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-CONVERGENCE AND H-CONVERGENCE OF LINEAR ELLIPTIC OPERATORS

NADIA ANSINI, GIANNI DAL MASO, AND CATERINA IDA ZEPPIERI

Abstract. We consider a sequence of linear Dirichlet problems as follows(div(u) = fin ,

u H1

0(),

with () uniformly elliptic and possibly non-symmetric. Using purely variational arguments we give

an alternative proof of the compactness ofH-convergence, originally proved by Murat and Tartar.

Keywords: linear elliptic operators, -convergence,H-convergence.

2000 Mathematics Subject Classification: 35J15, 35J20, 49J45.

1. Introduction

The notion of H-convergence was introduced by Murat and Tartar in [10, 12] to study a wide classof homogenization problems for possibly non-symmetric elliptic equations. Let L

(;Rnn) be asequence of matrices satisfying uniform ellipticity and boundedness conditions on a bounded open set Rn. We say that H-converges to matrix 0 L

(;Rnn) satisfying the same ellipticity andboundedness conditions if for every f H1() the sequenceu of the solutions to the problems

div(u) = f in ,

u H10 (),

(1.1) Pe

satisfy

u u0 weakly in H1

0 () and u 0u0 weakly in L2

(;Rn

),where u0 is the solution to

div(0u0) = f in ,

u0 H10 ().

The notion of -convergence was introduced by De Giorgi and Franzoni in [5, 6] to study the asymptoticbehavior of the solutions of a wide class of minimization problems depending on a parameter >0, whichvaries in a sequence converging to 0. Let (X, d) be a metric space and let F : X R be a sequence offunctionals, we say that F (d)-converges to a functional F0 : X Rif for all x Xwe have

(i) (liminf inequality) for every sequencexd

x in X

F0(x) liminf0

F(x);

(ii) (limsup inequality) there exists a sequence xd

x in Xsuch that

F0(x) lim sup0

F(x).

Preprint SISSA 04/2012/M (March 2012).

1

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2 N. ANSINI, G. DAL MASO AND C.I. ZEPPIERI

It has been proved that when is symmetric, the equation (1.1) has a variational structure since itcan be seen as the Euler-Lagrange equation associated with

F(u) =1

2 (x)u u dx

f udx,

or, equivalently, as the solution to the minimization problem

min{F(u) : u H10 ()}. (1.2) i:min

Therefore, in this case (1.2) provides a variational principle for the Dirichlet problem (1.1) and theconvergence of the solutions of (1.1) can be equivalently studied by means of the -convergence, withrespect to the weak topology ofH10 (), of the associated functionals F or in terms of the G-convergenceof the uniformly elliptic, symmetric matrices () (see De Giorgi and Spagnolo [7]).

In this paper we consider the equivalence between H-convergence and -convergence in the possiblynon-symmetric case. To every elliptic matrix L(;Rnn) we associate a suitable quadratic integralfunctionalF :L2(;Rn) H10 () [0, +) (see (2.12)) and we consider the -convergence with respectto the distanced defined by

d((, ), (, )) = H1(;Rn)+ div( )H1()+ L2() .

We prove (Theorem 3.2) that the H-convergence of to 0 is equivalent to the (d)-convergence ofthe functionals F corresponding to to the functional F0 corresponding to 0. In [2] this result wasproved using compactness properties ofH-convergence [10, 12], while in the present paper the equivalenceis obtained as a consequence of a general compactness theorem for integral functionals with respect to(d)-convergence [1]. Moreover, as a consequence of the results proved in [1], we also give an independentproof (Theorem 3.1) of the compactness ofH-convergence based only on -convergence arguments.

2. Notation and preliminaries

In this section we introduce a few notation and we recall some preliminary results we employ in thesequel. For any A Rnn we denote by As and Aa the symmetric and the anti-symmetric part of A,respectively;i.e.,

As := A + AT

2

, Aa := A AT

2

,

where AT is the transpose matrix ofA. We use bold capital letters to denote matrices in R2n2n. Thescalar product of two vectors and is denoted by .

Let be an open bounded subset of Rn. For 0 < c0 c1 < +, M(c0, c1, ) denotes the set ofmatrix-valued functions L(; Rnn) satisfying

(x) c0||2, 1(x) c11 ||

2, for every Rn, for a.e. x , (2.1) ellipti

or, equivalently, satisfying

(x) c0||2, (x) c11 |(x)|

2, for every Rn, for a.e. x . (2.2) ellipti

Note that (2.1) (or (2.2)) implies that

|(x)| c1 for a.e. x ,

and that necessarily c0 c1. To not overburden notation, in all that follows we always write in placeof(x).

Given M(c0, c1, ) we consider the (2n 2n)-matrix-valued function L(; R2n2n) havingthe following block structure

:=

(s)1 (s)1a

a(s)1 s a(s)1a

. (2.3) Sigma

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-CONVERGENCE AND H-CONVERGENCE OF LINEAR ELLIPTIC OPERATORS 3

Notice that is symmetric. Moreover, the assumption M(c0, c1, ) easily implies that is uniformlycoercive (see [2, Section 3.1.1] for the details); specifically, there exists a constantC(c0, c1)> 0, dependingonly on c0 and c1, such that

w w C(c0, c1)|w|2, (2.4) unif-co

for every w R2n

, and a.e. in .If we consider the matrix-valued functions A, B, C L(;Rnn) defined as

A= (s)1, B= (s)1a, C= s a(s)1a, (2.5) notaz-c

the matrix can be rewritten as

=

A B

BT C

. (2.6) Sigma-c

We notice that, for a.e. x , the matrix belongs to the indefinite special orthogonal groupS O(n, n);i.e.,

J= J a.e. in , with J=

0 II 0

, (2.7) prop_si

where I Rnn is the identity matrix (see [2, Section 3.1.1]). Moreover, taking into account the symmetryof, it is immediate to show that (2.7) is equivalent to the following system of identities for the block

decomposition (2.6):

ABT + BA = 0

AC+ B2 =I ,

CB + BTC= 0

a.e. in . (2.8) conditi

Conversely, one can prove that, ifM L(;R2n2n) is symmetric and has the block decomposition

M=

A B

BT C

, (2.9) blockM

with A, B, C L(; Rnn), A and C symmetric, and det A = 0, then the first two equations in(2.8) imply the third one, and (2.8) implies that M is equal to the matrix defined in (2.3) with= A1 A1B (see [2, Proposition 3.1]).

Throughout the paper the parameter varies in a strictly decreasing sequence of positive real num-bers converging to zero. Let () be a sequence in M(c0, c1, ) and consider the sequence () L(;R2n2n) defined by (2.3) with = . Let Q : L

2(;Rn) L2(;Rn) [0, +) be the qua-dratic forms associated with ; i.e.,

Q(a, b) :=

a

b

a

b

dx. (2.10) general

Their gradients grad Q : L2(; Rn) L2(;Rn) L2(;Rn) L2(;Rn) are given by

grad Q(a, b) = (Aa + Bb, BTa + Cb), (2.11) differen

whereA, B, andC are as in (2.5) with = . We also consider the quadratic forms F : L2(;Rn)

H10 () [0, +) defined byF(, ) := Q(, ) (2.12) Feps

For every, H1

(); we consider the sequence of constrained functionalsF, : L

2

(;Rn

)H10 ()

[0, +] defined as follows

F, (, ) :=

F(, ) , if div= ,

+ otherwise,(2.13) constr-

where, denotes the dual paring between H1() and H10 ().

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Given a symmetric matrix M L(;R2n2n), we consider the quadratic functionals QM :L2(;Rn)L2(;Rn) [0, +) and FM :L2(; Rn) H10 () [0, +) defined by

QM(a, b) := M a

b a

b dx and FM(, ) := QM(, ) . (2.14) QM-FMConsidering the block decomposition (2.9), the gradient ofQM is given by

grad QM(a, b) = (Aa + Bb,BTa + Cb) . (2.15) differen

Finally, for every , H1(); we consider the constrained functional F,M

: L2(;Rn) H10 () [0, +] defined as follows

F,M

(, ) :=

FM(, ) , if div= ,

+ otherwise.

Let w be the weak topology of L2(;Rn) H10 () and let d be the distance in L2(;Rn) H10 ()

defined by

d((, ), (, )) = H1(;Rn)+ div( )H1()+ L2() .

The following result is proved in [1, Corollary 2.9].

-conv-Qe Theorem 2.1. Let()be a sequence inM(c0, c1, ). There exist a subsequences of, not relabeled, anda symmetric matrixM L(; R2n2n), such that the functionalsF defined by (2.12) (d)-converge tothe functionalFM defined in (2.14). Moreover, M is positive definite and satisfies the coercivity condition(2.4).

The following result is a consequence of [1, Theorem 3.3] and of the stability of -convergence undercontinuous perturbations.

100 Theorem 2.2. Let () be a sequence in M(c0, c1, ) and let M L(;R2n2n) be a symmetric,

positive define matrix satisfying(2.4). Assume that the functionalsF defined by (2.12) (d)-converge to

the functional FM defined in (2.14). Then, for every , H1(), the functionals (F, ) defined by(2.13)(w)-converges to the functionalF, defined by

F,(, ) :=

FM(, ) , if div= ,

+ otherwise.

For the readers sake, here we briefly recall a fundamental tool we employ in what follows, theCherkaev-Gibiansky variational principle [3] (see also Fannjiang-Papanicolaou [8] and Milton [9]) , whichwill be presented in the notational setting which is suitable for our purposes. Loosely speaking, thisvariational principle amounts to associate to the two following Dirichlet problems

div(u) = f in ,u H

10 (),

div(Tv) = g in ,v H

10 ().

(2.16) EL-sys

with f, g H1(), a quadratic functional whose Euler-Lagrange equation is solved by a suitable com-bination of solutions to (2.16) and of their momenta. We set

a := u and b:= Tv . (2.17) abeps

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weakly in L2(; Rn) L2(;Rn). By (2.11) and (2.15), considering only the first component, we get

A(a+ b) + B(u v) A(a0+ b0) + B(u0 v0), (3.6)

A(a b) + B(u+ v) A(a0 b0) + B(u0+ v0) (3.7)

weakly in L2

(;Rn

). Since by (2.5) A(a + b) +B(u v) = u + v and A(a b) +B(u+ v) = u v, from (3.6) and (3.7) we deduce that

u+ v A(a0+ b0) + B(u0 v0), (3.8)

u v A(a0 b0) + B(u0+ v0) (3.9)

weakly in L2(;Rn). Hence, adding up (3.8) and (3.9) entails u Aa0+ Bu0 in L2(; Rn), which

gives u0 = Aa0+ Bu0 by (3.1). This implies

a0 = 0 u0, (3.10) a*equal

with

0 := A1 A1B. (3.11) sigma

Since diva = f, by (2.16) and (2.17) we get that diva0 = f. Hence, (3.10) implies that u0 is the

solution to

div(0u0) = f in ,

u0 H10 ().

(3.12) Dirichl

So far we have proved that for every f H1() the solutions u of (2.16) converge weakly in H10 ()

to the solution u0 of (3.12) and their momenta u converge weakly inL2(;Rn) to0u0. Thus, to

conclude the proof of the H-convergence of () to0 it remains to show that 0 belongs toM(c0, c1, ).To this end, let u H10 () and choose

f := div(0u); (3.13) fw

in this way the solution u0 of the equation (3.12) coincides with u.Let Cc (). Using u as a test function in the equation div(u) =fand then passing to

the limit on we get

f u0 dx= lim0

fu dx= lim0

(

u u) dx

+

0u0 u0dx, (3.14) test1

where to compute the limit of the last term in (3.14) we appealed to the strong L2() convergence ofuto u0. On the other hand, since by (3.12)

f u0 dx=

(0u0 u0) dx +

0u0 u0 dx

from (3.14) we deduce that

lim0

(u u) dx=

(0u0 u0)dx, (3.15) div-cur

for every Cc (). Hence, choosing 0, combining (3.15), the first condition in (2.1), and theequality u = u0, we have

(0u u) dx c0liminf0

|u|2 dx c0

|u|2dx,

the second inequality following from u u0 = uin L2(;Rn). Since this inequality holds true for

every Cc (), 0, we get that

0u u c0|u|2 a.e. in , (3.16) ellip-1

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for every u H10 (). Using the second condition in (2.2), we find

(0u u) dx c11 liminf

0

|u|2 dx c11

|0u|2dx,

since

u

0

u0

= 0

u in L2(;Rn). From the previous inequality we deduce

0u u c11 |0u|

2 a.e. in , (3.17) ellip-2

for every u H10 (). Finally, (2.2) follows from (3.16) and (3.17) by taking u to be affine in an open set .

We now prove that T H-converges to T0 . Subtracting (3.9) from (3.8) gives v Ab0 Bv0

weakly in L2(; Rn); the latter combined with (3.1) imply thatv0 = Ab Bv0. We deduce then

b0 = v0, (3.18) conv-tr

where

:= A1 + A1B. (3.19) sigmati

Sincedivb = g by (2.16) and (2.17), we get divb0 =g , so that (3.18) implies that v0 is the solution

to div(v0) = g in ,

v0 H10 ().

(3.20) Dirichl

As in the previous part of the proof, this implies that T H-converges to . We want to prove that= T0.

To this end, we argue as in the previous step. Let u, v H10 (). We choose f :=div(0u) andg:= div(v) and we consider the corresponding solutions u and v of (2.16). Since u coincides withthe solution u0 of (3.12) and v coincides with the solution v0 of (3.20), the H-convergence of entails

u u0 = u weakly inH10 () and u 0u0 = 0u weakly inL

2(;Rn),

while the H-convergence of (T) yields

v v0 = v weakly inH10 () and

Tv v0 = v weakly inL

2(; Rn).

Let Cc (); using v as test function in the equation for u, we get

f(v) dx=

(u v) dx +

u vdx.

Therefore, appealing to the strong L2() convergence ofv to v and usingv as a test function in (3.12),we obtain

lim0

(u v) dx=

f(v) dx

0u v dx

=

0u (v) dx

0u v dx=

(0u v)dx. (3.21)

Moreover, arguing in a similar way, using now u as test function in the equation for v, it is easy to

show thatlim0

(Tv u) dx=

(v u)dx. (3.22) lim2

Then (3.21) and (3.22) yield

(0u v) dx=

(u v) dx for every Cc ().

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Arguing as in the previous proof of (2.2) we deduce from this equality that

0 = a.e. in ,

for every , Rn. This implies that = T0 a.e. in which concludes the proof of the theorem.

Given 0 M(c0, c1, ), the matrix 0 and the functionals Q0, F0, and F,0 are defined as in (2.6),(2.10), (2.12), and (2.13) with = 0.

nce-Theo Theorem 3.2. Let()be a sequence inM(c0, c1, )and let0 M(c0, c1, ). The following conditionsare equivalent:

(a) H-converges to 0;(b) T H-converges to

T0;

(c) F (d)-converges to F0;

(d) F, (w) to F,0 for every, H

1().

Proof. The equivalence between (a) and (b) follows immediately from Theorem 3.1. The implication (c) (d) is given by Theorem 2.2. The implication (d) (a) is obtained in the proof of Theorem 3.1. Itremains to prove that (a) and (b) imply (c). By Theorem 2.1 we may assume that F (d)-converges to

FM whereM L

(;R2n2n

) is a positive definite, symmetric matrix satisfying the coercivity condition(2.4).

To prove that M S O(n, n) we consider the block decomposition (2.9). In Theorem 3.1 we provedthat 0 = A

1 A1B and T0 = = A1 + A1B; hence, we immediately deduce that

ABT + BA = 0 a.e. in . (3.23) rel1

It remains to prove the second condition in (2.8). Let us fix f , g H1() and let u, v, a, b be as in(2.16) and (2.17). By (2.11), (2.15), (3.4), and (3.5) using only the second component we get

BT(a+ b) + C(u v) = u Tv B

T(a0+ b0) + C(u0 v0) (3.24)

BT(a b) + C(u+ v) = u+ Tv B

T(a0 b0) + C(u0 v0) (3.25)

weakly in L2(; Rn). Then, adding up (3.24) and (3.25) we get

u B

T

a0+ Cu0weakly in L2(; Rn); on the other hand, since u= a a0 weakly in L

2(;Rn), we obtain

a0= BTa0+ Cu0.

Since in the proof of Theorem 3.1 we already showed that a0 = (A1 A1B)u0, we finally obtain

(I BT)(A1 A1B) u0 = Cu0 a.e. in .

Therefore, suitably choosing fas in (3.13) and arguing as in the proof of Theorem 3.1 we can easilydeduce that

(I BT)(A1 A1B) = C a.e. in , for every Rn,

thus, by the arbitrariness of Rn, we get

(I BT)(A1 A1B) = C a.e. in .

The latter combined with (3.23) leads to

AC+ B2 =I a.e. in . (3.26) rel2

Eventually, by (3.23) and (3.26) we can apply [2, Proposition 3.1] and we deduce that M SO(n, n) a.e.in and that M is equal to the matrix defined in (2.3) with = A1 A1B. Since we have also0 = A

1 A1B, we conclude that M = 0.

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Acknowledgments. This material is based on work supported by the Italian Ministry of Education,University, and Research under the Project Variational Problems with Multiple Scales 2008 and by theEuropean Research Council under Grant No. 290888 Quasistatic and Dynamic Evolution Problems inPlasticity and Fracture.

References

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AZ [2] N. Ansini and C. I. Zeppieri, Asymptotic analysis of non-symmetric linear operators via -convergence.SIAM J. Math.

Anal. (2012) to appear.

CG [3] A. V. Cherkaev and L. V. Gibiansky, Variational principles for complex conductivity, viscoelasticity, and similar prob-

lems in media with complex moduli. J. Math. Phys. (1), 35(1994), 127-145.

DM [4] G. Dal Maso, An Introduction to -convergence, Birkhauser, Boston, 1993.

DeG [5] E. De Giorgi, Sulla convergenza di alcune successioni di integrali del tipo dellarea.Rend. Mat. 8 (1975), 277294.

DeGF [6] E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei Rend. Cl. Sci. Mat.Fis. Natur. 58(1975), 842850.

DeGS [7] E. De Giorgi and S. Spagnolo, Sulla convergenza degli integrali dellenergia per operatori ellittici del secondo ordine.

Boll. Un. Mat. Ital. 8(1973), 391411.

FP [8] A. Fannjiang and G. Papanicolaou, Convection enhanced diffusion for periodic flows.SIAM J. Appl. Math. 54 (1994),

333408.M [9] G. W. Milton, On characterizing the set of possible effective tensors of composites: the variational method and the

translation method. Comm. Pure Appl. Math. 43(1990), no. 1, 63-125.

Mu [10] F. Murat,H-convergence. Seminaire dAnalyse Fonctionelle et Numerique de lUniversite dAlger, 1977.

T76 [11] L. Tartar, Quelques remarques sur lhomogeneisation. Proc. of Japan-France seminar1976 Functional analysis andnumerical analysis, 469482, Japan Society for the Promotion of Science, 1978.

T77 [12] L. Tartar, Cours Peccot au College de France. Paris 1977.

T [13] L. Tartar, The general theory of homogenization. A personalized introduction. Lecture Notes of the Unione MatematicaItaliana, 7. Springer-Verlag, Berlin; UMI, Bologna, 2009.

(Nadia Ansini) Dipartimento di Matematica G. Castelnuovo, Sapienza Universita di Roma, Piazzale AldoMoro 2, 00185 Roma, Italy

E-mail address, Nadia Ansini: [email protected]

(Gianni Dal Maso) SISSA, Via Bonomea 265, 34136 Trieste, Italy

E-mail address, Gianni Dal Maso: [email protected]

(Caterina Ida Zeppieri) Institut fur Angewandte Mathematik, Universitat Bonn, Endenicher Allee 60, 53115Bonn, Germany

E-mail address, Caterina Ida Zeppieri: [email protected]

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