haddadou.pdf

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Electronic Journal of Differential Equations, Vol. 2006(2006), No. 137, pp. 111.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

ftp ejde.math.txstate.edu (login: ftp)

A PROPERTY OF THE H-CONVERGENCE FOR ELASTICITY

IN PERFORATED DOMAINS

HAMID HADDADOU

Abstract. In this article, we obtain theH0e -convergence as a limit case of the

He-convergence. More precisely, if is a perforated domain with (admissible)

holes T and denote its characteristic function and if (A, T) H

0e

A0, weshow how the behavior as (, ) (0, 0) of the double sequence of tensors

A

= ( + (1 ))A

is connected to A0

. These results extend thosegiven by Cioranescu, Damlamian, Donato and Mascarenhas in [3] for the H-convergence of the scalar second elliptic operators to the linearized elasticity

systems.

1. Introduction

The notion ofH-convergence was introduced by Murat and Tartar [7, 8, 9] forthe second-order elliptic operators (non necessary symmetric) and extended to thecase of holes by Briane, Damlamian and Donato in [2]and called H0-convergence.Cioranescu, Damlamian, Donato and Mascarenhas [3] obtain the H0-convergenceas a limit case of the H-convergence with a vanishing coercivity constant in the

holes.In this work, we show that a similar property holds for the linearized elasticity

systems, namely between the He-convergence studied by Francfort and Murat in[6]and its generalization to the case of holes, denoted by H0e -convergence, whichhas been developed by Donato and El Hajji in [5]. The He-convergence dealswith the convergence of the solutions of a system of linearized elasticity whosetensor coefficients{A} are equibounded and uniformly definite positive. The H0e -convergence treat the same problem in a perforated domain with a tractioncondition on the holes for which uniform Korn estimates hold.

Let us briefly describe here the main results of this paper. Let a boundedopen subset ofRn, {T} a sequence of (admissible) holes, denote = \ Tthe perforated domain and the characteristic function of . Let also{A

} a

sequence of linearized elasticity tensors on such that ( A

, T)

H0e

A

0

. We prove(Theorem4.1) that if we set for every >0

A = (+ (1 ))A a.e. in

2000 Mathematics Subject Classification. 35B40, 74B05.Key words and phrases. Homogenization; H-convergence; linearized elasticity system;

perforated domains.c2006 Texas State University - San Marcos.

Submitted July 6, 2006. Published October 31, 2006.

1


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2 H. HADDADOU EJDE-2006/137

and if A He-converges to a tensor A (for some subsequence), then A A0

strongly inLp() for any p 1, and weakly in L(). Moreover, under suitableassumption (see (4.3) below), we have also (Theorem 4.2)

(A0(A, T)) in the sense

u u

strongly in H10 ()n,

Ae(u) A

e(u) strongly in L2()nn

and (Theorem4.3)

(A(,)(0,0)

A0) in the sense

u u

0 weakly in H10 ()n,

Ae(u) A

0e(u0) weakly in L2()nn,

where u, u and u0 are the solutions of (2.2), (3.5) and (2.4) respectively. This

results can be resumed by the following commutative schema:

AHe A

(A, T) H0e

A0.

The definition and the main properties of the H0e -convergence are recalled inSection 2. In Section 3 we give some preliminary results and in Section 4 we stateand prove the main results.

2. The H0e -convergence

We use the following notation: IfA= (Aijkl)1i,j,k,ln is a forth order tensors and Rnn, we set

A =

1i,j,k,ln

Aijklpq,

. = 1i,jnijij ,||= (

1i,jn

|ij|2)

1

2 ,

is a domain ofRn, ifFis a set of matrices fields, FS= {MF s. t. M is symmetric}, {} and {} denote a two strictly decreasing sequence converging to zero, if v = (v1,. . . , vn) is a vector valued function and = (ij)1i,jn is a secondorder tensor of variable x = (x1,. . . ,xn), we set

(v)ij = vi

xjDxjvi,

e(v) =1

2(v+t v),

(div )i= ij

xj;

for two real numbers and such that 0< < , Me(,, ) is the set of thetensorsA = (Aijkl)1i,j,k,ln defined on , such that a.e. on we have

(i) Aijkl L(), for any i,j,k, l= 1, . . . , n,

(ii) Aijkl= Ajikl= Aklij , for any i,j,k, l= 1, . . . , n,(iii) ||2 A., for any symmetric matrix ,


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EJDE-2006/137 A PROPERTY OF THE H-CONVERGENCE 3

(iv) |A| ||, for any matrix ,

iffH1()n and u H10 ()n, we set f, u=f, uH1()n,H1

0()n .

Let us recall first the main results concerning the H0e -convergence introduced by

Donato and El Hajji [5]. We introduce the perforated domain

= \T,

where T is a sequence of compact subsets of and set

V= {v H1()n s. t. v= 0 on }.

In the following, we denote bythe outward normal unit vector on the boundaryof and . the extension by 0 from to and set = .Definition 2.1 ([5]). The set T is said to be admissible (in ) for the linearizedelasticity (or e-admissible), if and only if:

(i) Every L weak -limit point of{

} is positive a.e. in ;(ii) there exists a positive real C, independent of , and a sequence {P} of

linear extension operators such that for each

P L(V, H10 ()n),

(Pv)| =v, v V,

e(Pv)0, Ce(v)0, , v V.

(2.1)

We denote by P the adjoint operator ofP, which is defined from H1()n to

V with P given for every fH

1()n by

v V, P f, vV ,V =f, PvH1()n,H10 ()n .

Definition 2.2 ([5]). Let A Me(,, ), T be e-admissible in . The pair(A, T) is said H

0e -converge to the tensor A

0 Me(, , ) and denoted by

(A, T) H0e A0 if and only if for each function f H1()n such that f f

strongly in H1()n, the solution u of

div (Ae(u)) = P f in ,

(Ae(u))= 0 on T,

u = 0 on ,

(2.2)

satisfies the weak convergence

P(u)u0 weakly in H10 ()

n,

Ae(u)A0e(u0) weakly inL2()nn,(2.3)

where u0 is the unique solution of the problem

div(A0e(u0)) = f in ,

u0

= 0 on .

(2.4)

Remark 2.3. (1) In [5]the definition is given for fixed f.

=f. The two definitionsare equivalent in view of [5,Proposition 2].(2) In the case where T = , this definition reduces to the definition of the He-convergence[6].

This notion of convergence makes sense in view of the following compactnesstheorem:


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Theorem 2.4 ([5]). LetA Me(,, ) andT be e-admissible in. Then thereexists a subsequence, still denoted by {}, and a tensor A0 Me(

C2 , , ), such

that the sequence{(

A, T)}

H0e

-converge to A0.

Remark 2.5. The fact that A0 belongs toMe( C2

, , ), does not appears explicitlyin the statement given in [5], but can be easily deduced with the same argumentsas that used in the non perforated case.

Let us recall also a property proved recently in [4].

Theorem 2.6 ([4]). Let{A} Me(,, ) and{B} Me(, , ) such that

AHe A0 and B

He B0. Assume that there exists two functions h, h0 L1()

such that

|A B| h h0 strongly inL1().

Then

|A0(x) B0(x)| h0(x) a.e. in.The following proposition completes a result given in [5]:

Proposition 2.7. One has

(1) If{v} is a bounded sequence inH10 (), then

(v v weakly inH10 ()n)

P(v|) v weakly inH

10 ()n

.

(2) If (, ) is a sequence ofR+ R+ such that (, ) (0, 0) and {v

} is a

sequence inH10 () bounded independently of and , then

(v v weakly inH10 ()n)

P(v |) v weakly inH

10 ()n

.

Proof. Suppose that

P(v|) v weakly inH

10 ()n. (2.5)

Observe first that

v =P(v|

). (2.6)

On the other hand, since {v} is a bounded sequence in H10 (), there exists a{} {} and w H10 ()

n such that

v

w weakly inH10 ()n. (2.7)

But|

| 1, hence there exists 0 L() and{} {} such that

0 weakly in L(). (2.8)

Passing to the limit (in D ()) in (2.6) by using (2.5), (2.7) and (2.8), we find

0w= 0v.

Taking now into account the fact that (in view of Definition 2.1) 0 > 0, weobtainw = v. This, together with (2.7), implies that the whole sequence P(v

|

)

converge weakly tov . We refer to [5] for the converse implication. The proof of (2)follows by the same arguments.


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3. Preliminary results

In this paper,{A} is a sequence of fourth-order tensors of Me(,, ) and {T}

is a sequence of holes e-admissible in such that

AH0e A0. (3.1)

Set, for every >0,

A = (+ (1 ))A a.e. in . (3.2)

Since, for fixed > 0, A Me(min(1, ), max(1, ), ), in view of the com-pactness properties ofHe-convergence, there exists a subsequence {m}of{}and

A Me(min(1, ), max(1, ), ) such that AmHe A as m 0. Hence, for

every >0, the set

W ={A; {m}mN {}s. t. AmHe A} (3.3)

is not empty. Let{f}be a sequence in H1()n such that

f f quadstrongly in H1()n (3.4)

and let A be in W. Let u and u the solutions of

div(Ae(u)) = f

in ,

u = 0 on (3.5)

and div(Ae(u)) = f in ,

u = 0 on (3.6)

respectively. We consider now the following sets:

U = {u : u is the solution of (3.6) for some A W},

V = {The set of weak limit points ofAe(u) in L

2()n as 0}.(3.7)

One has the following result:

Lemma 3.1. One has

V = {Ae(u) : A W andu is the solution of (3.6)}.

Proof. It is clear that, if A W and u is the solution of (3.6), then Ae(u)belongs to V. On the other hand, let v V. Then, there exists a subsequence{m}of such that

Am e(um ) v weakly inL

2()n, (3.8)

as m 0. But the compactness property of the He-convergence shows that thereexists a subsequence {m} of{m} and a forth-order tensor A such that

AmHe A.

This implies in particular

Am e(um ) Ae(u) weakly inL

2()n.

This, together with (3.8), gives v = Ae(u).


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Remark 3.2. Let us show that in view of Theorem 2.6, there exists {m} {}

and for all >0 a tensor

A such that

AmHeA. (3.9)

Let us also point out that in Theorem4.1we will consider a more general situation,

where for every >0, there exists{} and a tensor A such that AHe A.

Let us prove (3.9). Using the diagonal subsequence procedure and the compact-ness property of the He-convergence, one extracts a subsequence {m} of{}suchthat, for every Q+, one has

Am He-convergences to a limit A. (3.10)

Since a.e in one has

|Am1 Am2

| |1 2|, 1, 2 Q+,

Am1 Me(min(1, 1), max(1, 1), ),

Am2 Me(min(1, 2), max(1, 2), ).

Then, from Theorem2.6, it follows

|A1 A2 | 2

max(1, 1) max(1, 2)

min(1, 1) min(1, 2) |1 2|.

This implies that the mapping Q+ A L() is uniformly continuous.

Hence, it can be extended to a mapping (still denoted by A) defined anduniformly continuous on all R+ (sinceQ

+ is dense inR

+).

Let nowbe a strictly positive real and {s} be a sequence ofQ+which convergesto as s . Then, there exists a sub-subsequence{m} of{m}such that

Am He-converges to some A. (3.11)

In view of Theorem 2.6 this give, together with (3.10) and the fact that |A

m

Ams

| | s|, the following inequality:

|A As | 2

max(1, ) max(1, s)

min(1, ) min(1, s) | s| a.e. in .

Using the continuity of the mappingA onR+ and passing to the limit in thisinequality as s , one finds

A= A, a.e. in .

The uniqueness of the limit implies then that the whole subsequence Am He-converges to A, for every >0.

The following results state some a priori estimates that we will need in the

following:Proposition 3.3. Letu andu the solutions of (2.2)and(3.5)respectively. Then,there existsc >0 independent of and such that

P(u| ) PuH1

0()n c(

1/2 + |f, P(u| ) u

|

1/2),

e(u)L2(T)nn c(1 + 1

2 |f, P(u | ) u

|

1/2),

Ae(u) Ae(u) L2()nn c(

1/2 + |f, P(u| ) u

|

1/2).

(3.12)


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Proof. Observe that the variational formulations of problems (2.2) and (3.5) are

w H10 ()n, Ae(u).e(w)dx= f, P(w| ) (3.13)

and

w H10 ()n,

Ae(u).e(w)dx +

T

Ae(u).e(w)dx=f, w

respectively. Then, for every w H10 ()n, one has

A(e(u) e(u)).e(w)dx +

T

Ae(u).e(w)dx=f, P(w| ) w.

In particular, for w = u Pu, this gives

A(e(u) e(u)).(e(u) e(P

u))dx +

T

Ae(u).(e(u) e(P

u))dx

=f, P

((u P

u

)| ) u P

u

.Using that Pu| =u

, one deduces

A(e(u) e(u)).(e(u) e(u

))dx +

T

Ae(u).e(u)dx

=

T

Ae(u).e(Pu)dx f, P

(u)| u.

In view of the fact that A Me(,, ), this gives

|e(u) e(u))|2dx +

T

|e(u)|2dx

|T Ae(u).e(P

u)dx| + |f, P(u | ) u

|.

(3.14)

Using the Youngs inequality, one obtains T

Ae(u).e(Pu)dx

T

|e(u)||e(Pu)|dx

2

T

|e(u)|2dx +

2

2

T

|e(Pu)|2dx

2

T

|e(u)|2dx +

2

2

|e(Pu)|2dx.

But, taking w= Pu in (3.13), one finds

|e(Pu)|dx c1,

withc1 > 0 independent of and . Then T

Ae(u).e(Pu)dx

2

T

|e(u)|2dx + c2,

where c2 > 0 independent of and . This, together with (3.14), gives

|e(u)e(u))|2dx+

2

T

|e(u)|2dx c3(+|f, P

(u| )u|). (3.15)


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with c3 > 0 independent of and . From this, (3.12)ii) follows immediately.Moreover, since u|

u H10 ()n, Definition2.1 shows that

e(P(u| u))0, Ce(u u)0, .

Hence, by virtue of the Korn inequality, (3.15) gives also (3.12)i).

On the other hand, since A = A a.e. in and A

= A

a.e. in T, one has

|Ae(u) Ae(u)|2dx

|Ae(u) Ae(u)|2dx + 2

T

|Ae(u)|2dx

2

|e(u) e(u)|2dx + 22

T

|e(u)|2dx,

which, together with (3.12)(i) and (3.12)(ii), gives (3.12)(iii).

Proposition 3.4. Letu0 the solution of (2.4). Then

supuU

u u0H10

()n c 1/2,

supvV

v A0e(u0)L2()nn c 1/2.

(3.16)

Proof. Let be u in U. This means that there exists A W such that u isthe solution of (3.6). But the fact that A is in W implies that there exists asubsequence{m} of such that A

m He-converges to A. Hence, the solutionu

of div(Am e(u

m )) = f

m in ,

um = 0 on (3.17)

satisfies as m 0

um u weakly inH10 ()n,

A

m

e(u

m

) Ae(u) weakly inL2

()

nn

.

(3.18)

Estimate (3.16)(i): By Lemma2.7,(3.18)i) implies that, for every fixed >0,

Pm(um |m ) u weakly inH

10 ()n. (3.19)

Hence, by (3.4), one has

limm0

fm , Pm(um |m

) u= 0. (3.20)

From this and (3.12)(i), it comes

limm0

Pm(um |m) PmumH1

0()n c

1/2. (3.21)

But, (3.19) and (2.3)i) imply

Pm(um |

) Pm(um) u u

0 weakly in H10 ()n.

This gives, by using the weak lower semi-continuity of the H10 -norm,

u u0H1

0()n lim

m0Pm(um |m

) PmumH10

()n ,

where u0 is the solution of (2.4). Hence, (3.21) gives

u u0H1

0()n c

1/2.

This is still valid for every u U, which implies (3.16)(i).


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Estimate3.16(ii): From (3.12)(iii) and (3.20), it comes

limm0

Am e(um ) A

m e(um)H10

()nn c1/2.

But (2.3)(ii) and (3.18)(ii) imply, as m 0, that

(Am e(um ) A

me(um)) (Ae(u) A0e(u0)) weakly inL2()nn.

In view of the weak lower semi-continuity of the L2-norm, these two last relationsgive

(Ae(u) A0e(u0))L2()nn c

1/2,

for every u U. Hence,

supuU

(Ae(u) A0e(u0))L2()nn c

1/2,

which, together with Lemma3.1,gives the claimed result.

4. Main resultsTheorem 4.1. LetA W. Then, the solutionu of (3.6) satisfies as 0

u u0 strongly inH10 ()

n,

Ae(u) A0e(u0) strongly inL2()nn,

(4.1)

whereu0 is the solution of (2.4). Moreover, one has the convergence:

p [1, [, A A0, (4.2)

strongly inLp() and weakly inL().

Theorem 4.2. Letf, f be inH1()n satisfying (3.4). Suppose that

>0, f, v= 0, v H10 ()n, v= 0 on. (4.3)

Then, as 0,u u

strongly inH10 ()n,

Ae(u) A

e(u) strongly inL2()nn,(4.4)

whereu andu are the solutions of (2.2) and (3.5) respectively.

Theorem 4.3. Letf, fH1()n satisfying (3.4) and (4.3). Then, as(, )(0, 0)

u u0 weakly inH10 ()

n,

Ae(u) A

0e(u0) weakly inL2()nn,(4.5)

whereu0 andu are the solutions of (2.4) and (3.5) respectively.

To prove these results we use similar arguments as those used in [ 3]. Before

giving these proofs, we recall the following lemma:

Lemma 4.4 ([3]). Let {m} be a sequence of L2(). Suppose that there exists

, L2() such that

m strongly inL2loc(),

m N, |m| a.e. in.

Then, m strongly inL2().


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Proof of Theorem4.1. Observe first that (4.1) follows immediately from Proposi-tion3.4. On the other hand, taking in (2.2) and(3.5), f =f

.= div(A0e(x))

with D() and RnnS

, then (2.4) reads

div (A0e(u0)) = div(A0e(x)) in ,

u0 = 0 on .

This implies, in view of the fact that A0 Me( C2 , , ), that u

0 = x. This,together with (4.1), gives

u x strongly inH10 ()n,

Ae(u) A0e(x) strongly inL2()nn.

Taking now D() such that = 1 on and where , one obtains

u strongly inH1()n,

Ae(u) A0

strongly inL2

()

nn

.

(4.6)

On the other hand, one has almost everywhere in ,

|A A0| |A Ae(u)| + |Ae(u) A

0|

| e(u)| + |Ae(u) A0|.

Then, using (4.6), one gets

A A0 strongly inL2()nn,

for every . Since |Am| ||, this gives by Lemma 4.4 written form= Am (with m 0),

A A0 strongly inL2()nn.

By the symmetric properties ofA andA0

, this convergence is still valid for everymatrix Rnn. Thus

A A0 strongly in L2()nn.

From this convergence and the fact that AL() , one obtains convergence(4.2).

Proof of Theorem4.2. From hypothesis (4.3), Proposition 3.3 and the fact thatP(u |

) u = 0 in , it follows that

u| uH1

0()n P

(u| ) PuH1

0()n c

1/2,

Ae(u) Ae(u) L2()nn c

1/2.

Passing to the limit as 0, one obtains (4.4).

Proof of Theorem4.3. (i) Under hypothesis (4.3), Proposition3.3 gives

lim(,)(0,0)

P(u | ) PuH1

0()n = 0

and the fact that AHe A0 implies

Pu u0 0 weakly inH10 ().


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Hence, by passing to the weak limit in H10 () as (, ) (0, 0) in the followingequality:

P

(u|

) u0 = (P

(u|

) P

u) + (P

u u0),

one deducesP(u|

) u0 weakly in H10 ()n. (4.7)

On the other hand, using (4.3) and Proposition3.3, one gets

P(u| ) PuH1

0()n c

1/2,

e(u)L2(T)nn c.

This implies

uH10

()n P(u| )H

1

0()n c

1/2 + PuH10

()n ,

e(u)L2(T)nn c.

Since Pu is bounded independently of in H10 ()n, one deduces that u is

bounded independently of and in H1

0 ()

n

. This, together with (4.7)and Propo-sition2.7,gives(4.5)i).

(ii) Using Proposition3.3, the fact that AHe A0 and

Ae(u) A

0e(u0) = (Ae(u) Ae(u)) + (Ae(u) A0e(u0)),

one obtains the convergence4.5(ii).

Acknowledgments. This work was finalized when the author was visiting theLaboratoire Jacques-Louis Lions at the Paris University Pierre et Marie Curie,supported by a grant of the Algerian Ministry for the higher education and scientificresearch. The author is very thankful for the hospitality during this visit. Theauthor is also grateful to P. Donato for the useful discussions and advice.

References

[1] A. Bensoussan, J. L. Lions, G. Papanicolaou; Asymptotic Analysis for Periodic Structures,North-Holland, Amsterdam, 1978.

[2] M. Briane, A. Damlamian, P. Donato; H-convergence in Perforated Domains, in Non-linear

Partial Differential Equations and Their Applications, College de France seminar vol. XIII,D. Cioranescu & J. L. Lions eds., Pitman Research Notes in Mathematics Series, Longman,

New York, (391) (1998), 62-100.[3] D. Cioranescu, A. Damlamian, P. Donato, L. Mascarenhas; H0-convergence as a limit-case

of H-convergence, Advances in Math. Sci. and Appl., 9, n.1 (1999)319-331.

[4] P. Donato, H. Haddadou; Meyers type estimates in elasticity and applications to H-convergence, To appear in Advances in Math. Sci. and Appl., 16, n. 2 (2006).

[5] M. El Hajji, P. Donato; H0-convergence for the linearized elasticity system, AsymptoticAnalysis 21 (1999), 161-186.

[6] G. A. Francfort, F. Murat; Homoenization and Optimal Bounds in Linear Elasticity, Arch.Rational Mech. Anal.,94 (1986),307-334.

[7] F. Murat;H-convergence, Seminaire dAnalyse Fonctionnelle et Numerique, 1977/1978, Univ.

dAlger, Multigraphed.[8] F. Murat, L. Tartar;H-convergence in Topics in Mathematical Modeling of Composites Ma-

terials, ed. A. Cherkaev and R. Kohn, Birkhauser, Boston, (1997), 21-34.[9] L. Tartar,Cours Peccot au College de France, 1977.

Hamid Haddadou

Institut Nationale de formation en Informatique, BP 68, Oued Smar, El Harrach, Al-

ger, Algerie

E-mail address: h [email protected]

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