo)=
J
R'
duo*
x
axt + P,
8210
-Hst( -) -da:
ax,
E
(1.35) Note t h a t due t o Theorem 6.2, Ch.
I,
the elements o f the matrices H s t are
bounded functions. Denote by J, the set of all vectors z E Z" such t h a t by J: denote the set of all z E
Z"such that E ( Q
E(Q
+ z) c R' and
+ z ) n dR' # 0. Then
E,(u') - (mes Q n w)Eo(uO)=
It is clear t h a t t h e 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 E.
dR' and 6 is o f order
Therefore using Lemma 1.5, Ch. I, we deduce t h a t
Consider now
K 2 , i.e. the second sum in the right-hand side of (1.36).
Denote by 0'' the set formed by the cubes
E(Z
+ Q ) , when z takes values in
J,. Set
The vectors y t ( x ) are constant on each o f the sets E ( Z
+ &). We have
(1.38)
It is easy t o see that
(1.39) Taking into account the Poincark inequality in
E(Z
+ Q ) we obtain (1.40)
Let us estimate the sum
K2.
Due t o (1.33) the last integral in (1.38) vanishes
since yt are constant on each o f the sets
E(Z
+ Q ) for z E J,.
It follows that
11. Homogenization of the system of linear elasticity
132
Therefore taking into account (1.39), (1.40) we obtain 1K21
I C8E l l ~ o I l $ ( * , r )
(1.41)
*
The inequality
IIu0 IIHW I C9(llf011L2(*)
+ 11~ollH3/l(an))
and (1.31), (1.36), (1.37), (1.41) imply estimate (1.27).
Theorem 1.3 is
proved.
0
Let us consider now the convergence of the stress tensors, i.e. o f matrices whose columns are
o,"(x)
x du'
Apk(-) E
ax,,
p = 1, ..., 12
,
(1.42)
where u' is the solution of problem (1.1).
The stress tensor corresponding t o the homogenized problem (1.14) has the form duo
O;(X) E APk6
axk
,
p=l,
..',1 2 .
(1.43)
In the homogenization theory the matrices with columns o,", 00' are also referred t o as weak gradients or flows. In the next theorem it is assumed t h a t
and t h a t
Theorem 1.4. Suppose t h a t the conditions of Theorem 1.3 are satisfied. Then
133
$1. Mixed problem in a perforated domain
+ IlfO - Yll.
lPO- @cllH1/2(re)]
-t
where c is a constant independent o f
E,
(1.44)
1
and the matrices GP'J(<) are defined
by the formulas
< E Q nw , for < E Q\w .
GPs(<)= A P 8 ( (+ ) Api aN" - (mes Q n w)&'
for
ati
GPs(()= -(mes Q n w ) k 8 Moreover, if
f' = fo, W
= cPo then
u,"(x)--+ (mes Q n w)uOp(x) weakly in L2(R) as
Proof.
(1.45)
E
+0
.
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
Therefore taking into account (1.43) we get
o,"(x)- (mes Q n W ) U ; ( X ) =
This equality and (1.29) imply (1.44). The weak convergence of u,"(x)t o (mes Qnw)uOp(Z) for follows from (1.44) and the fact that convergence
I.
auo
J
Theorem 1.4 is proved..
QC
= cPo
GP"(<)d(= 0 which implies the weak
Q
G p " ( - )- -t 0 in L2(R)as E
ax,
f" = fo,
--f
0 by virtue o f Lemma 1.6, Ch. 0
11.Homogenization of the system of 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 t h e Neumann problem in a perforated domain
R' 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 t h e conormal derivative of a rapidly oscillating corrector
x duo E axs
which has the form E N ' ( - ) - outside a neighbourhood of do. Therefore in order t o clarify the main ideas used in the proof of 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 t h a t 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
a ax;
consider the following Neumann problem
x auc
L c ( u e )= - (aI3(-1 "
R
E
-)axj
= f in R
where ( u l , ...,un) is the unit outward normal t o are smooth functions in
where
K ~ , = K const ~
R",1-periodic in [,
> 0.
The functions
f
,
dR. It is assumed that aij([)
and such that
and 'p are sufficiently smooth and
satisfy the condition of solvability of problem (2.1)
J n
fdx =
J
an
'pdS.
(2.2)
We define the functions N p ( t ) ,p = 1, ..., n , as solutions of the problems
135
$2. Boundary value problem with Neumann conditions
N P ( J ) is
1-periodic in
t,
J
N P ( t ) d t= 0 .
Q Set
hhk =
J
[uh"((s)
Q
Nk + uhi 1-dati
dt ,
Q = {X : 0 < x3 < 1 , j = 1,...,n} . As a n approximation to the solution of problem (2.1) we take the function 6, = u'(x)
+ E N ' ( - ) -, dUO(X)
2
(2.4)
ax,
where uo is the solution of the homogenized Neumann problem
duo = 6'3.. duo u; = cp
auA
on dR
ax
(2.5)
.
In analogy with (1.16) simple calculations show that CE(uC - 6,) = -EU'~N'
a3uo
axiax,ax
Let us define the functions
Then
a
- 6 ) = E -(ukh
CE(UC
axk
+
N i s ( ( ) as solutions of the
problems
d3U0
ax,axiaxj
-
11. Homogenization of the system of linear elasticity
136 where
(2.11)
Lemma
2.1.
Functions &"t) defined by
1 n
cu'a(t)cif = o
a Qis ( f ) = 0 at;
J
s;
( 2 . 1 1 ) satisfy the relations
,
,
i , s = 1, ...,n ; s = l,...,n ;
aJ3(<)dij =0,
(there is no summation over j ) .
tlt E R', s , j = 1 ,...,n ,
(2.12) (2.13) (2.14)
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 Qilt2 the set
Multiplying (2.13) by (, - tl and integrating over
8
J
0=
*i a ( < ) ( t i - t l ) d t
we get for each
j
=
Q:, t 2
J
=
c P ( ( ) ( t z - tl)d& -
J
a3'(()d( .
Qil
3i2
t2
Setting tl = t z - 1 and taking into account (2.12) we obtain (2.14) for t = tz. Lemma 2.1 is proved. 0 Lemma 2.2. Let a"(() be functions in H ' ( Q ) 1-periodic in ( and satisfying the conditions (2.12)-(2.14). Then for any v E H ' ( R ) the following inequalities are valid
(2.15) where c is a constant independent of
E,
v.
Proof. Denote by 1; the set of all indices z
+
p ( ~ ( z Q ) ,80)2
E.
Set
01
=
u
(EZ
+
-t Q) Q ) . It is easy to see that
E Znsuch that
E(Z
ZEl:
o
=
J n\n1
-
J anl
Therefore
d dXi
X
- (aik(--))vdx =
J
a"u;vdS -
an
aiku;vdS -
J
R\RI
dV
aik- dx
dXi
,
k
= 1, ..., n
c R,
11,Homogenization of the system of linear elasticity
138
J
aik(;)u;vdS 2 =
an =
J
aiku,vdS
+J
an
&
aik
n\nl
dx
.
(2.16)
We clearly have
(2.17) Let us estimate t h e first integral in the right-hand side of (2.16).
It is easy t o see t h a t
an1 consists o f ( n- 1)-dimensional faces o f the cubes
+ Q) for some z E If. Denote by o:,..., ojI ) the ( n - 1)-dimensional faces of the cubes E(Z + Q ) for z E If such that ojk is parallel t o the hyperplane xj = 0 and lies on an1,j = 1, ..., n. We thus have
E(Z
j=l
a=l
Denote by q; the cube E(Z
+ &) whose surface contains the set a;. It is
obvious t h a t among the cubes q;, j = 1, ..., n; s = 1, ...,13, there cannot be more than 2n identical ones. Due t o the condition (2.14) for any n; we have
J
atkv,dS =
0:
since u, = 0 for
J
aJkv3 dS = o
,
(2.18)
0:
i#j
(there is no summation over j ) . Set
~ ( x =) (mesq;)-’
J
vdx
,
9:
X E $ ,
j = 1 , ...,n ; s = 1 , ..., l J .
Thus the function ~ ( x is) constant on each surface a;. Taking into account (2.18) and the fact that not more than 2n cubes q; can have a non-empty intersection we obtain
139
$2. Boundary value problem with Neumann conditions
(2.19) Here we have also used the inequality
J
1v - r]l2dx 5 c3.5
0:
J
1vvl2da:
(2.20)
9:
which can be proved if we pass to the variables [ = E
- ~ X and
apply Proposi-
tions 3 and 4 of Theorem 1.2, Ch. I , in the domain R = &-I$. The relations (2.16), (2.17), (2.19) imply (2.15). Lemma 2.2 is proved. Therefore due to (2.7)-(2.10) we have (2.21)
where Fl, F t , F2 are bounded uniformly with respect t o Setting w = u ' -6 obtain from (2.21)
E.
+ vc where vC is a constant such that nJ
w dx = 0 , we
+ F2wdS1] . auo
Applying Lemma 2.2 to v = -20 and using the Poincare inequality (1.5), Ch. I, we find that
8x9
II. Homogenization of the system of linear elasticity
140
Thus we have actually proved Theorem 2.3.
f, cp are smooth functions satisfying the solvability conditions for problems (2.1), (2.5). Then there is a constant q'such that Suppose t h a t u', uo are solutions o f problems (2.1), (2.5) respectively, and
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
R' denotes a perforated domain of type II defined dRc is a union o f dR and the surface of the cavities
In the rest o f this section in $4, Ch. I. The boundary
s, c R . In
R'
we consider the boundary value problem o f Neumann type for t h e
system o f linear elasticity:
d x du' Lc(uE= ) - (Ahk(--) -) = ~ ( x in) dxh E d2k a,(u') = O on S, = dR'\dR , a c ( u c )= $f
on dR
w,
1
(2.22)
J
.
It is assumed t h a t the elements of the coefficient matrices Ahk satisfy the same conditions as the coefficients of the system ( l . l ) , f' E L2(Rc), @ E L2(dR) and satisfy the conditions of solvability for problem (2.22), i.e.
J (f',rl)dx = J ( V , r l ) d S vv E
n*
an
3
'$2. Boundary value problem with Neumann conditions where
R is the space of
141
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 (2.23)
(mes Q n u)+(uo) = ?,bo where 6(uo)
"hahk
&Lo
-, the matrices Ahkare defined by the formulas (1.3), axk
',1 E L2(dR), fo E L2(R), satisfy the solvability conditions (mes Q n u)-1
J (GO,
V ) ~ S=
an
J
(fo, q ) d x
VV E
R.
R
It is important t o note that the factor mes Q nu appears in t h e Neumann a R . This factor is equal t o 1 if R' coincides with R (see $2.1, formula (2.5)). To characterize the closeness between functions fo, ',1 and j ' , 1,' we conditions on
introduce the following notation.
L2(dR) the scalar products (f,v ) p ( n . ) , ($, v),p(an) define continuous linear functionals on H'(R'), and therefore f and 1c, can be considered as elements of the dual space H1(R')*. Let us denote the norms of the respective functionals as IlfllH1., ll11,llH1*,i.e. For any vector valued functions
f
E L2(Rc), 11, E
{ (f,v)Lz(ne),v E H1(R'),
Ilf llH1* =
IIOIIH1(Re)
= 1)
7
U
II$IIH1*
=
SUP
{(11,,")L2(an),v E
H'(R'),II~IIHl(n.) = I} .
v
G(x) = uo + &cp(X)N$(-)- . & ax, x
duo
(2.24)
Here uo is the solution o f problem (2.23), N 8 ( t ) are solutions of problems (1.4), cp(x) is a truncating function which satisfies the following conditions:
142
Homogenization of the system of linear elasticity
x E R1 such that p(x,dR,) L where cl, c are constants independent of
E;
R1 is defined
,
(2.25)
C ~ E
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 uc is considered in t h e per-
Rc but neighbourhood o f dR.
forated domain
the matrices
X
N " ( - ) are in general not defined in E
a
The main result of this section is Theorem 2.4. Suppose t h a t
f'
E
Then the solutions
L2(R'), fo E H'(R), @ t i E , uo
E L2(dR), 6 ' E H3l2(dR).
of problems (2.22), (2.23) respectively satisfy t h e
following inequality
L
t (2.26) 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
143
$2. Boundary value problem with Neumann conditions
Lemma 2.5. ) the following conditions The matrices a i d ( (satisfy
J
ai"(()d(= 0 ,
i , s = 1, ...,n , (2.27)
Qnw
-aa
ati
is
in w , s = l , ...,n ,
(()=O 1.
u,aid = u,A'" on
J
dw ,
s = 1, ..., n
,
(2.28)
a j d ( ( ) d i j= (mes j?j - mes Q n w ) P
,
(2.29)
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 wnQ:,
Ajqd( t2
/
(j
- t l ) E , where
E
n w , (tl < t z ) ,we obtain
is the unit matrix, and integrating over
( ( j - tl)ukAkqdS.
(2.30)
a(Q:, t2 n w )
Each integral in (2.30) over d(Q;,,,
n w)
can be represented as a sum of
integrals over the sets
and the integrands are 1-periodic in (2.30) that
ST, T #
j,
= 1,
..., n , it follows from
II. Homogenization of the system of linear elasticity
144
(there is no summation over j ) . Setting tl = tz - 1 in this equality and taking into account (1.3) we find that
It follows from the definition of
ajs t h a t
= (mes S,3)A3"- (mes Q n u)ka = A
.
A
.
= (mes 2; - mes Q n u)&" . Lemma 2.5 is proved.
0
Remark 2.6.
If the domain Re is not perforated, i.e. w = R",Rc = R , then aj"(()d& = 0 , since mess; = mesQ n u = 1 (see Lemma 2.1).
/
A
.
9; Lemma 2.7. Let ol,...,oZn be (n- 1)-dimensional faces of the cube EQ = { z : 0
~ ,=j 1, ..., n}. Then each u E H1(eQ) satisfies the inequality
< zj <
145
$2. Boundary value problem with Neumann conditions
(2.31) where c is a constant independent of
Set o1 = {q = 0) n EQ, o2 = { x 2 = 0} n E Q , S1 = &-lo1, - E -1 6 2 . Consider the points = (O,yZ,y3, ...,yn), ?j2 = (y2,0,y3, ..., yn)
Proof.
s
2
i, j, E.
6'
on thefacesS1, SzofcubeQ. Thesegmentg(t,y2,y3, ...,yn) = t$'-k(l-t)i2 for t E [0,1] belongs t o Q. It is easy t o see t h a t for any v E H ' ( & ) we have v2(g(l,
~ 2 ,
Y,)
- v2 (g(0,
~ 2 ,
Yn))
=
Integrating this equality with respect t o y2, ..., yn from 0 t o 1 we obtain the estimates
J
v2dS-
J
v2ds
X
= - , C = const &
J
&+-I) v2
J Ivl(Vpldl, Q
sz
s1
,
do - &-(,-I)
01
v2 do 01
5CPE
J
IvI IV,vl dx
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
dR1 which is the boundary of the domain Ol given (4.3),Ch. I. The domain R1 depends on E and its boundary 6'01 consists of the (TI - 1)-dimensional faces of the cubes E ( Z -k Q), z E T,. Denote by C T ~..., , o) the faces o f the cubes E ( Z -k Q ) for z E T, parallel t o the hyperplanes xj = 0, j = 1, ...,n , and laying on 801. Then
functions defined on the set by formula
II. Homogenization of the system of linear elasticity
146 The cube
E(Z
+ Q ) , z E T,, on whose boundary lies the set ojs is denoted by
q;. It is easy t o see t h a t among g;, j = 1, ...,n , s = 1, ..., Zj, the number of
the identical cubes is not greater than 2n. Lemma 2.8. Let u E H'(R). Then
Ill'llH1(n)
ll'llLz(ani)
where c is a constant independent of
Proof. According (n -
(2.33)
7
E.
dR1 consists of the sets ofi, and each ofi is an 1)-dimensional face o f the cube q i . The boundary dR is a smooth t o (2.32)
surface, therefore each cube q; possesses a face os,jsuch t h a t os,j is parallel
0 and oSj is the orthogonal projection along t h e axis OX^(^,^) of a surface S8,j C dfl which is given by the equation
to the hyperplane
2,
xrn(j,*)=
= $m(?rn)
and
CIE
where constants cl,
7
j.m
E
osj
I 15 - yI I C ~ Efor c2,
~ 5
~
I $M
7
mm
=~m ( j ~,3)
~
~
E os,j, y E S8,j ,
M do not depend on
E,
s, j .
Denote by Qs,j the set formed by the segments orthogonal t o o,j and connecting the points o f a,,j and Sd,,. Then using a suitable diffeomorphism mapping
QS,j to
EQ and taking into account Lemma 2.7 we find t h a t
2
IIuIILz(u,,,)
5 c ( ~ ~ u ~ ~ ~ z ( S+s ~, ,~) u ~ ~ ~ ~ ( Q s , , ) ) *
Therefore by Lemma 2.7 we get 2
+
5
.
~ ~ u ~ \ L z ( u'1 ~( )~ ~ u ~ ~ ~ z ( S , ~, ,~)u l ~ ~ l ( Q , , , ) )
Summing these equalities with respect t o s, j we obtain estimate (2.33), since due t o the smoothness o f dR there is an integer and such that each
Ic independent of
E
can have a non-empty intersection only with a finite
number of Q{,$which is not greater than
Ic.
Lemma 2.8 is proved.
Lemma 2.9. Let the matrices y h k ( z )E
Lm(dfl1)be such that
0
$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), functions uo E
147
y = const. Then for any vector valued
H3(R),w E H'(R) the inequality
holds with a constant c independent o f
Proof. Consider a function
E.
r ( x ) defined almost everywhere on
dRl by the
for m u Ia
Obviously r ( x ) is constant on each
0;". Therefore
setting
and taking into account (2.34) and the Poincarh inequality in a;" we obtain
J
anl
lrI2dS =
CCJ n
11
j=1 s=1
a:
lrI2ds =
II. Homogenization of the system of linear elasticity
148
It follows from Lemma 2.8 that
J
lr12dSI czEZrZlu011;3(n)
(2.36)
1
anl where c2 is a constant independent of It is easy to see that
E.
(2.37) Due to (2.36) and Lemma 2.8 we have
L
.
C3EY 11~011H3(n) l14IH’(n)
(2.38)
Let us estimate the second integral in the right-hand side of (2.37). Define the vector valued function ~ ( x on ) do1 by the formula T(x) = (rnesqm)-’
J
1 w dx for x E om
.
s!n
Therefore
J
Iw-qf’dS=
anl
5
55 1
Iw-qI2dS
5
j = 1 s=l
g :
CCJ
n 11 C ~ E j=1 a = l
9:
I ~ w l ~ d5xC
J
~ E
IVw12dx.
(2.39)
n
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 ZJ = w. By the definition of r we have
149
$2. Boundary value problem with Neurnann conditions
J
(yhkvh auo - r ) d S = 0
.
4 Therefore since q(x) is constant on each c i we find by virtue of (2.39), (2.36) and Lemma 2.9 that
Ic77+’/2
lluollH3(n)llWllHl(n,
*
This inequality together with (2.37), (2.38) yields (2.35). Lemma 2.9 is proved.
0
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 of Theorem 2.4. Let us apply the operator
ii is given by (2.24). Then
C, to the vector valued function
uc - 6 , where
11. Homogenization of the system of linear elasticity
150
Let us define the matrices Nhk(E)as weak solutions of the following bound-
a r y value problem
$2. Boundary value problem with Neurnann conditions
151
152
II. Homogenization of the system of linear elasticity
153
$2. Boundary value problem with Neumann conditions
du" duo Q,(u"- G) = (1 - v ) u ~ A ' ~ - (1 - cp)Aiju; "
ax + 43; + ~ $ 7on
Set w = u" - 4
+ q E , where 7'
dXj
+
dR' .
(2.43)
is a rigid displacement such! that
ER .
( w , ~ ) H I ( ~= . ) 0 for any
Due to the boundary conditions for uo, uEand the fact that cp = 0 in R\Rl
it follows from (2.41), (2.43) that
+J
J
( u ; A l j ( l -duo ( ~ ) ~ , w ) d ~((mesQnw)-'1Clo-1Cl',w)dS. + an
s.
(2.44) 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 Rl,p(x,aRl) 2 of
E.
ax
:
x E
C ~ E and } IsVql 5 c , where c , c1 are constants independent Therefore by Lemma 1.5, Ch. I, we obtain
5 cz IlwllH1(n*)&"2lluollH3(n)> where c2 is constant and does not depend on
(2.45) E.
It follows from (2.42) that
5
C3E l l w l l H 1 ( n e , ) lluollH3(n)
(2.46)
7
where c3 is a constant independent of E. Taking into account (2.27), (2.28) and setting
cy
= mes Q n w we get
11.Homogenization of the system of linear elasticity
154
+J
(a-1+0
- +‘“,w)dS =
an
(2.47) It should be noted here that in the integral over (aRl)\S, the normal v is exterior t o to
dR1, whereas in the integral over dR1n S,,
the normal v is exterior
0‘. T h e last two integrals on the right-hand side of (2.47) can be estimated
by C@
11~011H3(n) llwllHl(n.,
similarly to (2.45). Let us introduce the matrices
phk(t)setting
ah’([) in w
,
in P \ w . Then p h k ( ):
=
a (--) in
It follows from (2.47), (2.48) that
(2.48)
$2. Boundary value problem with Neumann conditions
155 (2.49)
+J
(2.50)
(+o-v,~)~s.
an
The integral identity for uo yields t h a t
Therefore by virtue of Lemma 1.5, Ch. I, mesQnw
an
= (mes &\w)
/
anl
dUO
(uhAhk -,w)dS
a xh
+
53
,
(2.51)
We thus obtain from (2.50), (2.51) t h a t
(2.53)
yhk = (mes &\w)Ahk - phk
in Lemma 2.9. It is easy to verify that conditions (2.34) are satisfied for y h k . Indeed due to (2.48) and (2.29) we have
156
J
11. Homogenization of the system of linear elasticity
-
yhkdS = tn-l(mes &\w)Ahk
J
= En-'(mes &\w)Ahk
-
J
J
phkdS-
q\s*
ohm
phkdS =
urnsc
ahkdS - (mes .; nI .)Ahk " =
a,"\&
J
= En-'(mes &\w)Ahk - cn-' c-1
= En-l(mes &\w)Ahk
a h k ( ( ) d S- (mes
n sC)Ahk =
(u,m\S,)
- (mes Q n w ) a h k -
- c"-'(mesE-'(a~\Sc)
~ ) =A en-' ~ ~(mes &\w - mes E-'(ur\sc) + + mes Q n w - mes c-l(u;~"n s,))Ahk= o , - (mes u r
ns
since
mes&\w t m e s Q n w = 1 , mesc-'(ur\Sc)
+ rnesc'(@
n Sc)= 1 .
W e conclude from (2.52), (2.53) and Lemma 2.9 t h a t lJ2l
I c [Ell2
IbOIIHJ(*)IIWIIHl(n.)
+ E l l 2 IlfOllLz(n)IIWIIHl(n.)
+ I I 4 O - @IIH'* IlWIIH'(n.,]
t
(2.54)
*
It follows from (2.44), (2.45), (2.46), (2.49), (2.54) that ~ ~ e ( W ) ~ ~ ~5z (c1P )[&'I2 11uo11H3(n) IIWllH1(R*)
+
+ E l l 2 Ilf011L2(n)llWllHl(n.) + llfO - f C l l H' * l l ~ I l H 1 p + )
+ I I 4 O - PIIH1. IIWllH'(n.)]
(2.55)
*
From the well-known results on the smoothness of solutions of elliptic boundary value problems we have lluollH3(n)I c2(llf011Hl(n)
+ l14°11H3/1(an)) ,
since dR is a smooth surface and fo E H'(R),
(2.56)
4 ' E H312(dR).
Therefore by virtue of Theorem 4.4, Ch. I , the inequalities (2.55), (2.56) yield (2.26). Theorem 2.4 is proved. 0
157
$2. Boundary value problem with Neumann conditions
2.5. Estimates for Energy Integrals and Stress Tensors
1.3 and 1.4 on the convergence
Slightly modifying the proof o f Theorems
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 instead of
(2.26)
(1.15).
2.1Q (On the convergence of energy integrals). 2.4 are satisfied and EE(u'),Eo(uo) are defined by (1.25), (1.26). Then
Theorem
Suppose t h a t all conditions of Theorem
IEE(uE) - (mes Q n u)Eo(uo))I
5
+ 11$011~3/2(an))+ l l f o
[E1'z(llfoll$l(n)
+ (1I@O11L2(an)+ IlfollL2(n)) (1l.f'
- f"llH1* +
- fCIIH'*
where C is a constant independent of
+
E ; u E , uo
- +'IIH'*
- @'IlH'*)]
7
+ (2.57)
are solutions of problems
(2.22),(2.23) respectively. The proof of this theorem in the main repeats t h a t of Theorem 1.3. However, slight modifications should be made. In particular we consider the solutions uo and u' such that
J
(ue,q)dZ =
ne
J
n*
(u0,q)ds = 0
,
vq E
R.
(2.58)
This choice of uc and uo is possible since solutions of problems
(2.22),
(2.23) are defined t o within a rigid displacement. In this case one can take qE = 0 in (2.26),and use the estimates 0 llH'(n)
11u011H3(R)
5 C ( ~ ~ f o ~ ~+L ~2~( @~ o) ~ ~ L 2 ( a n ) ) 9
5 c(llfolIH'(n)
+
11$o11H3/2(aS2))
9
which are well known from the theory o f elliptic equations (see Similarly t o Theorem
1.4 we establish
[l]).
II. Homogenization of the system of 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
o!(x),o:(z) be defined by the formulas (1.42), (1.43). Then
where c i s a constant independent o f E , t h e matrices GPq are defined by (1.45). Moreover
oP(x) 4 (mes Q n w)oOp(z) weakly in L2(R) as
E +0
2.6. Some Generalizations For the homogenization o f eigenvalues and eigenfunctions related to 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 ,Cc(uc)- pe(x)u' = f' in o,(uc) = $f
on dR
,
Re ,
o,(u') = 0 on S,
,
(2.60)
and also the corresponding homogenized problem
L(uo)- po(x)uo = f o in (mes Q n u ) & ( u o )= $O where operators L ,
P, E L"(R'),
PO
,
on dR
,
E are the same as in problems (2.22),
(2.61) (2.23), the functions
E L"(R) are such t h a t (2.62)
52. Boundary value problem with Neumann conditions and constants
Q,
cl, c2, c3 do not depend on
159
E.
In Theorem 2.4 we established the closeness of solutions of problems (2.60), (2.61) when pc
= 0, po = 0.
If we introduce a parameter characterizing the
closeness o f pe 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 ( ( , ~ )is 1-periodic in 5 and satisfies the Lipschitz condition with respect t o z E
R
uniformly in <, i.e.
p ( t , z) E QR. x 0) in terms of
Lemma
1.6, Ch. I.
Let po(z) = ( p ( . , z ) ) ,where ( p ( . , z ) ) is defined by (1.23), Ch. I, and is
equal t o the mean value o f p ( [ , z )with respect t o
t.
It follows from Lemma 1.6, Ch. I, t h a t for any vector valued functions u,o
E H'(R') we have
Ic-5 l l U l l H l ( W ) IIUIIHl(n*)
3
c = const .
(2.63)
Indeed, set g(<,z)= ( p ( < , z ) - p o ( z ) ) x w ( t ) in Lemma 1.6, Ch. I, where
xw(<)is the characteristic function of the domain w with a 1-periodic structure. It is easy t o see t h a t g(<,x)E L(Rnx !=I), J g(t,z)dt = 0. Consider the extensions P,u, P,v of u , v t o the domain R Q
which were constructed in Theorem 4.2, Ch. I. Then
Note t h a t the set fl\R1 belongs t o a 6-neighbourhood o f dR and 6 is of order
E.
Therefore applying Lemma 1.6, Ch. I, t o estimate the first term in
II. Homogenization of the system of linear elasticity
160
the right-hand side of this inequality, and Lemma 1.5, Ch. I, to estimate the second term, we obtain qc
5 C1E IIPeUIIHl(C2) IIPeVIIHl(fi)
*
This estimate together with (4.17), Ch. I, yields (2.63). 2 Therefore the functions p ( - , s ) and po(z) are close in the sense of the & inequality (2.63). In a more general situation we shall characterize the closeness of pE and PO by the norm
IllPlll =
SUP U,v
{
P(U,VldZI
?
1
IIuIIHl(n*)= 1 1 ~ 1 1 H l ( n ~=) 1
7
(2.64) where the supremum is taken over all vector valued functions u , w in H'(R'). Relation (2.64) implies that for any u , w E H ' ( 0 " ) we have (2.65)
It is easy t o see that estimate (2.63) implies Lemma 2.12.
2
Let p'(z) = p ( - , s ) , &
IllPr
PO
= ( p ( . , s ) ) , p ( t , z ) E i ( I R " x fi). Then
- Poll1 < = 7
where c is a constant independent of
(2.66) E.
Theorem 2.13. Suppose that f' E LZ(R'), fo E H'(R), $f E LZ(aR),go E H312(dR),
E C'(n) and u e , uo are the solutions of problems (2.60), (2.61) respectively. Then PO
(2.67)
$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 to the proof of Theorem 2.4. An approximate solution of problem (2.60) is sought in the form
ii = U O
duo + ~ c p N "xE ( -) , ax,
where uo is the solution of problem (2.61), N" are the same as in Theorem 2.4. Applying the operator Cc - p e l to uc - fi we obtain Cc(UC -
where .Fl,
a) - p c ( U = - a )
=
q ,e, e, are defined by the formulas (2.42)
c = (pc
-
PO)^'
+ E P ~ Q N: . ax, aUo
and (2.69)
For oc(uc- 6 ) the formula (2.43) remains valid. Setting 20 = 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 fl\fl, for the solution uo of problem (2.61). Further changes in the proof of Theorem 2.4 are obvious.
$3. Asymptotic expansions for solutions of boundary value problems 163 $3. Asymptotic Expansions for Solutions of Boundary Value Problems
of Elasticity in a Perforated Layer
3.1. Setting of the Problem Consider a domain
RE o f the form
where w is an unbounded domain with a 1-periodic structure satisfying t h e Condition
B
of $4.1, Ch.
I,
E
> 0 is a
small parameter, and
E - ~ is
an integer
number. Set
If w In
# R",then RE is a
perforated layer.
0' we consider the following boundary value problem
u c ( i , o )=
W ( i ) on
uE is 1-periodic in
Here Ah'([)
ro,
u c ( i , d )= i p 2 ( i ) on
,
rd
1
I
I
i = (q, ..., xn-l) .
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 of problem (3.1) under suitable assumptions on
f , Q1,
a2are 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"),
@3
E Cm(Rn-'),
f , @j
II. Homogenization of the system of linear elasticity
164
Our aim is t o find an asymptotic expansion for the solution uc 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
Here we reproduce the results obtained in [87].
k > 0 the solution
u' of problem (3.1) can be represented
in the form k U E ( 2 )=
+
81
PA(? ,&)Y:(2,&)
&'
E
m=O
I=O
E k p k ( & , 2)
where PA((,&)are n x n matrices such that
PL(E,E) = PL,(t) and PA,, are 1-periodic in (,
d
+ PA,(t)+PAz(i,tn- -) &
7
PAl(()and PAz(i,En - f ) define boundary lay-
ers, the components of the vectors Y;(x,E)are polynomials in
.E
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 {z : 0
< x , < d}, the remainder IIpk(E,x)IIHi(fie)
with a constant
5 Mk
Mk, independent
p k ( ~ , z satisfies ) the inequality
k = 1,2,*** 7
1
of
E.
3.8. Formal Construction of the Asymptotic Expansion We seek the solution o f problem (3.1) in the form uc =
2 c €'
I=O
(+I
Na(t)DD"va(x) ,
t = €-lz.
(3.2)
In contrast t o Chapter I, here for the sake o f convenience we use the following notation
a, takes the values 1, ..., n; N a ( ( )are matrices whose elements are 1-periodic in
t ;v , ( x ) = (vf, ...,vi) 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
a (v(x,€-%)) = av(x7 t ) + &-1 %x, ~
dXi
Xi
we obtain the formal equality
Here we used the following notation
at;
0,
[ =e-lx,
11. Homogenization of the system of linear elasticity
166 for ( a )=
1,
5 c N,(t,;)D% - d
where
for ( a )> 0 and
E
~($1,
E
rd ,
$3. Asymptotic expansions for solutions of boundary value problems 167
for ( a ) = 0. Let us represent
Na(<) in the form
NOI(t) = N,o(E)4where
mo + mo
>
N : ( [ ) are matrix valued functions 1-periodic in t , N i , Nz define bound-
ary layers near the hyperplanes Set
2,
= 0 and
2,
= d respectively.
N: = E , N t = N t = 0, where E is the unit ( n x n)-matrix. Denote
a
a
T,"(5)= (AkawN:2...011+ ( t )A) " * W - ~:2...01,(o + %k atj +
AOIlW
(oN:3...*l 7
z:(E)= ~ k A k a 1 ( E ) N ~ 2 . . . a t P( t=) 0 , 1 , 2 7
7
where N z = 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
< E aw , N : ( E ) is 1-periodic in t , J N , O ( ~ ) ~ E= o ,
a(N:) = - Z z ( t ) on
(a)= 1,2, ...
(3.5)
Qnw
.
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
168
J
i,
N : ( t ) is l-periodic in
where h i , h: are ( n x n)-matrices with constant elements chosen in such a way t h a t the inequalities
s=o,1,2
s =
,...)
d d
-, - - 1, ... , &
(3.9)
&
hold with constants Cp, C;,
K:,
K:
independent o f
E.
(3.6), (3.7) and existence o f the h i can be proved by induction on the basis o f Theorem
Existence o f the solutions for problems constant matrices h;,
8.4, Ch. I. Note t h a t because o f the boundary condition in (3.7) on the hyperplane
&,
d = - , the matrices N," and h i depend on &
from the l-periodicity in
N : ( t ) on
E
t
E.
If d is a multiple of E it follows
of the matrices A h k ( J )t h a t the dependence of
is determined by the relation
d N:(t) = fi:ci,tn - );
9
53. Asymptotic expansions for solutions of boundary value problems 169 where
fi:([)
are solutions of the corresponding sequence of problems of type
(3.7) in w(--oo, 0) with the boundary conditions
N:(t,O)= -NZ(t,O) + kz . Obviously the matrices Having thus defined
fiz,i t do not depend on E .
NZ, p = 0,1,2,, let us substitute v, in (3.1). We get
the formal equalities
nc, c c , a*(?)= c c +N:(t c c ( h i + N i ( ; , ;))D% W
f ( z ) 2i'
zE
ho,D"vc(z)
2-2
(+I
I=O
(3.10)
00
(hL
€I (+I
I=O
&
3
00
@(3)
1=0
,O))D%, d
(3.11)
(a)=/
It is easy t o verify t h a t here
hO, = hO,l = 0
, hi = hi = E
(3.12)
(c3W)\(I'o U r d ) are satisfied due t o the boundary conditions in (3.5),(3.6), (3.7) for the matrices N,, NA, N:. Note t h a t by virtue of (3.4)the constant matrices h:,,, are defined by the
and the boundary conditions on
formulas
Comparing these equalities with
(1.3)we conclude that h$ = aij,i , j =
1, ...,n, i.e. h:j are the coefficient matrices of the homogenized elasticity system. Let us seek vc in the form of a series
c 00
v,(z) =
€jVj(X)
.
(3.13)
j=O
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 of linear elasticity
170
0 0 0 0
Therefore by virtue of (3.10) we find that (3.14) Consider the first equality in (3.11). Due to (3.13) it is obvious that
=
c c c
h;Z)"Vk-,
Ek
k=O
.
m=O ( a ) = m
By virtue of (3.9) we have ?
I I N ; ( ; ,o ) I I ~ ~ , Z ( ~ , ) L ce-
, y,c = const > o .
Therefore the first equality in (3.11) yields
a'(?)2
c c c k
00
Ek
k=O
m=O ( a ) = m
In the same way we find that
hiDaVk-m
I
(3.15) =o
I "
(3.16) k=O
m=O ( a ) = m
Equating the terms of the same order with respect to E in (3.14), (3.15), (3.16) we get the following recurrent sequence of problems for y(x):
$3. Asymptotic expansions for solutions of boundary value problems 171
& ( x ) is 1-periodic in
P,
j = 0,1,2,
I
... .
Here
j = 1 , 2 , ... . Existence of
Vj follows from Theorem 6.5, Ch. I, when w = IR" and the
coefficients of the elasticity system are independent of
E.
3.3. Justajication of the Asymptotic Expansion. Estimates for the Remainder In the previous section we constructed a formal asymptotic expansion for t h e function u' which is the solution of problem (3.1). This asymptotic ex-
pansion has the form (3.2) where
No = N:
+ NA + N:,
N,, NA, N: are
solutions of problems (3.5), (3.6), (3.7) respectively, u, 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
(3.19) where L.
(3.20) j=O
In t h e next theorem we give an estimate for t h e remainder term of the asymptotic expansion for t h e solution u' of problem (3.1). Theorem 3.1. Let u' 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
172
IIdk)- U=llHl(fi.) I M k E k + 1 ,
(3.21)
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, p = 0,1,2.
Lemma 3.2.
The solutions N,P, p = 0,1,2, o f problems (3.5), (3.6), (3.7) satisfy the inequalities
J
(lvm;)12 + IN,(;)I - ) dx < - M," ,
A*
2
ll~;(;)llHl,z(~d) X llm;)llH1,2(Fo)
where ME,
cj, -yj
,
Ise-? , I cze-
(3.22) (3.23)
21
(3.24)
7
are positive constants independent of
Proof. Let us establish (3.22) 0,1,2,
p = 0,1,2
E.
for p = 0. By induction with respect t o (a)=
... we obtain from Theorem 6.1, Ch. I, t h a t
J
( I V ~ X ( O I+~I N , o ( ~ ) I ~ I ) ~C, E
Qnw
Changing the variables of
2
=
E<
and taking into account the 1-periodicity
N,"(t) and the fact t h a t the domain
h. contains
not more than (d
+ 1 ) ~ - "cells ~ ( +z Q nu), z E Z",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 t h a t
J t(O,
where
9)
( I V ~ N ; ( O+ I ~1 ~ ; ( 5 ) 1 ~ ) I d t ~a
M , is a constant independent of
E.
Passing t o the variables
in this inequality we obtain (3.22) for p = 1 since and the domain z =
(2,O).
he contains exactly ~ - " + l
domains
x =
E(
N:(E) are 1-periodic in [ E(Z
+ h(0, i)), z E Z", E
$3. Asymptotic expansions for solutions ooff boundary value problems 173 Estimate (3.22) for p = 2 is proved in the same 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 1 / 2 ( t ~ H'I2(f'd). )H112(i!d). , Lemma 3.2 is proved. of the norms in H H1I2(t0), Proof of Theorem 3.1. Let us show that the vector valued function u ( ~given ) by (3.19) is the solution of the problem . ~ , ( u ( ' ) )= f
+
+
E ~ + ~ ~ OE () Z ,E ~ + '
d k ) ( ; , 0 ) = a'(?)+ ~ ( ~ ) ( ; ,= d )a'(?)
a
-gm(z,E ) in R' ,
\
axm e k + + ' d 1 ( i , e on ) ro,
+ E ~ + ' ~ , ( $ , E ) on
C = ( U ( ~ )= ) V , E ~ + ~ ~ ~ ( Z ,on E )
u ( ~ ) ( xis) 1-periodic in f
,
r d
>
(3.25)
(anc)\(rou r d ) ,
,
1
where n
Mo, MI, Mz
(3.26)
are constants independent of
E.
Then estimate (3.21) would follow directly from Theorem 6.5, Ch. I. Consider first the boundary conditions for ti@)("). We have
C
k+l
~ ( ~ ) ( f ,=d )
E'
f=O
C s=o
ht
(a)=f
k
=
C
cc S
E'
t=O
(+t
Ck
+ 7,' =
E'V"V,
j=O
x,=d
I +
hX'DD*V,-,
x,=d
II. Homogenization the system system of of linear linear elasticity elasticity Homogenization of the
174
where where
(3.
^.
and hz are assumed t o be zeros if the length of the index larger than k 1. Due to the conditions (3.18) we have
Q
is negative or is
+ 7: .
(3.29)
+
2k+l
u ( k ) ( ? , d )=
+C
k
C C
hZ,D"V,
Taking into account the smoothness of V, in the layer { z : 0 5 z, 1. d } we conclude from (3.23), (3.28), (3.29) that u ( ~ ) ( ? , = ~ 2 ( ? ) + & k f l 2 9 2 ( ? , € ) and the second inequality (3.27) is satisfied. In the same way we prove that u ( ~ ) ( ? , O ) = @ I ( ? + ~ ' + + ' d , ( ? , . z )and that the first inequality (3.27) is satisfied. Let us now calculate o,(u(~)) on d q r o u rd). Setting E = .z-'z, due to (3.3) we have
(3.30)
$3. Asymptotic expansions for solutions of boundary value problems 175 Here we used the equality B,(() = 0 for
t E & - l (bR‘\(ro 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
+
A*lW ( t ) N a , . . . a k + z ] D a v ( k ) ( x )
C
+ &k+l
AWWN a 3 . . . a k + 3 ( E ) D a V ( k ) ( x )
7
(a)=k+3
t = &-12
’
(3.31)
Since N,“, N:, 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 kA2
(3.32)
+
Ek+l
c
(a)=k+3
A a1w ( ~ ) N o s . . . o * + 3 ( ~ ) D ,v(k) (2) .
Let us transform the expression (3.32) setting ( a ) 2 k 3. We have
+
i:= 0 for
C,(u(”) =
c c s=o
k+2
(+I
j=O S
c k
k+2
k0
(3.33)
&
hO,DD*
(a)=s-j
j=O
&j<
+ 7,“
ft:
=
= h: for ( a ) 5
Ic + 2,
II. Homogenization of the system of linear elasticity
176
s=k+3
j=o
s=k+3
j=o (.)=s-j
(a)=.-j
Therefore it follows from (3.17), (3.18), (3.33) t h a t
(3.34) where
(3.36) Estimate (3.26) holds due to the inequalities (3.22), Lemma 3.2 and the smoothness of V,. 0 satisfies (3.25). Theorem 3.1 is proved. It is obvious t h a t Remark 3.3. It follows from the estimate (3.21) and the equalities (3.19), (3.20) that u'(x) =
=
u(k)(x)
+&k+lQ(X,€)
=
c c N$) c
k+l
k
€'
I=O
(+I
&
j=o
. € T V j ( X )
+&k+'q(x,€)
,
(3.37)
53. Asymptotic expansions for soJutions of boundary value problems 177 where JIq(z,E)JIHl(fie) u"(2) =
5M
with a constant M independent of
c c c k
k
&*
t=O
j=o (a)=t-j
N,(qD"1/, &
+E"+'q1(s,E)
E.
Therefore
,
where llq1(x,E)llL2(fi.)I Ml, and constant
Ml does not depend on E , N , = 0 for negative (a).In particular we obtain for k = 0 llu'(5) - VollL2(fiC)
I CE
7
where C is a constant independent of
E.
It also follows from (3.37) for k = 0 t h a t
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 of order
E
for the remainder
term, whereas without the boundary layers we can only get an estimate of order
E
~
as/ in ~ Theorem 1.2 with
= a', fo = f" (see estimate (1.15)).
II. Homogenization of the system of linear elasticity
178
$4. Asymptotic Expansions for Solutions of the Dirichlet Problem for the Elasticity System in a Perforated Domain Here we consider asymptotic expansions in
E
for solutions o f the Dirichlet
problem for the elasticity system in a perforated domain
R' with
a periodic
structure. The displacement vector is assumed t o vanish on t h e surface o f t h e cavities
S,.
Similar asymptotic expansions for solutions o f t h e Dirichlet problem for t h e equation AuE = f in a perforated domain
RE were
obtained in [52], where
the estimates for the remainder term were proved in the case
f E C,"(R).
In order t o justify t h e asymptotic expansion, when f ( x ) is sufkiently smooth and may be non-vanishing in a neighbourhood of
dR, we construct boundary
layers which exponentially decay in x with the increase o f the distance from z to
aa.
4.1. Setting of the Problem. Auxiliary Results Consider a perforated domain domain o f
R" with
by the vectors z E class C' and
RE = R n E W , where
w is an unbounded
a 1-periodic structure, i.e. w is invariant under the shifts
Z".It is
assumed here t h a t Q\w contains a surface of
R is a smooth bounded domain.
Note that in this section we do not impose any restrictions on the smoothness o f w. In R' we shall study the following Dirichlet problem
U'
= 0 on
dR',
f E C"(R) ,
where Ah'(<)are ( n x n)-matrices o f class E ( n l , n2) and their elements a$(<) are 1-periodic in (. The aim o f this section is t o justify the asymptotic expansion
=c M
uE(x)
1=0
<=
&-I2
,
&It2
c
(+I
N c r ( & , ( ) D a f ( ,x )
$4. Asymptotic expansions for solutions of the Dirichlet problem
179
for solutions of problem (4.1). Here a,ID" are t h e same as in $3, N o ( [ ) are matrices of the form N , = NL
+ NZ, where the elements of N t are functions 5
defined in w and 1-periodic in (, the elementscof N ~ ( E -) , decay exponentially
Re with on f. in
the increase of t h e distance from
E
z t o dR, N:, N:
do not depend
Let us now prove some auxiliary propositions t o be used below for the justification o f the asymptotic expansion (4.2). Lemma 4.1. For any vector valued function w II4IL2(n.)
E HA(R') the following inequalities are valid
I ME lle(4llLqn*)
IIV'IUIIL2(n.)
5
(4.3)
I
Ile(w)IlLqn.)
where M is a constant independent of
(4.4)
I
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 H;(R) such that 6 = 20 in R', 6= 0 in R\Rc. Let us prove (4.3).
Obviously we can assume t h a t
R"\Re. Denote by T' the set o f all + Q ) n R # 0 and consider the function W ( [ )=
20
is defined in
R"
and vanishes in
z E Z" such t h a t
E(Z
~ ( E J ) .
Taking into
account the properties of aw and the fact t h a t 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
J (Z+Q)nw
IW(t)124 I c
J
I V W 2 4.
(4.5)
(Z+Q)nw
Summing up these inequalities with respect t o z E
T"
variables x = E [ we obtain (4.3). Lemma 4.1 is proved. Lemma 4.2. Let U ( z ) E H1(Rc)be a weak solsution of the problem
0'
I
and passing t o the 0
II. Homogenization of the system of linear elasticity
180
r
7
(4.7) 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
J R C
b2J r
5
I V W I ~ ~~z~
n*
+J
~ f ~ l ~ d a : (fi,f')dz
1fol2da:+ & z
Q
J
n*
n.
+J
J
7
I V Q I ,~ ~ ~
R'
(f',fi)dX
+ €2 J
lviplzdz]
,
nz
K3 do not depend on E. Therefore estimates (4.7) are valid, since w = U - i p . Lemma 4.2 is proved.
where the constants Kz, 0
The next theorem shows in particular that the solutions of problem (4.6) have the form of a boundary layer in the vicinity of
dR, provided t h a t @ = 0
54. Asymptotic expansions for solutions of the Dirichlet problem on
181
( d P ) n R and f'(z), i = 0, ..., n , rapidly decay in Re with the growth of
the distance from x t o 30.
C'(n) such t h a t T = 0 in a neighbourhood of 8 0 , r 2 0 in R, (Vr1 5 M = const. It is assumed that E is so small t h a t there is a subdomain 0' c R whose closure consists of the cubes EQ+EZ with z belonging t o a set T, c Z", and dR' lies in the neighbourhood of 80 where T = 0. Consider a scalar function r(x) E
n'
Theorem 4.3. Let U(x) be a weak solution of problem (4.6) with @(x)= 0 on
(aW)\dO
(i.e. Q, E H'(R",S,). Then E-2
J
Iu12exp(fi)dx+ &
nennl
where
J
67 IVU12exp(-)dx &
+
n*nW
K , 6 are positive constants independent of E .
Proof. Set
v = (epT- 1)U in the integral identity (3.15), Ch. I, for U(x),
where p = const
Since
T
> 0 is a
parameter t o be chosen later. We have
= 0 outside R',
we find by virtue o f (3.13), Ch. I, t h a t
II. Homogenization of the system of linear elasticity
182
(4.10) Due t o the Korn inequality for vector valued functions w E H1(Qnw, a w n
Q ) (i.e.
20
= 0 on (dw) n Q ) we have
J
( I ~ I+ ' I
I
v C ~ I ~ ) M ~ ~
Qnw
J
lec(w)l2dt .
(4.11)
Qnw
This inequality follows from Theorem 2.7, Ch. I, if we extend
w as zero t o
Q\w and note that Q\w contains a surface o f class C'. Passing t o the variables
t = E - ~ X in (4.11)
we obtain for any w, = E(Q
w)
+ E Z C R' n R', z E T', the following inequalities
J
IUl2eXP(P~)dz5
n
WZ
(4.12)
54. Asymptotic expansions for solutions of the Dirichlet problem
U
Therefore since p = -we find from (4.13) that 2 d G E
183
II. Homogenization of the system of linear elasticity
184
+J
Ie(U)l2dz .
(4.15)
n* Choosing u sufficiently small and independent o f E and taking into account U
-we
obtain from (4.15), (4.14), (4.13) the estimate (4.9). 2 Theorem 4.3 is proved. 0
that p =
,m& .- - -
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 a w , w is 1-periodic in [ ,
(4.16)
where A h k ( [ )are matrices o f class E(rcl, Q ) , w = (w1, ...,wn)*, 3j E L2(Qn u), 3 3 are 1-periodic in l , j = 0,1, ..., n. A weak solution of problem (4.16) is defined as a vector valued function
w E$ ( w ) = k i ( w ) n H1(Q nu, Q n 8u)which satisfies the integral identity (6.2), Ch. I, for any v €$ ( w ) . Theorem 4.4. There exists a weak solution of problem (4.16) which is unique and satisfies the estimate
c n
5
~~wI!H1(Qnw)
j=O
IIF’llLz(QW)
*
185
54. Asymptotic expansions for solutions of 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
c
c
00
.ii"(.) g
(+l
I=O
X
Na(c,t)DDolf(x) , [=-
(4.17)
E '
in the equations (4.1). Formal calculations similar t o those o f 53.2 yield
L"(.ii.(Z))
2
g c 6'
I=O
(,)=I
H L 1 ( t ) D 0 ( f (gX )f(x)
,
(4.18)
where
t A""(()
a
- Naz...a,(t)t A"1"2(t)N03...a.(() ,
atj
Let us seek
3
L2
.
N , ( t ) in the form N , = N,O(t) t NA([), where N , ( t ) are
matrices whose elements are l-periodic in ( functions belonging t o the elements of from
2
to
X
~
(w), and
N : ( - ) decay exponentially with the increase o f the distance
dR.
G
We introduce the notation
T,OEI, T,'=O,
11. Homogenization of the system of linear elasticity
186
where I is the unit matrix. Define the matrices NE(J) as weak solutions of the problems
N,"(()= O on bw
,
N,O(t) is l-periodic in
t .J
The matrices N t ( J ) are defined as weak solutions o f the problems (4.21)
J
NA(() = -NE(() on a(&-lfl").
Existence of N,, NA can be easily proved by induction with respect t o
I
on the basis o f Theorems 4.3, 4.4, Ch. I. Lemma 4.5.
2
The matrices N i ( -) satisfy the following inequalities &
llv,NillL~(n.) + ((N:llLZ(n.) I c, (4 2 j=o,1,
'5
7
(4.22)
01
where C, are constants independent o f
Proof. Relations (4.19),
E.
X
X
&
&
(4.20), (4.21) show that N , ( - ) = A':(-)
are solutions of the following boundary value problems:
&(No) = &-'I
in R'
,
No = 0 on aR"
,
(4.23)
Let us use induction with respect t o 1. For 1 = 0 it follows from (4.23) due to (4.7) with
+ N ~ ( E-)2& ,
= 0 that
§4. Asymptotic expansions for solutions of the Dirichlet problem
where C, is a constant independent o f
187
E.
Let 1 = 1. By virtue of (4.7), (4.24) we get
+ E-l IlLqn.) I M z ( E IIVz~ollL2(n.)+
IlV,Na, IPa1
IlLW)
I Ml (IlvzNollL2(n.,
IINoIlLZ(nc))
IINollL2(nc))
These inequalities and (4.26) imply that for lc IIL2(n') I 'al...akE-'
llvzNal...ak
I\Nal...aklIL2(n')
I
cC71...a,
5
k
.
1 we have
;
(4.27)
'
Suppose now that the inequalities (4.27) hold for t h a t they also hold for
,
kI 1 - 1.
Let us show
= 1.
It follows from (4.25), (4.7) t h a t
I c 1 [ IlVr~a2...a1 Ilv(n.) + E-l IINaz...allIL2(n.)
llvz~al...al11;2(n.)
+ E-l IPa1...a1
IILqn.)
+
IINa3...at11~2(ne)]
I cz [E IlV~a2...a1 IIL2(n*)+
+ II~a2...alllLZ(n.) + IINa3...aIIIL2(n')]. kI 1 - 1 we obtain
Therefore due t o (4.27) for
(4.27) for
k = 1.
The elements of the matrices N,"(() are 1-periodic in (. Therefore estimates (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.
0
Lemma 4.6. The elements o f matrices N:(a, subdomain
Ro such
that
2
-) are of boundary layer type, i.e. for any
E noC R the following inequalities are valid
(4.28) where C,, 7 are positive constants independent of
k f . Consider a domain
a'
such that
E.
0' c R , noc R'
and the distance
II. Homogenization of the system of linear elasticity
188 between of
E,
and
Ro and dR' is larger than
0'
consists of the cubes
The parameter
E
K
> 0,
where
h:
is a constant independent
+ z ) , z E T , for some subset T c Z".
E(&
is assumed so small t h a t
R'
with the above properties exists.
C'(n), r f 1 in Qo, r = 0 outside the --neighbourhood o f Qo, IVr1 5 C K - ' , C = const. Let us construct a scalar function r(x) such that r E K
2
Using the induction with respect t o s = 0,1,2, for a = (a1, ...,a,) satisfy the inequalities
X
... , let us prove t h a t N : ( -E)
where C,, 6 are positive constants independent of
E.
Let us first show t h a t (4.29) holds for the matrix N,' which is a solution of the problem 2
, NJ(-)&
L c ( N J )= O in
X
= -N,O(-) on a R c . E
X
N : ( - ) = 0 on dR'\dR, we can apply the estimate (4.9) of Theorem & 4.3 t o N i . We get Since
E-2
J
INA12exp(fi)dx &
n*nw
2 (I
J
+J
IVzNJ12exp(-)dx 87 E
R'
5
IV,N,'I2dx.
n* This inequality together with (4.22) implies (4.29) for
N t . Fix a positive
NA with a = (a1, ..., a l ) , 1 2 s - 1. Let us show that (4.29) holds for NA , with 1 = s, where NAl,.,ae
integer s and suppose that (4.29) is valid for all
is a solution of the problem
a
d
Lc(NAl,,,a,)= &-'Aa1' -N:2,,,aa
-(Ak"'N~Z...018) -
dXj
k
- & - ~ A ~ Ia3...aS ~ Z Nin* a" , X
N:l...,8($ = -NO
X
on do' 2
.
Taking into account the fact that N , ( - ) = 0 on &
Hlc\Xl and using the
estimate (4.9) of Theorem 4.3 applied t o NAl,,,,,,we obtain
189
$4.Asymptotic expansions for solutions of the Dirichlet problem
J
E-2
R"V
I K1
+
[l
+ J
67
l~;l...,,12
exp(-)d. E
n*nw
lvzN:l...a8 I2d. t E2E-2
E-4&2
J
67
lvz~:l...,a12 exp(-)dz E
J
67
n*nw
+
6T
lvzN:z...ae I2 e x p ( 7 ) d .
n.nw
IN:,...,fexP(p.
I
+ E-2
J
IN;z...a.12 exP(-)dz ST &
nwni
1
.
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 NL1...,#. The estimates (4.28) follow from (4.29), since
7
f 1 in
Ro. Lemma 4.6 is
proved.
0
Theorem 4.7. Let u'(z) be a weak solution of problem (4.1) with
c c 8
u:(.)
=
(+l
d
? I : ( = . )
l=O
where
N, = Nz
&1+2
+ NA, N,,
I
c N,(paf(4 c N,O(p"f(4 7
1=0
f E C"+'((S2).Set
1
(,)=I
(4.30)
NA are weak solutions of problems (4.20), (4.21)
respectively. Then
113.).
- uC(4IIH'(n.) 5 COEa+2Ilfllc.t~(n),
(4.31)
IIv:(z)
- Uc(z)IIH1(n*nn0)I C~E"+'Ilfllc+z(n) ,
(4.32)
Ro is a subdomain of R such that not depend on E ; C1 may depend on RO.
where
h f . Let us apply the operator
C, t o u:
noc R, the constants Co, C1 do - u'. Assuming
N , = 0 in (4.17)
for (a)2 s we obtain in the same way as (4.18) that
L,(uZ - u C )= Ed+1
2
AOlk
ol,...,astl=i
a
@k
,I
...O . + l
Naz-'O'tlD
f+
11. Homogenization of the system of linear elasticity
190
c n
+
a], ...,a,+2=1
7p...a,+2 f in 0' , A"'"' N"3...as+2
u: - u ' = 0 on dR'
I
Note that, because of (4.22), the matrices
x
d x N a ( - ) , - Na(-)are E
atj
E
Lz(Rc) norms of the elements of the
bounded by a constant independent of
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) it suffices to observe that
No = N:
the inequalities (4.28). Theorem 4.7 is proved.
+ N:
and
N i satisfy 0
191
55. Some generalizations for the case of perforated domains $5. Asymptotic Expansions for Solutions of t h e 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 :
A2u'(x) = f(z) in R'
auc
,
,
u ' = - = 0 on dR' dU
(5.1)
and obtain a complete asymptotic expansion for solutions of this problem.
R' is a perforated domain of type I with a periodic structure described in 54.1, f(x) is a sufficiently smooth function in R; u is the outward normal. Here
We seek the asymptotic expansion for the solution o f problem (5.1) in the form
u: =
c N,(&,[)DDLIf(x) ,
c
(+I
k 0
[ = &-lX
,
(5.2)
where DD", 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
Hi(R') the following inequality is satisfied
E - 2 )I'UIILZ(*.)
where
E2(v) = (-
+
d2V
dxidzj
E - l Ilvv(lL2(n*)I
Ml llEz(")IlL2(n.)
aZv '", -)dx;dxj
Proof. Obviously it is sufficient
>
(5.3)
MI is a constant independent of E .
C,"(R'). Set v = 0 in Rn\Rcand denote by T' t h e set o f all z E iz" such t h a t E ( Z + &) n R # 8. Consider the function W ( [ )= v(e[). Since W = 0 in W \ w , the Friedrichs inequality for each o f the sets w, = z Q yields t o prove (5.3) for v E
+
II. Homogenization of the system of linear elasticity
192
Summing these inequalities with respect t o z E
T' and passing t o the
variables x = E<, we obtain (5.3). Lemma 5.1 is proved. Let @ E
H2(Rc),f j E L2(Rc), j = O,1,2, ...,n.
We say t h a t U(x) is a weak solution o f the problem
A'U(x) = f o
af' +axi
in Re ,
au-- a@ U=@, au
on
du
(5.4)
anE,
if W = U - @ belongs to H;(R") and satisfies the integral identity
+J
(fov
- f' 5 ) d x dXi
n' for any
v E H,2(Rc).
Denote by @(w) the completion with respect t o t h e norm IIvII@(snw)of the functions and
v(t) such that v E
C2(G), w = 0 in a neighbourhood o f dw
v(E) is 1-periodic in t. Here w is an unbounded domain with a 1-periodic
structure, the same as in s4.1. We say t h a t w is a weak solution o f the problem
Aiw = F o ( t )
aw av
w=-=O
dFj +in atj on d w ,
w(<) is 1-periodic in
t,
w
,
I
J
where Fj E L2(wr l Q),F j ( < ) are 1-periodic in and satisfies the integral identity
t ,j
= 0, ...,n, if w E @ ( w )
$5. Some generalizations for the case of perforated domains
for any
193
v E &(u).
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 ( 2 ) of problem (5.4) sdtisfies the following inequalities
II. Homogenization of the system of linear elasticity
194
J
It follows t h a t
Since W = U - 0, this inequality implies (5.8). Due t o
(5.3) we have IIWIIL2(nq 5
M 1 E Z IIEz(W)IlL2(n.) >
IIVWIIL2(n.) I MlE IIEz(W)IlL2(n.) . From these inequalities and (5.11) we obtain (5.9), (5.10), since Lemma 5.2 is proved. Let r(z) be a function of class Cz((sl)such t h a t
7
W = U - CP. 0
= 0 in a neighbourhood
a R , 7 2 0 in R. Consider a subdomain R' defined just before Theorem 4.3 and assume t h a t 7 = 0 outside 0'.
of
Theorem 5.3.
acp
U ( z )be a weak solution of problem (5.4),CP = - = 0 on aR'\aR (i.e. av cp E HZ(Rc, aRc\an)).Then Let
(5.12)
195
55. Some generalizations for the case of perforated domains where K O
> 0, 6 > 0
can depend on
are constants independent of
E.
(Note that
KO
and
S
52' and Il~(~)llczcfi,.)
Proof. For any v(x) E H,2(RC)the function U ( x )satisfies the integral identity
Set v = (ep'
-p
J
- 1)U, where p > 0 is a
d2U
n*
a2r
U -eprdx-p2
axiaxj
parameter t o be chosen later. W e have
J
n c
a2u
a7 a7
-U -
axjazj axi ax, e p T d x t
(5.13) Since
T
= 0 outside of a', we obtain by virtue of the Holder inequality
(5.13) that
and
II. Homogenization of the system of linear elasticity
196
In the same way as in the proof of Theorem 4.3 to obtain (4.14), we find that
J
IU12epTdx5
J
IVU12ep7dx,
~ 1 . 2
nCnW
(5.15)
Wnnl
where p = u/I
E,
u E (0,l) is a constant
Since U ( x ) can be approximated in the norm of H2(R")by functions vanishing in a neighbourhood of ane\Xl it follows that inequality similar to (5.15) holds for the first derivatives of U ( x ) , i.e.
J
J
IVU12eFTdx5 K2e2
nwnl
1 ~ 2 ( ~ ) 1 2 e o. ~ d x
(5.16)
nwnl
Estimates (5.15), (5.16) yield
J
IU12ep7dx5 K3c4
ncnnl
J
where K2, K 3 are constants independent of From (5.14), (5.16), (5.17) we obtain
J
IE2(U)12epTdxI c e p e
n*
(5.17)
,
n*nW E.
+
J
IEz(U)12epTdx
ncnnl
+ c,(p t p2)c2 J
(
+ C,e2 J
n*nnl
n*nw
+
IE~(U)I~~'T~~
(J
lfo12ep7dx)1'2
nennl
IE2(U)12ep7dx
$5. Some generalizations for the case of perforated domains
197 (5.18)
n* where p = u / K e . If we choose u sufficiently small but independent of
c , we
get from (5.18) the following inequality
J
IE2(U)12eMTdx I ~
nc
+ Mzc2
1
ljol2epTdx
€
4
n*nnl
+
f ' f ' e p T d x M3
WnW
where
J
~
+
1
IE2(U)12dx ,
(5.19)
na
Ml, M2, M3 are constants independent of c .
Estimate (5.12) follows from (5.19), (5.16), (5.17). Theorem 5.3 is proved. 0
5.2. Justification of the Asymptotic Expansion for Solutions of the Dirichlet Problem for the Biharmonic Equation
f E C8+4((s2) in (5.1). Let us seek an asymptotic expansion for the solution o f (5.1) in the form (5.2) where N,(E, [) = N:([) + N:((E, t), X N,(t) are l-periodic in (, NA(e, -) are functions o f boundary layer type in E R", which decay exponentially with the increase of the distance from x to dR. Suppose t h a t
It is easy t o verify t h a t
Therefore
+
2€-3
a
- A, N,D"
at;
af + 4€-2 a IN, D" a2f -
axi
atj ati
+
axiaxj
198
11. Homogenization of the system of linear elasticity
+ E - ~ A ~ N , Vf~+A~, - l 2P atj
Aaf + N,V"A: f .
ax
(5.20)
From (5.2), (5.20) we obtain
(5.21)
where S, is the Kronecker symbol. Let us define the functions N , ( E , ~as ) weak solutions of the following boundary value problems
NO
AiNo = 1 in E - ~ Q ' , No = -= 0 on
au
a(~-'n')
(5.22)
199
$5. Some generalizations for the case of perforated domains
a
Nal...a, = - Nol ...a, = 0 on av
a(&-'Q").
On the basis of Theorem 1.3, Ch. I, we can easily prove by induction t h a t
N a ( [ )exist. Let us show t h a t periodic in
t
N , ( E , ~= ) N:
+ NA, where
N : ( [ ) are functions 1-
and belonging t o & ( u ) ; N:(E, f ) are of boundary layer type in
0". Set
j=O,l,
m24,
Define the functions N : ( [ ) as solutions o f the following boundary value problems w
,
8N,o N , = = z -- 0 on dw
,
A i N : ( t ) = T,O([) in
N , ( [ ) is 1-periodic in
Obviously Theorem 1.3, Ch. I, guarantees the existence of In the domain Dirichlet problems
&-*nC define
the functions
.
I
(5.23)
N,([).
N: as weak solutions o f the
II. Homogenization of the system
200
Obviously
of linear elasticity
N , = N," -k NA.
where the constants
M, do
not depend on
E;
j = 0,1.
This lemma is proved by induction in the same way as Lemma 4.5. Lemma 5.5.
X
The functions N ~ ( E -) , are of boundary layer type, i.e. for any subdomain such that !=lo c IIN:(E,
where
R
&
no
the following inequalities hold X
c,
,)lla(nonn=) I exP(-7&-')
(5.26)
7
C,, 7 are positive constants independent o f E, (C, and 7 may depend
no). Proof. The estimate (5.26) on
is obtained in the same way as (4.28) in Lemma
4.5. Let us indicate the main steps of the proof. Consider a subdomain some z E
Z", and
R' c R
which consists of the cubes
E(Z
+ Q ) for
let ~ ( x E) C2(O)possess the same properties as in the
proof o f Lemma 4.5. 5
The function N;l.,,,m(~,-) &
. IS
a weak solution o f the problem
55. Some generalizations for the case of perforated domains
Due t o
J
E-4
aN:
-= 0 on aW\aQ,we can apply av (5.12) for U = N:l,,.am we get
Since N o =
n*nn'
J
67
~~:~...,,~~exp(-)dx E +&-'
where ICz is a constant independent of
nennl
Theorem 5.3 t o
201
U = N:.
67 I V N ~ , . . . ~ , I ~E ~ X P+( - ) ~ ~
E.
From these inequalities and (5.25) we obtain by induction with respect t o
... that
m = 0,1,2, E-4
J
n*nn!
where the constant 7
+
67
~ ~ : , . . . , , , , ~ ~ e x p ( F ) dEz -
K,,,,,,,
~
J
nCnW
67 I V ~ N : , . . . , , I ~ ~ X P+( ~ ) ~ ~
does not depend on
= 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 of solutions of problem (5.1)).
Let
~ ' ( 5be )
).(:it
a weak solution o f problem (5.1) and let
=
2 I=O
c Ncr(€,
"Vf(a:)
(CI)=I
&
,
f E CSf4((s2),
0
202
11. Homogenization of the system of linear elasticity
where N a ( e , [ ) = N:
+ NA, N:,
Ni
are weak solutions of problems (5.23),
(5.24). Then
IIu'(x) - U:(X)IIH~(W) L C 1 ~ s +Ilfllc*+4(n) 3 7
(5.27)
llu'(x) - C(X)IIH~(~OW*) I C 2 ~ s +Ilfllcw(n) 3 ,
(5.28)
C1, C2 are constant independent of E , Ro is a subdomain of R such that fi0 c R, the constant C2 may depend on no. where
Proof.
It is easy t o see that by virtue of (5.21), (5.22) u:
-u ' is
a weak
solution of the problem
C
A'(uZ - u c )= &'+l -
&J+1
c
a1,...,a,+1=1
&
Y
1
d
~ E ~ A ~ N ~ ~ .Z)a1 . . ...~a.+l, + ~ ax,,
011 ,...,as+ 1=1
rn=s+1
a
4~~-[(A2Nal...o.+l)Z)a'..'a'+l
...,am=l
a4,
f+
f] -
$5. Some generalizations for the case of perforated domains 5+4
+ E5+1&2 C
m=a+l
n
C a s ,...,
Since N , = A ':
l&(.:
IINa5...om
o,=l
I
IILz(~*) IlfIIcs+4(ii) .
+ N:, we obtain by virtue o f (5.25) - UC)llL.l(W) I M E 5 + ' ( E 2
5 MI
203
IlfIlc.+,(n, ,
+ E3)
that
IlfIlc.+4(n)
I
M l , =~ const .
This estimate and (5.9), (5.10) imply (5.27). Inequalities (5.28) follow from (5.27), (5.26). Theorem 5.6 is proved.
0
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 st ruct u res. Suppose t h a t a subdomain
R' c R
is such that
a' c R and a' =
u d,
s=l
&,
B: are bounded domains of R" such that B: n Bj" = 0 for i # j . Suppose also t h a t ,:'I s = 1, ...,de, are closed sets r: c @ and for each 2, E C1(&) such t h a t v = 0 in a neighbourhood o f r:, the Friedrichs inequality where
holds with a constant C* independent o f Let
T(Z)
and s.
be a function in C 2 ( a )such that
5 M',
))T))~z(Q
E
where
M'
T
= 0 in Cn\s2',
is a constant independent of
E,
s.
Theorem 5.7. Let
U ( Z )be a weak solution of the boundary value problem
au d@ U = @ , - -- - on 30' , dv
au
T
2 0 in R,
II. Homogenization of the system of linear elasticity
204
dip
HZ(R'), ip = - = 0 on R' n dR', f j 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'. aip Suppose t h a t f j = 0, j = 0, ..., n in R'nR', ip = - = 0 on R'ndR' and av the domain Ro is such that noc R', p(dRO,80') 2 K > 0 with x independent of E . Then the solution U ( z )satisfies the inquality
where ip E
II UllHz(no"n') I
c
L where
[
II@IIH2(W)
+2
l 6
l l . P I l L ~ ( n ~ \ * ~ )e-G
j=O
,
(5.30)
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 ~ ( xsuch
~ ( x= ) 1 in Ro, ~ ( 2 = ) 0 outside the ~/2-neighbourhood o f Ro, the C2(n) norm of 7(2) is bounded by a constant independent o f E.
that
Consider now the system of elasticity. Suppose that the sets
rz,s = 1, ...,dc, are such that for each TJ E C'(&),
= 0 in a neighbourhood of E - l )l"llLz(E:)
l?:
the following inequality is valid
+ IIv"IILz(E:) 5 c; IIe(")llL2(B:)
where C; is a constant independent o f
a',
5 M;,
where
E.
let
T(Z)
7
(5.31)
E C'(O), ~ ( x= ) 0 outside
M; is a constant independent of E .
Theorem 5.8. let
U ( x ) be a weak solution o f the boundary value problem for the elasticity
system
U = ip on dR'
,
L2(R'), j = 0,..., n , ip E H'(R'), ip = 0 on R ' n 80' and matrices A h k ( x , e )belong t o the class E(nl,n2) with tcl,lc2 > 0 independent
where
fj
E
$5. Some generalizations for the case of perforated domains of
E.
Then for U ( z ) the estimate (4.9) holds with constants I(
depending only on C; in (5.31), Suppose t h a t the domain
fj
= 0, j
E.
> 0, S > 0
KZ.
n R', = 0 on 0' n doZ'and p(8Ro,dR') 2 ~d> 0, where IE is a constant
= 0, ...,n , in R'
Ro c R' is such t h a t
independent of
M;, n l ,
205
Then the solution U ( z ) satisfies the inequality
where C is a constant depending only on C;,
M:,
~ 1 K ,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 ~ ( z such ) that of
7
Ro, 1(7(lCl(n)is bounded
= 1 on
Ro, 7 = 0 outside the ~/2-neighbourhood
uniformly in
E.
11. Homogenization of the system of linear elasticity
206
$6. Homogenization of the System of Elasticity with Almost-Periodic CoefFicients In this section we consider homogenization o f solutions o f the Dirichlet problem for the system of elasticity with rapidly oscillating almost-periodic coefficients.
6.1. Spaces of Almost-Periodic Functions Denote by Trig IR" the space o f real valued trigonometric polynomials. Thus Trig R" consists o f all functions which can be represented in the form of finite sums
4Y)= y,E E
cc C€exP
{i(Y,E))
R", (y,E)
= yit;,
7
ct = c-( = const
.
(6.1)
The completion of TrigR" in the norm sup Iu(y)I is called the Bohr R"
space of almost-periodic functions and is denoted by A P ( R " ) (see [50],[51]).
The space of all finite sums having the form (6.1) and such t h a t
Q
= 0 is
0
denoted by Trig R". Let II,E
L;oc(EP).We say that M {$} is the mean value of II,,if
II,(&-'z) + M { $ } weakly in L 2 ( G ) as
E --+
0
for any bounded domain G c R". It is well known that for any function g E L:,,(R"), which is T-periodic in y, the mean value exists and is equal t o
J
1 M{g}= 5
g(y)dy
I
[OYTI"
[O,T]"={y
:
O l y j S T , j = 1 , ...,n } .
Thus each function belonging t o TrigR" possesses a finite mean value, and therefore we can introduce in TrigR" the scalar product defined by the for rnu la
$6. Homogenization of the system of elasticity
207
The completion of T r i g R with respect t o the norm corresponding t o t h e scalar product (6.2) is denoted by B 2 ( R " )and is called the Besicovitch space of almost-periodic functions. We keep the symbol
M { $ g } for the scalar product of the elements II,and
g in B2(R").
As before we say that a matrix (or vector) valued function belongs t o one
, P ( R " ) , if i t s components belong t o the of the spaces Trig R",B 2 ( R " ) A corresponding space. In this case the mean value is a matrix (or vector) whose components are the mean values of 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.
As usual e ( u ) denotes the symmetric matrix with elements e U ( u ) =
auj 1 (k+ -), 2 8%
dYl
where u is a vector valued function u(y) =
(u1
,...,u,).
Lemma 6.1. Suppose that f , g E Trig R", and u =
(u1,
...,u,)
E Trig IR". Then
F I E~ Trig R" such that F/h = Fhl, 1, h = 1, ...,n , there is a vector valued function w E Trig R" such t h a t Moreover for any functions
Proof. Note that
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 = (u1, due t o (6.6) we find that
...,u , ) , uj = E
4e’(yx).Then
This implies (6.4). Let us show now the existence of the solution of equations (6.5). Suppose that
$6. Homogenization of the system of elasticity
Obviously for each
# 0 the coefficients w:
209
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 ( # 0, c y = 0 ,
I , h = l,...,n. Then multiplying the equations (6.7) by W: and summing up with respect t o I from 1t o n we obtain h 2 1-1 - 5 IwcthI - 5 W(W(
=0 .
Therefore w: = 0. Let us replace
by -( in (6.7) and write the complex conjugate equation.
One clearly has w: = wkt, since ckh = C(.!'
Lemma 6.1 is proved.
0
Consider the Hilbert space of ( n x n)-matrices whose elements belong t o
Bz(LT)and denote by W
the closure in this space of the set
S = { e ( u ) : u = ( u l ,...,u,) E Trig R"}. Elements o f W will be denoted by e, Z, etc. The norm o f an element e E W is given by
M {e,je1j)'/2 = M { ( e ,e))'" . It should be noted that not every element e E W can be represented as e = e ( u ) with u E B2(1Rn).Nevertheless for every e E W there is a sequence o f vector valued functions { u 6 } with components in TrigR" and such t h a t
M {Ie - e(u6)12}+ o as 6
--.)
0.
6.2. System of Elasticity with Almost-Periodic Coeflcients. Almost- Solut ions Consider the system o f linear elasticity
11. Homogenization of the system of linear elasticity
210
a
621
aYh
auk
- ( A h k ( y )-)
dfj
= -, &j
E ( I E ~ , II E E I~ , K) .,~ = const. > 0, whose elements belong t o A P ( R " ) ,u = (ulr...,un),f j = ( fij, ..., fnj) are column vectors, f j r = f,j E A P ( R " ) . 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 ( R " )o f system (6.8) has yet been
where A h k ( y )are matrices of class
found. However we can construct the so-called almost-solutions u6 o f (6.8) with components in Trig R".This fact was established in [149]. Following [149] 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:
d
- ( u y ( y ) e j k ( u ) )= ayh
afij ~
&j
, I = 1, ..., n .
In the rest o f this paragraph we shall denote by v h t h e column vnh)*
o f the matrix r) with elements
vih.
(vlh,
...,
Then system (6.9) becomes
(6.10) where e k ( u) =
...,e n k ( U ) ) * .
(elk(u),
If the coefficients u k k ( y ) and the functions f , 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
{ ( M e( u) ,e ( 4 ) - (f,+,>} for any
w E +j(R"),where f is a matrix with elements
( M v)ih
hk aij v j k
Let the coefkients u:/
(6.11)
=0 flh
and
. be almost-periodic functions of class A P ( R " ) .
Then in analogy with (6.10), (6.11) we consider the system (6.12)
§6. Homogenization of the system of elasticity
21 1
and define a weak solution of (6.12) as the element E E W , 2 = { E i j } , which satisfies the integral identity
M ( M 2 , e ) - ( f , e ) }= 0
(6.13)
for any e E W .
It follows from Lemma 3.1, Ch. I, that the bilinear form M {(ME, e)} is continuous on W x W , i.e. (6.14) fot any 2,e
E W , since for a ( y ) E A P ( R " ) , f E B2(R")we have af E
I ( ~ ~ I I B ~ ( R ~5)
B ~ ( R and ")
SUP
R"
la1 I l f l l ~ 2 ( ~ n ) .
Moreover the condition (3.8), Ch. I, yields the inequality
for any e E W . By virtue of (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 Us(y) = (U,",..., U,") E Trig R" which approximate solutions of the system (6.9) in the sense of 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 of system (6.12). Then there exist sequences of vector valued functions U s E Trig R" and matrices gs E A P ( R " ) with columns g: = ( 9 3 , ...lg$), gfj = g:, j , l = 1, . . . l n , such that (6.16)
lim M (12 - e(U')12} + o
6-0
as 6 -+ 0, 6
> 0, and the integral identity
(6.17)
II. Homogenization of the system of linear elasticity
212
holds for any +(y) = (&, ...,$,,) E
Cr(Rn).
Proof. By
the definition of t h e space W we see that there is a sequence U 6 E TrigR" which satisfies the condition (6.17). Therefore due to (6.13), (6.14) we have
for any e E
W , where y(6) t 0 as 6 + 0.
Set = M e ( U 6 )- f ,
Since the elements a;,, of matrices Q6 belong to A P ( R " ) ,we can represent @ in the form Q6
where
= F6
+ G6 ,
(6.20)
a6,F 6 , G6 are symmetric matrices with
elements
ath,Fph, Gfih, F6 E
Trig Rn, G6 E A P ( R " ) , and lim M { lG6I2}= 0 6-0
.
(6.21)
Since
it follows from (6.19), (6.21) that
for any e E W , where n ( 6 ) --t 0 as 6 + 0.
According t o Lemma 6.1 there is a vector valued function w6 E Trig R" such that
a
aFp,
8Yh
ayh
-e,h(w6) = -
,
1 = 1,...,n
.
(6.23)
213
$6. Homogenization of the system of elasticity
Multiplying each o f these equations by wf and summing with respect to
I
from 1to n, we find by virtue of (6.3) and Lemma 6.1 that
(6.24) Therefore
M {le(w6)I2) I M {lF612} .
(6.25)
It follows from (6.20), (6.17), (6.21) that M { lF612} are bounded by a constant independent of 6. Therefore due t o (6.24), (6.25) we obtain
M {le(ws)12} -+ O as 6 -+ o .
(6.26)
Obviously by virtue of (6.20) we have
where g6 = e(w6) butions.
+ G6;and the equations (6.27)
hold in the sense of distri-
The convergence (6.16) is due to (6.26), (6.21), and the integral identity (6.18) follows from (6.27) and the conditions (3.2), Ch. I, for ubk. Lemma 6.2 is proved. 0 The vector valued functions U 6 are called almost-solutions of system (6.9) with almost-periodic coefficients. Let us now establish some other properties of the almost-solutions U s , which are essential for the study of G-convergence of elasticity operators with almost-periodic coefficients. Lemma 6.3. Suppose that f j , Ahk E A P ( R " ) ,2: is a weak solution of system (6.9), 2: E W , and U s (6 -+ 0) is a sequence of almost-solutions of system (6.9). Then for any sequence E -+ 0 there exists a subsequence € 6 -+ 0 as 6 + 0, such that E6(u6(;)
+ cs) -+ o
weakly in ~ ' ( 0, )
(6.28)
II. Homogenization of the system of linear elasticity
214 where
cg
is a constant vector,
rsP(E)+ M { A P k & - f p } weakly in L 2 ( R ) ,
p = 1, ...,n , (6.29)
a
X
-r 6 h ( E g ) +
axh
o
in the norm of H-'(R)
,
(6.30)
as 6 -+ 0, where
(6.31)
0 c R" is a bounded Lipschitz domain.
Proof. Taking
into account the inequality (6.4) o f Lemma 6.1, the fact t h a t
U s E Trig R",and the convergence (6.17) we obtain (6.32) where
I<
is a constant independent of 6.
Denote by G6'(x) the matrices whose elements are
G~:(x)
a
x
-U f ( - ) E TrigR" .
ax,
E
Note that the matrices G6icare not necessarily symmetrical. By the definition of mean value we have
lim
E'O
J
1 ~ 6 i ~ ( z ) 1 2 d= z
n
= lim E-10
J(
au6(C)
dU6( 2)
n
dYl
'
dYl
)dx =
(6.33) Similarly
lg6(z)12dx= (mesR)M{lg612},
lim E'O
n
&
where g6 are the matrices from Lemma 6.2.
It is obvious that
(6.34)
215
56. Homogenization of the system of elasticity
M e(U6)(f) + M { M ,(Us)} weakly in L2(R) as
E
-i
0. (6.35)
Moreover G6lC(x)-i 0 weakly in L2(R) as
aus x 0 since-(-)ETrigRnande'(~l€) 8Yl
E
E -+
,
0
(6.36)
+OweaklyinL2(R)as~-+OforJ#0.
Let V = {ql,$, ...} be a countable dense set in the Hilbert space of all matrices with elements in L*(R). For each 6 by virtue of (6.33)-(6.36)
for m
I 6-l,
we can find
€6
such that
m = 1 , 2 , ...; qm E V .
It follows from (6.32), (6.39) that the norms IIG6vC611Lz(n,are bounded by a constant independent of 6, and inequalities (6.40) imply that for any qm E V we have
J
(~69~6,qm)dx -+
o
as 6 -i o
.
n
Therefore G6lc6(x) -i 0 weakly in L2(R) as 6 -i 0 Set
where the constants
Cg
are chosen such that
(6.42)
II. Homogenization of the system of linear elasticity
216
J
X
V6(-)dX
n
E6
(6.43)
=0.
Then due to the Poincari inequality we have
where c is a constant independent of 6. Since the right-hand side of (6.44) is bounded in 6, it follows from (6.42), (6.44) that E6'V6'(?) + V weakly €6,
in H'(R) and strongly in L2(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'(R) c L2(R). By virtue of (6.43), (6.42) lished.
V
= 0. Thus the convergence (6.28)
is estab-
Since the elements u p of matrices Ahkare bounded, it follows from (6.17), (6.37) that the norms 1IM e ( L i 6 ) ( E ) / I L Z ( are n ) bounded by a constant independent of 6, and
lim M { M e ( V 6 ) }= M { M 2 ) .
6-0
Therefore we conclude from (6.41) that
M e ( U 6 )E6( z )
M { M 2 ) weakly in L2(R) as 6 -+ 0
To complete the proof of (6.29) it is sufficient to observe that
Therefore
X
I'6p(-)
are
€6
given by (6.31). Let us prove (6.30). For any $(x) = (&, (6.18) we obtain
.
..., &)
E C,"(R) due t o (6.31),
217
$6. Homogenization of the system of elasticity
This inequality together with (6.38), (6.16) implies (6.30). Lemma 6.3 is 0 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
a
- (Ahk(-)
&(ti')
axh
E
-)aau= xk
= f ( x ) in R , u E HA(R) , (6.45)
f E H-'(R), matrices Ahk(y)belong t o the class E(n1, Q ) , n l , n2 = const > 0, and their elements ahk(y) are almost-periodic functions of class
where
AP(R").
If matrices A h k ( 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
(a?(y) -k a:j"ejk(N:))dy 7
=
(6.46)
6 where
N," = (N:8, ..., dN!8
1 dNZs
- (2
ayj
+ -), dYk
is the s-th column of the matrix N q , ejk(N:) =
and the columns
N,' are 1-periodic solutions o f the system
d
a
ayh
ayj
- ( a F ( y ) e j k ( N i ) )= - -a!," Setting A:q = (a;:, ..., a::),
&I
=
,
(A;:, ...,i::),
(6.47) in vector form
=
J 6
1 = 1, ..., n
( A F ( y ) t Ahkek(N,'))dy,
.
(6.47)
we can rewrite (6.46),
II. Homogenization of the system of linear elasticity
218
belong t o AP(R"). It was shown above that for fixed q, s
Now let a(:y)
we can find weak solutions e"' E W (27' is a matrix with elements
a):
o f the
system (6.48) which is similar t o (6.12) with fj
= -(a{:, ...,a;:) = -A!'
.
(6.49)
Set
FI;:
= M {a:,"
and denote by
ahqthe
+ a:j"Z;;} , A!q
= M {Atq
+ AhkZY} ,
(6.50)
matrices with t h e elements ;I:,".
Theorem 6.4. Suppose that APQ(y) are matrices of class E(nl,riz),
6 1 ,= ~ const ~
> 0,
and
their elements aQ'(y) are almost-periodic functions belonging t o AP(R"). Then the sequence of operators
(
CC,(u) a A h"(') axh
&
au -)dxk
(& + 0)
is strongly G-convergent t o the elasticity operator
L
whose coefficients are
given by (6.50).
Proof. Let
us show t h a t there is a sequence 6 -+ 0 and matrices X
l , . . . , n , such t h a t matrices Ahk(-), I,'as 6 + 0, where
N:, q
ahkare matrices whose elements are defined by (6.50). &6
=
ahksatisfy the Condition N of $9, Ch.
virtue o f Theorem 9.2, Ch. I, this means t h a t
C,,
3 L as 6 + 0.
By
Due t o
the uniqueness of the strong G-limit (see Theorem 9.3, Ch. I) it follows that
C,
S L a s & -+o.
Fix q, s and consider the almost-solutions
U& = (U&,, ...,U:ns) of system
(6.48) constructed in Lemma 6.2. Set
N ~ s (=~(N6qa,-..rN69,~)* ) = &6('%(&)
X
i-'%)
7
(6.51)
$6. Homogenization of the system of elasticity
219
where c:# are constant vectors satisfying the condition (6.28) with U s = U,P,. Denote by N:(x) the matrices whose columns have the form (6.51). Let us verify that the matrices
(2
N i , APQ - , h ' q satisfy the Condition N as 6 -+ 0.
Indeed, the Condition N 1 follows from (6.51) and (6.28).
Consider the
Conditions N2, N3. Due t o (6.29)-(6.31)
A:'(:)
we have
+APX(:)
d ~ N i ~ (-+ xM{APkEg,"+Az4} )
(6.52)
weakly in LZ(R),
in the norm o f H-l(R), as 6 ---t 0. These relations show that Conditions N2 and N3 are satisfied, since due t o
(6.50) t h e expression in the right-hand side o f (6.52) is equal t o 6.4 is proved.
,@.
Theorem 0
II. Homogenization of the system
220
o f linear elasticity
57. Homogenization of Stratified Structures
7.1. Fonulas for the Coeficients of the Homogenized Equations. Estimates of Solutions Consider a sequence
{L,}
o f differential operators o f t h e linear elasticity
system
a
ax;(@(p(X),xi, ...,2,)
&(u)
dU
-)axj , > 0 independent
belonging t o class E ( I c nZ) ~ , with constant 53, Ch. I). Here
E
is a small parameter,
E
of E ,
2
(see
E (0,l); the elements of matrices
AF(t,y) are bounded (uniformly in E) measurablefunctions of t E R', y E R" with bounded (uniformly in
function in Cz(!=l) such t h a t 0
...,yn; p(x) is a scalar 1, (Vpl 2 const > 0; R is a bounded
first derivatives in
E)
5 p(x) 5
y1,
smooth domain. Let us also consider the following system o f linear elasticity
E(u)
d dU =( k j ( q 9 ( X ) , X 1 , ...,2") -) dX; 8x3
=f
,
whose coefficient matrices belong t o E(k1,Zz) and k l , stants which may be different from
~ 1 n2; ,
k2
are positive con-
the elements of the matrices
a ' j ( t , y ) are bounded measurable functions o f t E R',y E R",possessing bounded first derivatives in
y1,
...,yn.
In this section we consider the following Dirichlet problems
L,(u') = f
in
R , u" = GJ
E ( u ) = j in R , u
f E H-'(R) ,
=
on
dR ,
on ~ 80,
(7.3) (7.4)
a) E H q d n ) .
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 p ( x ) (see [go]). Here we obtain estimates for the difference between the displacements u" and u,the corresponding stress tensors and energies. We establish explicit
§7. Homogenization of stratified structures
221
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
2.
k
as
E +
0, and give
The corresponding spectral problems are studied in $2, Ch. Ill. Let t h e matrices N,'(t,y),
~ ( Y) t = ,
t
J [cpr(y)ipk(y)~:'(T, Y)]-lcpp(Y) 0
M;At 7 Y)
t
=J { 0
-
M $ ( t , y ) be defined by the formulas
AY(T,Y))
cpj (Y
)A!
+ A;(T,Y)
(7, Y) [cpl(Y )cpk(Y
-a
( a p u ( T , Y)
-~
( 7y))dT ,
,
)A%, Y )] -l YP(Y 1('WTY) -
' y ~ ~ ~ ,) } d ~ (7.5)
where (cpl(y) of
,...,cp,(y)) = (3 ,..., a ')= Vcp, B-' 8Yl
8Yn
is the inverse matrix
B . It will be proved in Lemma 7.5 t h a t the matrix [(pl(y)(pk(y)A,k'(~,y)]-'
exists and t h a t its elements are bounded functions (uniformly in
E).
To characterize the closeness between solutions of problems (7.3), (7.4) we introduce a parameter 6, setting
Theorem 7.1. Let u", u be the solutions of problems (7.3), (7.4) respectively, and u E Then the following estimates hold
H'(0).
II. Homogenization of the system of linear elasticity
222
where
..
depend on
auc
.
A:' -, 9'
rf E
axj
au , z = 1,...,n , the constants q,,c1 do not = A'3 ,
axj
Proof. Define v,(x) as the solution o f the problem
au axj
,Cc(vc) = 0 in R , v, = N;((P(x),x) - on dR
,
v, E H'(R).
(7.9)
Then it is easy to calculate that
Therefore
(7.10) T h e right-hand side of this equation is understood as a n element o f Let us show using the definition of
a
- - - Mi',
at
+ aYia(X,E)
,
S,, N,", MG
H-'(R).
that
(7.13)
0 7.
223
Homogenization of stratified structures
IP;.(x,E)I 5
] a , ( x , ~ )5 I c&, stants c2, cg, c4 do not depend on E. Indeed, we obviously have where
c&,
Multiplying these equations by
(pk
Iai..(x,~)\
I c46,,
and the con-
and summing them up with respect t o
1-
aMfs Ic, we obtain (7.11) due t o the inequality 5 c6,. aY1 Setting k = i in (7.14) we find by virtue of (7.5) that
a
-kf;",(cp(x),z)= (~;'~jA~[cplcpkA,k']-'cp,(A~" - AY) t ax;
This equality implies (7.12). According to the formulas (7.6), (7.5) we have
Let us estimate the H-'(R)-norm of the right-hand side of (7.10). For any column vector $ = ($1,
...,$,)
E
(?,-(a),due to (7.13),
au -)dx a$
-,
ax, ax;
(7.11) we obtain
-t
224
II. Homogenization of the system of linear elasticity
Therefore, taking into account (7.12) and the definition of 6, we find that
where c5 is a constant independent of E . Let us estimate the second term in the right-hand side of (7.10) in the norm of H-'(R). Using the definition of 6, we get
It thus follows from (7.10), (7.14), (7.15) that
$7. Homogenization of stratified structures where
c7
is a constant independent o f
E
225
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 (7.17) where
c8
is a constant independent o f
We now estimate the norm wc
where
$E
= V& - d c
,
E.
(1?&IIHl(n). Set
au
19' = *cN3f((P(x),x) dzj 7
= 1 in the 6,-neighbourhood of dR,
neighbourhood of dR, $' E C-(S=l), 0 follows from Theorem 3.1, Ch.
I,
5 $' 5
= 0 outside the 26,1, 6,1V$cl
I const.
It
that
and therefore
I
l l ~ c l l & ~ ( * ) ClOC IIUll&qn)
+ c11 l l V ~ l I i ~ ( w ) 9
where w1 is the 26,-neighbourhood of dR. By virtue of Lemma 1.3, Ch. I,
IIvuII~z(wl) L C126'
IIUll&l(n).
Hence
5
~ ~ ~ c ~ ~ H 1c13s~/2 ( S I ) IIuII@(n)
.
(7.18)
Estimates (7.17), (7.18) imply (7.7). Let us now prove (7.8). It follows from (7.7) that du'-
ax,
du
dN'
axj
axj ax,
-du+ G ,
--+2
(7.19)
II. Homogenization of the system of linear elasticity
226
and thus the estimate (7.8) is valid. Theorem 7.1 is proved.
0
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 of Theorem 7.1 we have lluc
1
- u - N' a
ax,
H*(W
< Q 6 f / 2 ( l l f l l L 2 ( S l ) + II@IIH3/2(aSl)) 7 (7.20)
(7.21) where Q, c1 are constants independent of E . Estimates (7.20), (7.21) follow from (7.7), (7.8) due t o the inequality 11u11H2(n)
5 c2 (IlfllL2(Sl) + ll@llH3/2(aR))
7
(7.22)
which is known from the theory o f elliptic boundary value problems in smooth domains (see [l]). 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 u" and u by the formulas
Theorem 7.3. Let u", u be the solutions o f problems (7.3), (7.4) respectively, u E H 2 ( 0 ) . Then
IJ%(~")
-
5 c1(G)6?(11UtI&2(n)
+ IIuIIHz(Sl) ~ ~ u c ~ ~ H 1 ( S l ) ) (7.23)
§ 7. Homogenization of stratified structures where q ( G ) is a constant independent of
227
E.
Proof. For the sake o f simplicity we prove this theorem assuming the elements of the matrices
k? 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) t h a t
Taking into account
(7.19), (7.11) we find
II. Homogenization of the system of linear elasticity
228
(7.25)
Therefore it follows from (7.25) that
aMf8
I
au
c4 [ l l u l l & 2 ( C l ) 6 .
From (7.19) we obtain
auc
+ IIVU11~2(8G)GC + 'l"IIU1lh(n)]
'
(7.26)
§ 7. Homogenization of stratified structures
5
c 5 ' ~ ( ~ ~ u ~ ~llu\lH'(n) ~2(n)
+
~
~
229
"
~
+ ~ ~ V U ~ ~ & ( a +G )bfl ) ~
~
2
(
~
)
3
(7.27)
:c66f/* 11u11$2(n). where lpil I Since by the imbedding theorem we have IlVullp(ac)5 ~ l l ~ l l ~ for 2 ( any u E H2(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.
[7
Corollary 7.4. If the coefficients of system (7.3) are smooth, it follows from (7.22), (7.23) that
IE&(u")- EG(U)\ 5 c2(G)S:/2 (Ilfllil(n)
+ ll'll&D(8n))
a
Note that the matrix [cpkcp,A:']-' was used in (7.5) to define N i , M;'. Let us show that this matrix exists and its elements are bounded functions (uniformly in
E).
Lemma 7.5. , tcl, Let A ' j ( z ) , i , j = 1, ...,n , be matrices o f class E ( K K~ ~, ) where positive constants independent o f 2. Let 'p E C'(fi), ( V y ( 2 const.
are
> 0,
Vcp = ( 9 1 , ..., cpn). Then there exist two constants such that for any E R"
t
K3
K ~ K, ~ depending ,
It?I ([cpl(.)(P,(.)A"'(.)l-l~,t)
only on
I 6 4 1tI2,
K',
n2 and
K 3 , K4
'p,
>0 . (7.28)
hf Set. q i h = ( P i t h 4-cp& in (3.3), Ch. 1. Then aY"lihqj1
(Cpith
= 4'pp,(PpayTtitj
,
+ ( P h t i ) ( ( P i . t h+ ( o h t i ) = qih%h
=
~
)
230
II. Homogenization of the system of linear elasticity
Set
K ( s )= ( p , ( x ) ( p p ( x ) A p q ( x )Then . by (3.3), Ch. I, for any [ E R" we get
where the constants cl,
MI depend only on
nl,
x 2 , cp. It follows that
K-l
exists. Setting [ = K-lC we obtain
13
These inequalities imply (7.28). Lemma 7.5 is proved.
7.2. Necessary and Suflcient 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 the sequence {C,}
to the operator
2 in terms o f convergence o f certain combi-
nations of the coefficients o f Ce, and t o obtain for the coefficients o f
E explicit
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 Coip the space of bounded measurable functions g ( t , y ) , ( t ,y) E [0, 11 x 0, equipped with the norm
t varies over a set of full measure. By Clip we denote the space o f functions
CobP,j = 1, ..., n.
g ( t , y) such t h a t g ( t , y ) , 89 E ayj
§ 7. Homogenization of stratified structures
231
Lemma 7.6. Consider a family o f functions $,(t,y) bounded in
E
whose norms in Cotp are uniformly
E ( 0 , l ) . Then there exists a subsequence E’ + 0 and a function
@ E Covflsuch that &(t,y) for any y
E
-+
@(t,y) weakly in L2(0,1) as
E’ +
0
a.
Proof. Let V
be a dense countable set in L2(0,1). For a fixed
w 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’ + 0 such that
j
$et(t,Y)w(t)dt + q,(y)
uniformly in y
,
(7.29)
0
where @,,(y) is a function of y
E i=2, Since V is a countable set, one can use
the diagonal process to construct a subsequence L‘
--t
0 such t h a t (7.29) holds
w E V. Now let w be an arbitrary function in L2(0,1) and j -+ 00, vj E V .
for any
Let us show that there exists Q,(y) S,(y)
-+
Q,,,(y)
wj
4
w in L 2 ( 0 ,1) as
such that
uniformly in
a
as j
-+ 00
.
Indeed, it is easy t o see t h a t
Choosing
EO
sufficiently small in order that for
E’
< EO
we have
11. Homogenization of the system of linear elasticity
232
II
1
qUk(y) 1
1
J $Ei(t,Y)vkdt + Q ~ , ( Y ) - J $ c i ( t , y ) v j d t 5 6/2 , 0
0
~,(y) + qW(y) uniformly in y E
0
as j
+ 00
.
Choosing a sufficiently large j in the inequality
we find t h a t
1
+ct(t, y)w(t)dt
+ U,(y)
as
E’ +
0
0
uniformly in y
E
a.
Obviously Qw(y) is a bounded linear functional on w 6 Lz(O,l) for any
y E
a. Therefore
1
Qw(Y) =
J
Q(t,y)w(t)dt
7
0
where
@ ( ty) , E L2(0,1) for
any y E
a.
Thus
J + a l ( t , y ) w ( t ) d t + ] qt,y)w(t)dt 1
0
as
+0
0
for any w(t) E L2(0,1). The function @ ( t , y ) satisfies the inequalities
-c(y’-y”(P
5 @(t,y’)
-@(t,y”)
5 cly”y“(4,
(7.30)
$7. Homogenization of stratified structures
233
f weakly in L2(0,1) as E --t 0 and m 5 fs 5 M , then m 5 f 5 M for almost all t E ( 0 , l ) . Therefore correcting, if necessary, CP on the set o f measure zero we get iP E Co~@ due t o (7.30). Lemma 7.6 is owing t o the fact t h a t if fb
-+
proved.
0
Corollary 7.7. Let {&(t,y)},
E
E (0,1), be a family of functions, whose norms in C'vfl are
bounded uniformly in
E.
Then there exists a subsequence E' -+ 0 such t h a t
weakly in L'(0, 1) for any y E
0, j
= 1, ..., n , where
~
E C'@.
Proof. It follows from lemma 7.6 t h a t there is a subsequence E'
j = 1, ...,n weakly in L2(0,1) for all y E Obviously for any g E
= lim a'+O
JJ 1
0 where $,(pj
,
E Co3p.
CF(f2) we have
"' I (t,Y v(t)g(y)dydt = j
o n
Therefore p j ( t , y ) =
0 such t h a t
-+
-in the a yj
sense of distributions. Since q j , G E
Coiflthe last equality holds in the classical sense for almost all t . Lemma 7.8. Suppose t h a t the functions
&(t, y) are bounded in Coifluniformly with respect
E E (0, I ) , and t h a t &(t,y) y E 0. Then
to
---t
0 weakly in L2(0, 1) as
E -+
0 for every
11. Homogenization of the system of linear elasticity
234
t
J
Y)
~ c ( t ,
$e(T,
Y)dT
-+
0
0
in the norm of Co([O,11 x Moreover, if
0 ) as
E
--t
811,e . = 1, ..,, n , are also bounded ,
layj
I
in
Q c ( ~ ( x ) , z-+ ) 0 weakly in H'(R) as for any cp(x) E
.
0
E,
E
then
-+
0
C'(0).
Proof. The family
{Qe(t,y)}, E E ( O , l ) , is equicontinuous and uniformly
bounded in [O, 11xa. Therefore due t o the Arzela lemma there exists a function
$ ( t , y ) such that E' -+
-+
11, in
the norm of
Co(([O,l]x 0 ) for
a subsequence
0. Since
/ t
@,, =
$Q(T,
y ) d -~+ 11,(t,y) for all t , y E [0,1] x
0, and &(t,y)
0
-+
0
weakly in L2(0,1) as
E -+
0 for any fixed y E
Let us prove t h a t Qc(cp(z), z)
-+
0 , it follows that 11, = 0.
0 weakly in H'(R) as E -+ 0. Indeed, we
have already proved that Q , ( ( P ( z ) , L ) -+ 0 in the norm o f
L"(R) as E
a Moreover the derivatives -Q,(cp(z),z) are bounded uniformly in E .
axi
due t o the compactness of a ball o f L2(fl) there is a subsequence E'
a that - Q,~((P(z), x) ax;
7.8 is proved.
-+
-+
+
0.
Thus
0 such
~ ( xweakly ) in LZ(R),and therefore x = 0. Lemma 0
(7.31)
$7. Homogenization of stratified structures
235
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 t h a t the elements of the matrices AY(t,y), z,j = 1, ...,n , have norms in
C ' ~ puniformly bounded in
G-convergent t o t h e operator
Then t h e sequence
E.
t as E --+
{L,}is strongly
0 if and only if the following conditions
are satisfied
B,"(t,y)+h'(t,y), B?(t,y) + S"(t,y) weakly in L2(0,1) as
E
s=O,1,
,
. .
Z,J
-+ 0 for any y E
...,n
= 1,...,TI ,
E
(7.32)
0.
Proof. Assume first t h a t the conditions (7.32) in this case 6, + 0 as
,
are satisfied. Let us show t h a t
+ 0, where 6, is defined by formula (7.6). Indeed,
one can easily check that
'p,B,p" = 0 , cppBps= 0 , cppB,P= E 'ppAfld = (B,O)-'B:,
A! = -B; Therefore
'ppAps=
+ (B;)*(B;)-'B;,
,
(bo)-'bs , = -bs + (&)*(@)-1&s
.
II. Homogenization of the system of linear elasticity
236
- ( ~ i ( ~ , y ) ) * ( ~ o ( ~ , y ) ) - ' ~ s (-~ B'"(.r,y)] ,y) dT
.
(7.33)
Denote the integrands in the above formulas for N,'(t,y), M,"(t,y) by n:(t, y), rnt(t,y) respectively. By virtue of (7.32) we have
n:(t,y), rnL(t,y) as
E -+
o for
-+
(7.34)
0 weakly in L'(0, 1)
any y E fi.
According t o Corollary 7.7 it follows that
a
-a
t , Y)
+
0
7
a
-4
ayj a yj weakly in L2(0,1) as E
as
E
-+
-+
4 Y)
-+
0
0 for any y E S2
(7.35)
.
o for any y E s2.
Lemma 7.8 and (7.34), (7.35) imply that the matrices N,'(t, y), M,"(t, y),
a ??/ j
a
-N,'(t, y), -M,'(t, y ) converge t o zero in the norm o f E
-+
Co([O,11 x
j
a) as
0. Therefore due t o (7.6) we have
as ~ 4 0 .
&-+O
(7.36)
Moreover, it follows from Lemma 7.8 that (7.37) Taking into account (7.11), (7.36), (7.37) we find that
a
- M ; ( v ( z ) , z ) -+ 0 weakly in
at
'Pk
a
IV'Pl2
ask
L ' ( 0 ) as
E +0
,
(7.38)
since - -M,",((p(z),z)-+ 0 weakly in L2(R). Let us prove the strong G-convergence of L, t o C as E -+0. Set f = L ( u ) E H-'(R), iP = 0, u E C,"(R) in Theorem 7.1. Then estimates (7.7), (7.8) are valid. By virtue of (7.36), (7.37), (7.38) we have uc -+
u weakly in Hi(R), -yf
-+
3' weakly in
L2(R) .
87. Homogenization of stratified structures Now let us show t h a t the set {E(v),v E the convergence
L,
3 2
237
CF(R)} is dense in H-'(fi).
Then
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 =
v E H;(R), and for any f = e(w)E H-l(R), w E
(?,-(a), we have
k(v),
This means t h a t llg - fllH-l(n)
L c 1121 - wIIH;(n) .
w E Cr(R) close t o v in Hi(R) we get a functional f = E(w)close t o g in H-'(R). Therefore choosing
Let us now prove that the conditions (7.32) are necessary for the strong G-convergence of
L, t o 2. Suppose t h a t L,
3 k as
E
.+ 0.
Due to our assumptions about the coefficients of system (7.1) and Lemma
7.5, the elements of matrices & ( t , y), s = 0,1, ...,n , Bfj(t,y ) , i , j = 1, ...,n , belong t o C'vp and have norms in C'@ uniformly bounded in E. Therefore by virtue of Corollary
7.3 there exist matrices B o ( t , y ) , & ( t , y ) , @ ( t , y ) ,
s,i , j = 1, ...,n , with elements in C'@and such that for a sequence
E' +
0
we have
weakly in
L2(0,1) for any y E
a.
Set jiis
= (P)*(BO)-'fjs- B i s
,
2,s = l , . . . , n
.
(7.40)
Define the matrices f i J ( t , y ) , f i G ( t , y ) by the formulas (7.5) with A i j ( 7 , y ) replaced by . 3 ' j ( ~ , y )and define
& by (7.6)
with
N;, MG replaced by &,
kG.
The same argument t h a t we used a t the beginning o f the proof of this theorem shows t h a t
II. Homogenization of the system of linear elasticity
238
fi:'(cp(x),z) + 0 weakly in H'(R)
,
(7.41)
a
- AIi(p(z), z) + 0 weakly in L2(R) at as
E'
+ 0.
Let
ii E C,"(R). Denote by u' solutions of the following problems
Set
Similar t o the proof of Theorem 7.1 we obtain the inequalities
Therefore by virtue o f (7.41) we have u"
+ ii weakly in H'(R)
, y;, -+ Ti weakly in L*(R)
(7.42)
as E' + 0.
Denote by uo the solution of the problem
By the definition o f the strong G-convergence of we have uo = 4,
L, t o
2 and due t o
ahk= Ahk- almost everywhere in R. axk a3
a3
axk
an arbitrary vector valued function from C,"(R), it follows that almost everywhere in
-..
Since d is
Ahk = ahk
R.
Thus we have shown t h a t from any subsequence another subsequence
(7.42)
E' +
. .
E"
+ 0 we can extract
0 such t h a t relations (7.39) hold for
Bs = Bs,
B'J = B'J,s = 0, ...,n , z , = ~ 1, ...,n , where B s , B'J are expressed in terms I
.
.
of the coefficients of the G-limit operator
A
A
_
.
e by the formulas (7.31).
Since
3 7.
Homogenization of stratified structures
239
{ E " } is an arbitrary subsequence, it follows t h a t (7.32) is valid for Theorem 7.9 is proved.
E -+
0. 0
In the proof of Theorem 7.9 we have actually established Theorem 7.10.
C'@ are Suppose t h a t there exist matrices &(t, y), h'j(t,y),
Let the elements o f the matrices AY(t,y) be such that their norms in uniformly bounded in
E.
s = 0, ...,n , i , j = 1,
system (7.1).
...,n , such
efficient matrices AY(cp(x), coefficients
a'j(t,
a's
that (7.32) holds for the coefficients of
Then the sequence o f operators Cc corresponding t o t h e co-
x) is strongly
G-convergent t o operator
2 whose
y) have the form
,
= ( ~ ) * @ 0 ) - 1 ~ 5- B i s
i , s = l , ...)1 2 .
(7.43)
Let us consider some examples o f strongly G-convergent sequences
{LE}
which satisfy the conditions (7.32). Theorem 7.11. Suppose that the elements of the matrices A:j(z) o f class E(tcl,t c 2 ) have the form a ? , ( ~ - ' q ) , where a?,([) E A P ( R ' ) are almost-periodic functions of
E
E
R'.Then the sequence C, strongly
G-converges t o the operator
k whose
matrices of coefficients are given by the formulas
where (Aij) is by definition the matrix with elements m
Moreover estimates (7.7), (7.8) hold and 6, -+ 0 as
Proof. In the case under consideration x l . Set
E + 0.
we have A:f(t,y) = A'j(E-'t), p(x) =
II. Homogenization of the system of linear elasticity
240
Y;.(s)= (A"(s))-l
(A1j - A'j(s)) ,
+ A"( s
zij(s)= ~ i 1 ( ~ ) ( ~ 1 1 ( ~ )() ~- '1 -j A'+)) The elements o f matrices any almost-periodic f , g,
q, Z;j
SJ
- AS3
6..
.
are almost-periodic functions, since for
1
f 2 const > 0, the functions f g and - are also
f
almost-periodic.
It is easy t o see that
Obviously ( Z i j ) =
(5)=
0 and the elements o f N j ,
are uniformly
bounded and equicontinuous. Therefore 6,
M; converge t o zero as E
-+
-+ 0 in Theorem 7.1, since N j , 0 at any point z1E ( 0 , l ) .
L, t o 2 follows from the conditions (7.32), I -+ ( f ) weakly in L2(0,1) as E -+ 0 for f(-)
The strong G-convergence o f which hold due t o the fact that any almost-periodic
f. Theorem
E
0
7.11 is proved.
Let us consider some examples where the coefficients of We introduce a class A, consisting of functions Cf(Y)?
C depend on
z.
f (t, y) such t h a t for some
gf(t,Y) we have
j
f(s,y)ds - cf(Y)t = gf(t,y) .
(7.44)
0
The functions f ( t , y), -, af 8Yl
cj(y), dcf(y) , 8Yl
assumed t o be Holder continuous in y E
-
R
sf, 'gf, dYl
1 = 1,...,n , are also
uniformly in
t E [0,1], and such
that
d
Isr(t,Y)I + lay,gf(t,Y,I 5 4 1 where the constants Set
Q,
+ Itl)'-"
7
o do not depend on t , 0 E (0,1].
(7.45)
5 7.
Homogenization of stratified structures Obviously for any
f E A,
241
we have (f(.,y)) = cf(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 c = 1.
2. Consider a function f ( t ) of the form f ( t ) = M+cp(t), where A4 = const.,
f E A1, if N > 1; f E A, for any Q E ( O , l ) , if N = 1; f E A N , if 0 < N < 1. Ip(t)l I C(1+ Itl)-N, N
3. The sum c3
$1
+
$2,
= min(ol,ez), c
I /6 &
> 0.
where 1 , q
$1
We can easily check t h a t
E A,,
+2
E
A,
belongs t o
A,
with
E (0,1].
+
f ( ~ - ' ~ , y ) dI ~c~E,(T 1) , a = 0 , l ; I = 1, ..., n
,
(7.46)
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 y ~ and , dcf(y)/dy, = 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
242
where 0
< a < 6 < 1 and
X[a,b] is the characteristic function o f the interval
[a, 61.
L*(O,1) by linear combinations o f characteristic func1 aaf tions and taking into account t h a t f , a f / a y , are bounded, we get (; 0 Approximating v E
,y)v(s)ds + 0 as E
--t
For a given matrix
0. Lemma 7.12 is proved.
0
B(t,y) with elements B;j(t,y ) l e t ( B ( . , y ) )be the ma-
trix with elements (Bij(-,y)). Theorem 7.13. Let the elements of the matrices Aij have the form
t
= (p(z), y = ( 2 1 ,
and define for
...,,
n)
,
i, s = 1, ...,n , Ak' = {a;"j'}the following
matrices
1 Bf"(t,y )
=
G
t B y - , y) E
E
t (pj(Y)A'J(E,Y)[Q'(Y)Qk(Y)Ak'(E ' '
t
t t 7Y)]-1vp(Y)APs(;,Y) -AiS(; , Y ) .
I
(7.48) Suppose t h a t the elements o f
2
B'(T,Y),B " ( T , ~B'"(.r,y) ), belong t o A,
for some u E (0,1 . Then the sequence of operators t,corresponding t o the matrices A'j Q X) ,z) is strongly G-convergent t o the operator k whose
(T
coefficient matrices are
at"(,) = (B'(.,z))'(BO(.,,))-l(B'(.,2)) - (B'"(.,z)) .
(7.49)
243
$7. Homogenization of stratified structures Moreover, the number C E ~ where ,
6, used in Theorem 7.1satisfies the inequality 6, 5
the constant c does not depend on
E.
Proof. According t o Lemma 7.12 we have t B"(-,Y)4 (B"(.,Y)) , E
t
q 1 y ) --+ (B'j(.,y))
,
= 0,1,...,n
. .
2,J
,
= 1,".,12
a.
Lz(O,1) as E -+ 0 for any y E Therefore due t o Theorem 7.9 one can take
weakly in
&(t,y) = (B".,y)) s
=0,1,
Thus by virtue of by
,
S'j(t,y) = ( B q , y ) )
,
...,n , 2., J . = 1,...,n .
(7.43)the coefficients of the G-limit operator
E
are given
(7.49). Let us now show t h a t 6,
5 CE". It is easy t o see t h a t
(PP(Y).iP"(Y) = (BO(*, y))-'(BY, Y)) (Pj(Y)'wY)
Therefore, from
= (B'(.,y))*(BO(.,y))-'
1
*
(7.48)we see that the matrices Njc(t,y), Mfs(t,y) defined by
(7.5) can be written in the form t
N?(t,Y) =
J [BO(Z
- BY5 , Y ) ]
E 1Y) (B0(.,y))-'(B"*1Y))
0
7
7
&
E
d7
1
I
Denoting the integrands in (7.50) by nj(- , y ) , rnij(- ,y), respectively, we see t h a t the elements of n j ( t l y ) , rn;j(t,y) belong t o
A, and (nj(.,y)) =
(mij(.,y)) = 0. Thus by the definition of 6, and proved.
(7.46)we get 6, 5
CEO.
Theorem 7.13 is 0
244
II. Homogenization of the system
of linear elasticity
Corollary 7.14.
If p(z) = z1in Theorem 7.13, then the coefficients of the G-limit system are given by the formulas
245
58. Estimates for the rate of G-convergence
58. Estimates for the Rate o f G-Convergence o f Higher Order Elliptic Operators
8.1, G-Convergence of Higher Order Elliptic Operators (the n-dimensional case)
R c R" consider
In a smooth bounded domain
a differential operator of
the form C(U)
C
=
l45m
( - l ) a P ( a a p ( x ) V p u ),
xER ,
(8.1)
IOKm
where aap(x) are bounded measurable functions in a1
+ ... + an,u(x)is a scalar function in H,"(R). We say that a differential operator C :
R , a,/? E Z;,( a (=
H,"(R) + H-"(R) o f the form
(8.1) belongs t o the class E(Xo,X1, &), if its coefficients satisfy the following conditions
ZL E CF(R), where Ao, X l , A, are positive constants independent o f u. It follows from the last inequality (see (1341, [9])t h a t for any t E R" and
for any any
x E R we have
c
I4=IPI=m
where ~0 = const.
aap(x)S"tP
> 0, 5"
2 co ItlZrn 7
=
Thus every operator C of class E(X0, XI, A,) is elliptic. Now following [148] we give the definition for the strong G-convergence of a sequence of higher order elliptic operators.
We say t h a t a sequence of operators {Ck} of class E(X0,X I , A,) is strongly G-convergent t o the operator
k
o f class E ( i 0 ,
xl,i z ) , if for any X 2 i, (i,=
11. Homogenization of the system of 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
( L + X)u = f ,
t 00
t o the solution u of the problem
u E H,m(R);
(8.3)
and moreover, if the sequence of functions
converges in L2(R) weakly as
k
t 00
t o the functions
Here { a k p ( z ) }and { h a p ( x ) } are the matrices of coefficients of operators
Ck
and
E
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 ra(uk,&) t o I',(u,L)
in
L2(R) as k
4 00.
It is shown in [148] that the strong G-convergence o f Ck t o
E
as
is equivalent t o the following conditions, the so-called Condition N: There exists a sequence of functions { N , k ( z ) }such that
N1
N3
Ny"E H"(R), Ny" + 0 weakly in H"(R), IyI < m
C
lal=rn
Do(& - hop) + 0 in the norm of H-"(R) ,
;
k
t 00
247
$8. Estimates for the rate of G-convergence
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," we arrive a t a stronger condition (the so-called Condition N') which not only implies the
weak convergence o f uk t o u in
H,"(R) as k
-+ 00,
but enables us t o estimate
the difference between uk and u. We say that a sequence o f operators {&} E E(Xo,X1,X2) trices of coefficients { a $ ( z ) } ,
Icy[,
IpI 5
with the rna-
m , satisfies the Condition N' in 0,
if there exists an operator E E E ( i 0 , i 1 , A*) with the matrix of coefficients { Z l a ~ ( x )and } a family of functions N," E Hm(R),IyI 5 m , such t h a t
D a N t E L"(R)
N'1
for
Icy1
in the norm of L"(R),
5 m , IyI 5 m , D*Nt
0
la1 < m , )yI5 m ;
in the norm of H-">"(R), (for the definition of
-t
5 m as k
-t 00
H-"*"(R) see $9.2, Ch. I).
For the sake o f simplicity we assume t h a t the coefficients 2 1 , ~ are infinitely m o o th . Let us introduce the parameters which characterize the rate of convergence in the Conditions "1, N'2, "3. Set
II. Homogenization of the system of linear elasticity
248
Theorem 8.1. Let the Condition N' hold for the operators ck,
ji such t h a t for p
> ji
and s
k . Then there is a real constant
5 rn - 1 the solutions o f the
Dirichlet problems
satisfy the inequalities
(8.12)
Proof. For any operator L: E E(Xo,XI, A,) there is a real constant jl depending only on Xo, XI, Xz, and such t h a t if p > fi, then the solution w of the Dirichlet problem
(L:+ p ) w = f ,
f
E H-"(R),
w E H,"(R)
58. Estimates for the rate of G-convergence
249
satisfies the inequality
where c is a constant depending only on A,,
XI, Xz (see [9]). Let us choose
> p > 0 such that the solutions of the Dirichlet problems for .& + p , L + p satisfy (8.13) with a constant c the same for all k. p
operators
We shall use the following Leibnitz formulas (see [127]): (8.14)
(8.15)
where ( that
Pj
a
P
a1
)=(
Pl
)...(
an
pn
)=
each j = 1, ...,n.
5 aj for
a!
P!(a--P)!
, a ! = crl!
... an!, PI a means
Set
where u is the solution of problem (8.9) and N," are the functions entering the Condition N'. By virtue o f (8.14) we find (ck(U:),V) f
l4Im
IPKm
=
c 1
l49lPl
n
1
n
a$@u:D"vdx
atp@uVorvdx$
=
11. Homogenization of the system of linear elasticity
250
Denote by
50 the last integral. Then because o f (8.4) we have
where c1 is a constant independent of Ic. Transposing the indices y and
p
in the integral next t o the last one in
(8.16), we obtain
where 51, J z , 53 stand for the respective integrals on the left-hand side of the last equality. From (8.4) we have (8.19)
It follows from (8.6) and Lemma 9.1, Ch. I, that
$8. Estimates for the rate of G-convergence
251 (8.20)
Let us estimate 51. Using (8.15) we find
51 = J n
c
(i$-iafl)D~UDavdz =
bl=m IPlSm
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
(8.22)
for any v E H,"(fl), where
We obviously have U : - v k - Uk E H,"(R) and
II. Homogenization of the system of linear elasticity
252
: .1 5
- vk - ukllHm(f2) I c 3 [(ak
+ bf) + Y k ) IIuIlH2m(R) +
IlulIH2m+1(n)]
which implies (8.10), since for the solution u of problem (8.9) the following a
priori estimate is valid
One of the simplest examples, when the Condition N' holds, is provided
& with coefficients a $ ( e ) such that akp(s) + zl,p(z) in the norm of L"(R) as k + 00. In this case we can take N: G 0. Obviously the Conditions "1, "2, "3 by the sequence o f operators
are satisfied and
where c is a constant independent of
k,
$8. Estimates for the rate of G-convergence 6k =
max
253
.
Ila$ - &pIIw(n)
I46m IPISm
According t o Theorem 8.1 we have
Actually one can prove a stronger inequality in the case under consideration, namely:
- U I I H ' ( ~ )5 Kz6k I l f l l t 2 ( n ) s 5 m - 1, K1,K 2 = const . llUk
7
(8.25)
To obtain (8.25) we note t h a t in the proof of Theorem 8.1 the norm IlfII~l(n,estimates l l ~ l l ~ 2 ~ + in 1 ( (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 t h a t under our assumptions this integral can be estimated by 6k llUllH2"'(n) IIvIIHm(n)
*
Let us now consider a less trivial example, when the Condition N' issatisfied. Assume that the coefkients a$(z) of operators Lk depend only on z l , i.e. akp(z) = a$(zl).
Let the coefficients &p(zl) o f operator
\PI 5 m , o = (m,O, ...,0)
where 1 is such t h a t
Rc
L
be such t h a t for all la1
5 m,
we have
{z : 0
< x1 < I}.
Define the functions N j ( z , ) as solutions o f the equations
C
Jal=m
14=m such t h a t
Da(aL,D~~
pk= )
C
(al=m
~
~
(- a$) i ~
,
p
IPI I m , (8.27)
II. Homogenization of the system of linear elasticity
254
(8.28)
d" -N$(o)=o, dx;
s = 1 , ..., m - 1 .
(8.29)
It follows that the Condition N'3 is satisfied and /$) -+ 0 as k -+ m, -yk = 0 in (8.5), (8.7). By virtue of (8.26) the right-hand side of (8.28) tends to zero weakly in d" L 2 ( 0 ,I) as k -+ 00. Therefore -Ni(x1)-+ 0 in the norm of Co([O,11) as dx: k - + c o , s = O , l , ...,m - 1 . Owing to (8.28), (8.4) one has
and
0 as k
Thus the Condition N'1 is also valid. Let us consider the Condition N'2. We obtain due t o (8.28) that cyk -+
ii$
By
-+ 00.
- i,
= a!,,
2)"
Nj
+ akp - ii,p
=
virtue of (8.26) we have iikp(xl) - iop(zl)-+
0 weakly
in
L2(0,1).
Therefore
/
01
@$(zl) =
(h$(s) - i,p(s))ds -+ 0 in the norm of Co([O,Z]) .
0
d dXl
Since ii$(zl) - A,p(zl) = -@$(xl), we can assume in (8.6) that
255
58. Estimates for the rate of G-convergence
and ,@-) -+
0 as k + 00, c = const.
Now in order t o obtain an effective estimate for the closeness of
uk(2)
to
u ( z ) it is sufficient t o estimate IlvkllHm(n). We have
< Hm-'/Z(an)
5 C2a:l2 ~
~
~
5 C4a:l2 IlfllHl(f2)
~
~
IluII$zm(an) ~ ~ m - 5 l%":I2 (
~
IIuIIH2m+l(R) ~ )
5
Y
where cj are constants independent of Here we used the definition o f
(Yk,
Ic. the a prion' estimate (8.24), the in-
equality IIUIIL.(an)
5 c II4IH.-'(an)
IIUIIH.+f(an)
I
>0
Y
t >0
[9]),and the fact that Npk possess derivatives up t o the order m, which are bounded uniformly in k. Define the parameter 66, by 6 k = max {ak,,&'}. (see
Then, according t o Theorem 8.1, we obtain the inqualities lluk
- U(IHm-'(n) 5 C6:l2
IlfllHl(f2)
I
(8.31)
8.2. G-Convergence of Ordinary Differential Operators The results of the previous section are obviously valid for ordinary differential operators.
However in the latter case we can obtain more accurate
estimates. Here we prove some theorems in this direction.
II. Homogenization of the system of linear elasticity
256
Let 0 = ( 0 , l ) and let
Lk,
J? be ordinary differential operators of the form (8.32)
Theorem 8.2. Let U k , u be the solutions of the following Dirichlet problems
(Lk+ p ) u k where
c k ,
J?
=f,
( k + p)u = f
E H,"(o, 1)
(8.34)
are ordinary differential operators (8.32), (8.33). Then
where the constant c does not depend on
p=l,
216721
f , Ic,
...,m
and N i are the solutions of the equations
dmNk
iimp
akp
P=--dxm akm akm '
(8.38)
satisfying the boundary conditions d j Npk
X A l =0 ,
j = 0, ...,m - 1
(8.39)
58. Estimates for the rate of G-convergence
257
Proof. It follows from the above result for higher order elliptic equations, whose coefficients depend only on x l , that in order t o prove estimates (8.35), (8.36) we have only t o estimate the functions
z)k
which are solutions o f problems
(8.12), namely
For the functions
Moreover, if p
Npk we have
5 2m - 1 , it follows from
Sobolev's lemma t h a t
where c is a constant independent of u . Therefore due t o (8.24) we get (8.42)
Set
It follows from (8.41), (8.42) that
tdpk'l,b$!I 5 c2Ak
IlflILZ(o,i)
.
One can construct a continuous extension operator
(8.43)
P mapping any pair o f
numerical sets {a!"'),{ a ! ' ) } ,i = 0 , 1 , ..., m - 1 , into a smooth function cp(x) defined on [0,1] and such that
II. Homogenization of the system of linear elasticity
258
Obviously cp(x) can be defined by the formula (8.44) where e p ) ( x ) ,e { ' ) ( z ) are smooth functions which satisfy the conditions
Therefore
vk
is the solution of the Dirichlet problem
Lk(vk)
where
Vk
t P V k = 0 on (071) 7
vk
-Vk
E
H,"(o, 1) I
are the functions defined by (8.44) with a?) = a$!, all) = a$. By
vritue o f (8.43), (8.44) we get IIVkllH'"(0,l)
Set
wk
= 2)k
- (pk.
Lk(wk)
5 c3
{laf:A!l, IaI'pk'l} 5
maX
k 0 , ...,m-1
Then
w k
=Lk(9k)
9
c 4 A k IlfllL2(0,1)
.
(8.45)
is the solution o f the Dirichlet problem wk
E H,"(o, 1)
Using the inequalities (8.13), (8.45), we find that
5 c 5 l l v k I I H m ( o , l ) 5 '%Ak
IILk(qk)llH-m(O,l)
Hence
llWkllHrn
5 c 7 A k Ilfllp(o,l) and we finally
IIwkllHrn
It is clear t h a t
5 c8Ak 7k
=
'
,@)
IlfIlL2(0,1)
= 0;
IlflIL2(0,1)
.
obtain
*
&),(Yk
5
c&,
c = const, and therefore
estimates (8.10), (8.11) imply (8.35). Theorem 8.2 is proved.
0
Remark 8.3. Suppose that the coefficients
o f the operators
L k
have the form uk,(x) =
a,,(kx) where u p , ( ( ) 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
/
259
1
I rn - 1, (f)=
where p , q
estimates (8.35) become 1Iuk
f(l)d<. We also have Ak
0
- ullH*(0,1)5
C
~ ~ f ~ ~ L Z ( o , i )3
5
C
-
k
.
Moreover, the
= 0, ...,m - 1
where c is a constant indeDendent of k and f.
d d If Lk = - (ak((.) -), dx dx
-
reduced t o
Iluk - uoIILz(o,i) 5
d h = - (h(x) dx dx
c
-),
then the estimate (8.31) is
max XQ"
where C is a constant independent o f
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
du d d'u LI;(uk - U - N k -) = - (ak(X)Nk(X)7 ) dx dx dx Therefore
11. Homogenization of the system of linear elasticity
260 where So
5 uk(x) 5 M for L4w) = 0 ,
any
k
= 1,2,
R
R
- UIlL2(0,1)
2)k
is such t h a t
such that
on the coefficients of the operator
below. It is easy t o see t h a t
IIUk
the function
x E [O, 11 ,
Due t o (8.24) there exists a constant
The dependence of
... and
Ilell ax
1
2 will
be specified
< - ~ ~ f ~ ~ L z ( Thus ~ , l ) .we have
L~(OJ)
so
MR+1
5Ak 60
Ilflb(0,l)
+~~w~~~L2(o,1) I
where
In order t o estimate the norm
112)kllL2(0,1) we
apply the maximum principle,
which yields
since obviously
Thus
11%
- UIIL2(0,1)
MR+1
I (--
60
1
+ R + &)Ak
IlfllL2(0,*)
.
R > 0 in terms of the Soefficients of t h e d u dii du L.Squaring both sides of the equation i =f-- -
Now let us estimate the constant G-limit operator
and integrating it over [0,1], we obtain
dx2
dx dx
261
$8. Estimates for the rate of G-convergence
It follows t h a t
where
P > 0 is any constant such t h a t max x€[OJl
and therefore we can take
P2
R = - (1 + -) 60 6 : 2112
dii (z) 5 P.
Thus
112
W e finally obtain t h e inequality
where the constants M , R, So can b e easily calculated for t h e given coefficients o f t h e operators
&,
2.