- Email: [email protected]
A density result for the variation of a material with respect to small inclusions
C. R. Acad. Sci. Paris, Ser. I 342 (2006) 353–358 http://france.elsevier.com/direct/CRASS1/
Numerical Analysis/Calculus of Variations
A density result for the variation of a material with respect to small inclusions Juan Casado-Díaz, Julio Couce-Calvo, José Domingo Martín-Gómez Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, c/Tarfia s/n, 41012 Sevilla, Spain Received 18 March 2005; accepted after revision 7 December 2005 Available online 18 January 2006 Presented by Olivier Pironneau
Abstract We consider the family of materials obtained, via homogenization, by replacing a small portion, of size ε, of a fixed material by other materials. In a previous paper we have obtained a subset of the set of ‘derivatives’ of this family with respect to ε in ε = 0. In the present Note we prove that this set is, in fact, dense. This result can be applied, for example, to obtain optimality conditions for composite materials. To cite this article: J. Casado-Díaz et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006). 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved. Résumé Un résultat sur la variation d’un matériau en fonction de petites inclusions. On considère une famille de matériaux obtenus par homogénéisation consistant à remplacer une petite partie de matériau, de taille ε, par d’autres matériaux. Dans un article antérieur on a caractérisé un sous-ensemble de l’ensemble des « dérivées », par rapport à ε de cette famille, pour ε = 0. Dans cette Note on démontre que ce sous-ensemble est en fait dense. Le résultat peut être appliqué, par exemple, à l’obtention des conditions d’optimalité pour des matériaux composites. Pour citer cet article : J. Casado-Díaz et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006). 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.
Version française abrégée Il est bien connu que les caractéristiques d’un matériau (par exemple en conductivité électrique ou thermique) sont normalement associées à une matrice A (matrice de diffusion). Quand on considère des matériaux composites, il est intéressant de déterminer les variations de cette matrice quand on change des petites parties du matériau par d’autres matériaux différents. Ce problème se pose, par exemple, dans l’étude des conditions d’optimalité pour A. Dans ce cas on considère habituellement des petites variations de la matrice A (voir [1–3,8–10]). Un autre exemple est la construction des directions de descente qui permettent d’améliorer A en introduisant d’autres matériaux. Puisque le mélange des matériaux est bien caractérisé par la théorie de l’homogénéisation (voir par exemple [1,3,5,7,8,10]) et puisque la H-convergence est un procédé local, la question à poser est la suivante : Pour A, B1 , . . . , Bn matrices définies positives, θ1 , . . . , θn 0, avec ni=1 θi = 1, il s’agit d’obtenir l’ensemble D(A; θ1 , . . . , θn ; B1 , . . . , Bn ) de toutes les matrices D qui sont une limite de la forme E-mail addresses: [email protected] (J. Casado-Díaz), [email protected] (J. Couce-Calvo), [email protected] (J.D. Martín-Gómez). 1631-073X/$ – see front matter 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved. doi:10.1016/j.crma.2005.12.021
354
J. Casado-Díaz et al. / C. R. Acad. Sci. Paris, Ser. I 342 (2006) 353–358
Aε − A , (1) ε→0 ε où, pour chaque ε ∈ (0, 1), Aε est obtenue par homogénéisation, en mélangeant les matériaux correspondants aux matrices A, B1 , . . . , Bn avec des proportions respectives égales à 1 − ε, εθ1 , . . . , εθn . Dans un article antérieur, [2], on a montré (pour n = 1, mais le cas général est analogue) qu’un sous-ensemble de D(A; θ1 , . . . , θn ; B1 , . . . , Bn ) est donné par l’ensemble H(A; θ1 , . . . , θn ; B1 , . . . , Bn ) de toutes les matrices H , telles qu’il existe ω1 , . . . , ωn ⊂ RN , mesurables, disjoints, avec |ωi | = θi , i = 1, . . . , n, et tel que n Hξ = (Bi − A) θi ξ + ∇wξ dz , ∀ξ ∈ RN , D = lim
i=1
ωi
où wξ est la solution (unique à une constante additive près) de n n − div AχRN \ni=1 ωi + Bi χωi ∇wξ = div(Bi − A)χωi ξ i=1
in D RN ,
N ∇wξ ∈ L2 RN .
i=1
L’ensemble H(A; θ1 , . . . , θn ; B1 , . . . , Bn ) a été utilisé dans [2] pour obtenir des conditions d’optimalité sur A. De plus, on a aussi obtenu quelques éléments explicites de H(A; θ1 , . . . , θn ; B1 , . . . , Bn ). L’objectif de cette Note est de montrer le résultat suivant : Théorème 0.1. L’ensemble H(A; θ1 , . . . , θn ; B1 , . . . , Bn ) est dense dans D(A; θ1 , . . . , θn ; B1 , . . . , Bn ). La démonstration utilise un résultat bien connu dû à Dal Maso et Kohn [1,4,5] établissant que les matrices obtenues par homogénéisation périodique sont denses dans l’ensemble obtenu par homogénéisation quelconque. Ceci donne une approximation du quotient qui apparaît dans (1), mais on remarque que cette approximation n’est pas dans D(A; θ1 , . . . , θn ; B1 , . . . , Bn ) et donc, par exemple, elle ne peut pas être utilisée comme direction admissible pour un problème d’optimalité. 1. Introduction and main result It is well known that the characteristics of a material (for example, in heat or electric conductivity problems) are associated to a matrix A (the diffusion matrix). When we deal with composite materials, it is interesting to know the variation of this matrix when we replace a small portion of the material by other materials. This question arises, for example, in the study of optimality conditions for A, where to construct admissible directions, we consider small perturbations of A (see e.g. [1–3,8–10]). We also mention the construction of descent directions which permit to improve a material A by introducing other materials. Since the mixture of materials can be characterized via homogenization (see e.g. [1,3,5,7,8,10]) and this is a local process, themathematical question is the following: Given A, B1 , . . . , Bn definite positive matrices in RN , θ1 , . . . , θn 0, with ni=1 θi = 1, our problem is to obtain the set D(A; θ1 , . . . , θn ; B1 , . . . , Bn ) of all the possible matrices D which can be obtained as a limit of the form Aε − A D = lim , (2) ε→0 ε where, for every ε ∈ (0, 1), Aε is obtained via homogenization, by mixing the materials corresponding to the matrices A, B1 , . . . , Bn with respective proportions 1 − ε, εθ1 , . . . , εθn . In a previous article, [2], we have proved (for n = 1, but the general case is analogous) that a subset of D(A; θ1 , . . . , θn ; B1 , . . . , Bn ) is given by the set H(A; θ1 , . . . , θn ; B1 , . . . , Bn ) of all the matrices H , such that there exist ω1 , . . . , ωn ⊂ RN measurable, disjoint, with |ωi | = θi , i = 1, . . . , n, which satisfy: n Hξ = (Bi − A) θi ξ + ∇wξ dz , ∀ξ ∈ RN , i=1
ωi
where wξ is the solution (it is unique up to an additive constant) of n n − div AχRN \ ni=1 ωi + Bi χωi ∇wξ = div(Bi − A)χωi ξ i=1
i=1
in D RN ,
N ∇wξ ∈ L2 RN .
(3)
J. Casado-Díaz et al. / C. R. Acad. Sci. Paris, Ser. I 342 (2006) 353–358
355
The set H(A; θ1 , . . . , θn ; B1 , . . . , Bn ) has been used in [2] to obtain optimality conditions for A. Some explicit elements of H(A; θ1 , . . . , θn ; B1 , . . . , Bn ) have also been obtained in the mentioned paper. We observe that, using a dilatation, H(A; θ1 , . . . , θn ; B1 , . . . , Bn ) can also be defined as the set of matrices H , such that there exist λ > 0, and ω1 , . . . , ωn ⊂ RN measurable, disjoint, with |ωi | = λθi , i = 1, . . . , n, which satisfy: n 1 Hξ = (Bi − A) θi ξ + ∇wξ dz , (4) λ i=1
ωi
where wξ is the solution of (3). The goal of the present Note is to prove the following result: Theorem 1.1. The set H(A; θ1 , . . . , θn ; B1 , . . . , Bn ) is dense in D(A; θ1 , . . . , θn ; B1 , . . . , Bn ). The proof uses a well known result of Dal Maso and Kohn [1,4,5], which proves that the matrices obtained via periodic homogenization are dense in those which can be obtained by general homogenization. This provides an approximation of the quotient which appears in (2), but we remark that this approximation is not in D(A; θ1 , . . . , θn ; B1 , . . . , Bn ) and thus, for example, it cannot be used as an admissible direction for optimal problems. 2. Proof of Theorem 1.1 In the whole of the proof, we denote by C a generic constant, which can change from a line to another one. The proof is divided in three steps. Step 1. Let D be in D(A; θ1 , . . . , θn ; B1 , . . . , Bn ), and consider a sequence Aε of matrices obtained via homogenization, by mixing the materials corresponding to A, B1 , . . . , Bn with respective proportions 1 − ε, εθ1 , . . . , εθn , and such that (2) is satisfied. Using the result of Dal Maso and Kohn mentioned above, we can assume that for every ε > 0, there ωiε = k∈ZN (ωiε + k) (the exexist ωiε ⊂ Y = (− 12 , 12 )N , measurable, with |ωiε | = εθi , i = 1, . . . , n, such that defining n ε = AχRN \n ε + tension by Y -periodicity of ωiε ⊂ Y to RN ) M ε = AχRN \ni=1 ωε + ni=1 Bi χωiε , M ωiε i=1 Bi χ i=1 ωi i (remark that Mε = Mε in Y ), we have
Aε ξ = Mε ∇vξε + ξ dy, ∀ξ ∈ RN , (5) Y
where
vξε
is the solution of (it is unique up to a constant):
ε ∇vξε = − div M
n
div(Bi − A)χ ωiε ξ
in D RN ,
vξε ∈ H1 (Y ).
(6)
i=1
Here
H1 (Y ) Dε ξ =
1 (RN ), which are Y -periodic. A simple calculation shows denotes the space of functions of Hloc
n Aε − A 1 ξ= (Bi − A) θi ξ + ∇vξε dy . ε ε i=1
(7)
ωiε
Now, we define wξε as the solution of − div M ε ∇wξε =
n
div(Bi − A)χωiε ξ
in D RN ,
N ∇wξε ∈ L2 RN .
(8)
i=1
Then, from (4), the matrix H ε , defined by: n 1 H εξ = (Bi − A) θi ξ + ∇wξε dy , ε i=1
ωiε
(9)
356
J. Casado-Díaz et al. / C. R. Acad. Sci. Paris, Ser. I 342 (2006) 353–358
belongs to H(A; θ1 , . . . , θn ; B1 , . . . , Bn ). In order to prove Theorem 1.1, it is enough to show that (H ε − D ε )ξ tends to zero for every ξ ∈ RN , when ε tends to zero. Step δ > 0 such 2. For every ρ > 0, we denote Yρ = {y ∈ Y : dist(y, ∂Y ) > ρ}. Let us assume that there exists that ni=1 ωiε ⊂ Yδ . Taking into account that wξε satisfies − div A∇wξε = div g ε in D (RN ), with g ε = ni=1 (Bi − A) (∇wξε + ξ )χωiε , and that ∇wξε belongs to L2 (RN )N , we deduce that (a representative of wξε is) wξε = ∇KA ∗ g ε , with KA the fundamental solution of − div A∇. Since wξε satisfies (take wξε as test function in (8)) ∇w ε
ξ L2 (RN )N
√ C ε,
∇KA (z) = µ
Lt Lz , |Lz|N
with µ ∈ R, L ∈ MN non-singular, and the support of g ε is contained in ni=1 ωiε , we deduce: n
ε
ε ε δ
w (z) C , ∇wξε (z) C N , ∀z with dist z, ωiε > . ξ δ 2 δ N −1
(10)
i=1
We take ψ ∈ C0∞ (RN ) a cut-off function such that ψ = 1 in Yδ/2 ,
, supp(ψ) ⊂ Y
From (10) and (11), we deduce: ε
∇ w ψ − w ε 2 N C ξ ξ L (Y )
|∇ψ|
C δ
in RN .
(11)
ε . δ N −1/2
(12)
On the other hand, for every ϕ ∈ H 1 (Y ), we have:
M ε ∇ ψwξε ∇ϕ dy = M ε ∇wξε ∇ ψ ϕ − ϕ dz dy − M ε ∇wξε ∇ψ ϕ − ϕ dz dy Y
Y
+ n
Using (8), and ψ = 1 in n (A − Bi )ξ ∇ϕ dy.
Y
Y
M ε ∇ψ∇ϕ wξε dy. Y
ε i=1 ωi ,
i=1
Y
the first term on the right-hand side is
ωiε
The second and third terms on the right-hand side can be estimated by using (10), (11) and Poincaré–Wirtinger’s inequality. We then get:
n
ε
M ε ∇ ψw ε ∇ϕ dy − (A − Bi )ξ ∇ϕ dy
C N +1/2 ∇ϕ L2 (Y )N , ∀ϕ ∈ H 1 (Y ), ξ
δ i=1
Y
and then, from (6), we get:
M ε ∇ ψw ε − v ε ∇ϕ dy C ξ ξ
ωiε
ε δ N +1/2
∇ϕ L2 (Y )N ,
∀ϕ ∈ H1 (Y ).
Y
Taking here ϕ = ψwξε − vξε , we deduce ∇(ψwξε − vξε ) L2 (Y )N Cεδ −N −1/2 , which joining to (12) shows
2 1
ε ∇ wξ − vξε dy = 0, ∀ξ ∈ RN . lim ε→0 ε Y
From (7) and (9) we then conclude that (H ε − D ε )ξ tends to zero, for every ξ ∈ RN . Step 3. Let us now consider the general case.
(13)
J. Casado-Díaz et al. / C. R. Acad. Sci. Paris, Ser. I 342 (2006) 353–358
357
We fix δ ∈ (0, 1). Since in (5) and (6), ωiε can be replaced by ωiε + a, for every a ∈ RN , then, choosing a appropriately, we can assume that there exists C > 0 which does not depend on ε nor on δ, such that
n
ε (14) ωi ∩ (Y \ Yδ ) Cδε.
i=1
We take ωiε,δ
= ωiε
∩ Yδ ,
M
ε,δ
= AχRN \
n
ε,δ i=1 ωi
+
n
Bi χωε,δ .
i=1
i
ε,δ , the extension by As above, we also denote by ωiε,δ , the extension by Y -periodicity of ωiε,δ ⊂ Y to RN and by M ε,δ ε,δ Y -periodicity of M ε,δ χY to RN . We define wξ , vξ as the solutions of − div M ε,δ ∇wξε,δ = ε,δ ∇v ε,δ = − div M ξ
n
div(Bi − A)χωε,δ ξ
in D RN ,
N ∇wξε,δ ∈ L2 RN ,
(15)
div(Bi − A)χ ξ ωε,δ
in D RN ,
vξε,δ ∈ H1 (Y ).
(16)
i
i=1 n
i
i=1
From Step 2 (see (13)), we know
2 1
ε,δ ∇ wξ − vξε,δ dy = 0, lim ε→0 ε
∀ξ ∈ RN .
(17)
Y
Let us now estimate wξε − wξε,δ . Taking it as test function in the difference of (8) and (15) we get: n
M ε ∇ wξε − wξε,δ ∇ wξε − wξε,δ dx = (A − Bi )∇ wξε + ξ ∇ wξε − wξε,δ dx, i=1 ε ε,δ ωi \ωi
RN
which gives
ε
∇ w − w ε,δ 2 dx C ξ ξ RN
n
2 1 + ∇wξε dx.
ε i=1 (ωi ∩(Y \Yδ ))
From Meyers’ theorem [6] applied to (8), we know that there exists p > 2 such that n ε ε ∇w p N C div(Bi − A)χωε,δ ξ W −1,p (2Y ) + ∇wξ L2 (2Y )N , ξ L (Y ) i i=1 √ √ and as wξε satisfies ∇wξε L2 (RN )N C ε, we deduce ∇wξε Lp (Y )N C p ε. Thus, estimate (18) shows ε
∇ w − w ε,δ 2 N N Cδ (p−2)/(2p) ε 1/2 . ξ ξ L (R ) A similar reasoning proves ε
∇ v − v ε,δ 2 N Cδ (p−2)/(2p) ε 1/2 . ξ ξ L (Y ) These estimates and (17), then prove
lim sup ∇ vξε − vξε,δ L2 (Y )N Cδ (p−2)/(2p) ε 1/2 . ε→0
So, from (7) and (9), we deduce that
lim sup H ε − D ε ξ Cδ (p−2)/(2p) ,
∀δ ∈ (0, 1)
ε→0
and then (H ε − D ε )ξ tends to zero, for every ξ ∈ RN .
(18)
358
J. Casado-Díaz et al. / C. R. Acad. Sci. Paris, Ser. I 342 (2006) 353–358
References [1] G. Allaire, Shape Optimization by the Homogenization Method, Appl. Math. Sci., vol. 146, Springer-Verlag, New York, 2002. [2] J. Casado-Díaz, J. Couce-Calvo, J.D. Martín-Gómez, Optimality conditions for nonconvex multistate control problems in the coefficients, SIAM J. Control Optim. 43 (1) (2004) 216–239. [3] A.V. Cherkaev, Variational Methods for Structural Optimization, Appl. Math. Sci., vol. 140, Springer-Verlag, New York, 2000. [4] G. Dal Maso, R.V. Kohn, The local character of G-closure, Unpublished work. [5] V.V. Jikov, S.M. Kozlov, O.A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994. [6] N.G. Meyers, An Lp -estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 3 (17) (1963) 189–206. [7] F. Murat, L. Tartar, H -convergence, in: A. Cherkaev, R. Kohn (Eds.), Topics in the Mathematical Modelling of Composite Materials, in: Progr. Nonlinear Differential Equations Appl., Birkhäuser, Boston, 1997, pp. 21–44. [8] F. Murat, L. Tartar, Calculus of variations, in: A. Cherkaev, R. Kohn (Eds.), Topics in the Mathematical Modelling of Composite Materials, in: Progr. Nonlinear Differential Equations Appl., Birkhäuser, Boston, 1997, pp. 139–173. [9] O. Pironneau, Optimal Shape Design for Elliptic Systems, Springer-Verlag, New York, 1984. [10] L. Tartar, An Introduction to the homogenization method in optimal design, in: A. Cellina, A. Ornelas (Eds.), Optimal Shape Design, CIM/CIME, Summer School, Trôia, 1–6 June 1998, in: Lecture Notes in Math., vol. 1740, Springer-Verlag, Berlin, 2000, pp. 47–156.
Numerical Analysis/Calculus of Variations
A density result for the variation of a material with respect to small inclusions Juan Casado-Díaz, Julio Couce-Calvo, José Domingo Martín-Gómez Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, c/Tarfia s/n, 41012 Sevilla, Spain Received 18 March 2005; accepted after revision 7 December 2005 Available online 18 January 2006 Presented by Olivier Pironneau
Abstract We consider the family of materials obtained, via homogenization, by replacing a small portion, of size ε, of a fixed material by other materials. In a previous paper we have obtained a subset of the set of ‘derivatives’ of this family with respect to ε in ε = 0. In the present Note we prove that this set is, in fact, dense. This result can be applied, for example, to obtain optimality conditions for composite materials. To cite this article: J. Casado-Díaz et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006). 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved. Résumé Un résultat sur la variation d’un matériau en fonction de petites inclusions. On considère une famille de matériaux obtenus par homogénéisation consistant à remplacer une petite partie de matériau, de taille ε, par d’autres matériaux. Dans un article antérieur on a caractérisé un sous-ensemble de l’ensemble des « dérivées », par rapport à ε de cette famille, pour ε = 0. Dans cette Note on démontre que ce sous-ensemble est en fait dense. Le résultat peut être appliqué, par exemple, à l’obtention des conditions d’optimalité pour des matériaux composites. Pour citer cet article : J. Casado-Díaz et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006). 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.
Version française abrégée Il est bien connu que les caractéristiques d’un matériau (par exemple en conductivité électrique ou thermique) sont normalement associées à une matrice A (matrice de diffusion). Quand on considère des matériaux composites, il est intéressant de déterminer les variations de cette matrice quand on change des petites parties du matériau par d’autres matériaux différents. Ce problème se pose, par exemple, dans l’étude des conditions d’optimalité pour A. Dans ce cas on considère habituellement des petites variations de la matrice A (voir [1–3,8–10]). Un autre exemple est la construction des directions de descente qui permettent d’améliorer A en introduisant d’autres matériaux. Puisque le mélange des matériaux est bien caractérisé par la théorie de l’homogénéisation (voir par exemple [1,3,5,7,8,10]) et puisque la H-convergence est un procédé local, la question à poser est la suivante : Pour A, B1 , . . . , Bn matrices définies positives, θ1 , . . . , θn 0, avec ni=1 θi = 1, il s’agit d’obtenir l’ensemble D(A; θ1 , . . . , θn ; B1 , . . . , Bn ) de toutes les matrices D qui sont une limite de la forme E-mail addresses: [email protected] (J. Casado-Díaz), [email protected] (J. Couce-Calvo), [email protected] (J.D. Martín-Gómez). 1631-073X/$ – see front matter 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved. doi:10.1016/j.crma.2005.12.021
354
J. Casado-Díaz et al. / C. R. Acad. Sci. Paris, Ser. I 342 (2006) 353–358
Aε − A , (1) ε→0 ε où, pour chaque ε ∈ (0, 1), Aε est obtenue par homogénéisation, en mélangeant les matériaux correspondants aux matrices A, B1 , . . . , Bn avec des proportions respectives égales à 1 − ε, εθ1 , . . . , εθn . Dans un article antérieur, [2], on a montré (pour n = 1, mais le cas général est analogue) qu’un sous-ensemble de D(A; θ1 , . . . , θn ; B1 , . . . , Bn ) est donné par l’ensemble H(A; θ1 , . . . , θn ; B1 , . . . , Bn ) de toutes les matrices H , telles qu’il existe ω1 , . . . , ωn ⊂ RN , mesurables, disjoints, avec |ωi | = θi , i = 1, . . . , n, et tel que n Hξ = (Bi − A) θi ξ + ∇wξ dz , ∀ξ ∈ RN , D = lim
i=1
ωi
où wξ est la solution (unique à une constante additive près) de n n − div AχRN \ni=1 ωi + Bi χωi ∇wξ = div(Bi − A)χωi ξ i=1
in D RN ,
N ∇wξ ∈ L2 RN .
i=1
L’ensemble H(A; θ1 , . . . , θn ; B1 , . . . , Bn ) a été utilisé dans [2] pour obtenir des conditions d’optimalité sur A. De plus, on a aussi obtenu quelques éléments explicites de H(A; θ1 , . . . , θn ; B1 , . . . , Bn ). L’objectif de cette Note est de montrer le résultat suivant : Théorème 0.1. L’ensemble H(A; θ1 , . . . , θn ; B1 , . . . , Bn ) est dense dans D(A; θ1 , . . . , θn ; B1 , . . . , Bn ). La démonstration utilise un résultat bien connu dû à Dal Maso et Kohn [1,4,5] établissant que les matrices obtenues par homogénéisation périodique sont denses dans l’ensemble obtenu par homogénéisation quelconque. Ceci donne une approximation du quotient qui apparaît dans (1), mais on remarque que cette approximation n’est pas dans D(A; θ1 , . . . , θn ; B1 , . . . , Bn ) et donc, par exemple, elle ne peut pas être utilisée comme direction admissible pour un problème d’optimalité. 1. Introduction and main result It is well known that the characteristics of a material (for example, in heat or electric conductivity problems) are associated to a matrix A (the diffusion matrix). When we deal with composite materials, it is interesting to know the variation of this matrix when we replace a small portion of the material by other materials. This question arises, for example, in the study of optimality conditions for A, where to construct admissible directions, we consider small perturbations of A (see e.g. [1–3,8–10]). We also mention the construction of descent directions which permit to improve a material A by introducing other materials. Since the mixture of materials can be characterized via homogenization (see e.g. [1,3,5,7,8,10]) and this is a local process, themathematical question is the following: Given A, B1 , . . . , Bn definite positive matrices in RN , θ1 , . . . , θn 0, with ni=1 θi = 1, our problem is to obtain the set D(A; θ1 , . . . , θn ; B1 , . . . , Bn ) of all the possible matrices D which can be obtained as a limit of the form Aε − A D = lim , (2) ε→0 ε where, for every ε ∈ (0, 1), Aε is obtained via homogenization, by mixing the materials corresponding to the matrices A, B1 , . . . , Bn with respective proportions 1 − ε, εθ1 , . . . , εθn . In a previous article, [2], we have proved (for n = 1, but the general case is analogous) that a subset of D(A; θ1 , . . . , θn ; B1 , . . . , Bn ) is given by the set H(A; θ1 , . . . , θn ; B1 , . . . , Bn ) of all the matrices H , such that there exist ω1 , . . . , ωn ⊂ RN measurable, disjoint, with |ωi | = θi , i = 1, . . . , n, which satisfy: n Hξ = (Bi − A) θi ξ + ∇wξ dz , ∀ξ ∈ RN , i=1
ωi
where wξ is the solution (it is unique up to an additive constant) of n n − div AχRN \ ni=1 ωi + Bi χωi ∇wξ = div(Bi − A)χωi ξ i=1
i=1
in D RN ,
N ∇wξ ∈ L2 RN .
(3)
J. Casado-Díaz et al. / C. R. Acad. Sci. Paris, Ser. I 342 (2006) 353–358
355
The set H(A; θ1 , . . . , θn ; B1 , . . . , Bn ) has been used in [2] to obtain optimality conditions for A. Some explicit elements of H(A; θ1 , . . . , θn ; B1 , . . . , Bn ) have also been obtained in the mentioned paper. We observe that, using a dilatation, H(A; θ1 , . . . , θn ; B1 , . . . , Bn ) can also be defined as the set of matrices H , such that there exist λ > 0, and ω1 , . . . , ωn ⊂ RN measurable, disjoint, with |ωi | = λθi , i = 1, . . . , n, which satisfy: n 1 Hξ = (Bi − A) θi ξ + ∇wξ dz , (4) λ i=1
ωi
where wξ is the solution of (3). The goal of the present Note is to prove the following result: Theorem 1.1. The set H(A; θ1 , . . . , θn ; B1 , . . . , Bn ) is dense in D(A; θ1 , . . . , θn ; B1 , . . . , Bn ). The proof uses a well known result of Dal Maso and Kohn [1,4,5], which proves that the matrices obtained via periodic homogenization are dense in those which can be obtained by general homogenization. This provides an approximation of the quotient which appears in (2), but we remark that this approximation is not in D(A; θ1 , . . . , θn ; B1 , . . . , Bn ) and thus, for example, it cannot be used as an admissible direction for optimal problems. 2. Proof of Theorem 1.1 In the whole of the proof, we denote by C a generic constant, which can change from a line to another one. The proof is divided in three steps. Step 1. Let D be in D(A; θ1 , . . . , θn ; B1 , . . . , Bn ), and consider a sequence Aε of matrices obtained via homogenization, by mixing the materials corresponding to A, B1 , . . . , Bn with respective proportions 1 − ε, εθ1 , . . . , εθn , and such that (2) is satisfied. Using the result of Dal Maso and Kohn mentioned above, we can assume that for every ε > 0, there ωiε = k∈ZN (ωiε + k) (the exexist ωiε ⊂ Y = (− 12 , 12 )N , measurable, with |ωiε | = εθi , i = 1, . . . , n, such that defining n ε = AχRN \n ε + tension by Y -periodicity of ωiε ⊂ Y to RN ) M ε = AχRN \ni=1 ωε + ni=1 Bi χωiε , M ωiε i=1 Bi χ i=1 ωi i (remark that Mε = Mε in Y ), we have
Aε ξ = Mε ∇vξε + ξ dy, ∀ξ ∈ RN , (5) Y
where
vξε
is the solution of (it is unique up to a constant):
ε ∇vξε = − div M
n
div(Bi − A)χ ωiε ξ
in D RN ,
vξε ∈ H1 (Y ).
(6)
i=1
Here
H1 (Y ) Dε ξ =
1 (RN ), which are Y -periodic. A simple calculation shows denotes the space of functions of Hloc
n Aε − A 1 ξ= (Bi − A) θi ξ + ∇vξε dy . ε ε i=1
(7)
ωiε
Now, we define wξε as the solution of − div M ε ∇wξε =
n
div(Bi − A)χωiε ξ
in D RN ,
N ∇wξε ∈ L2 RN .
(8)
i=1
Then, from (4), the matrix H ε , defined by: n 1 H εξ = (Bi − A) θi ξ + ∇wξε dy , ε i=1
ωiε
(9)
356
J. Casado-Díaz et al. / C. R. Acad. Sci. Paris, Ser. I 342 (2006) 353–358
belongs to H(A; θ1 , . . . , θn ; B1 , . . . , Bn ). In order to prove Theorem 1.1, it is enough to show that (H ε − D ε )ξ tends to zero for every ξ ∈ RN , when ε tends to zero. Step δ > 0 such 2. For every ρ > 0, we denote Yρ = {y ∈ Y : dist(y, ∂Y ) > ρ}. Let us assume that there exists that ni=1 ωiε ⊂ Yδ . Taking into account that wξε satisfies − div A∇wξε = div g ε in D (RN ), with g ε = ni=1 (Bi − A) (∇wξε + ξ )χωiε , and that ∇wξε belongs to L2 (RN )N , we deduce that (a representative of wξε is) wξε = ∇KA ∗ g ε , with KA the fundamental solution of − div A∇. Since wξε satisfies (take wξε as test function in (8)) ∇w ε
ξ L2 (RN )N
√ C ε,
∇KA (z) = µ
Lt Lz , |Lz|N
with µ ∈ R, L ∈ MN non-singular, and the support of g ε is contained in ni=1 ωiε , we deduce: n
ε
ε ε δ
w (z) C , ∇wξε (z) C N , ∀z with dist z, ωiε > . ξ δ 2 δ N −1
(10)
i=1
We take ψ ∈ C0∞ (RN ) a cut-off function such that ψ = 1 in Yδ/2 ,
, supp(ψ) ⊂ Y
From (10) and (11), we deduce: ε
∇ w ψ − w ε 2 N C ξ ξ L (Y )
|∇ψ|
C δ
in RN .
(11)
ε . δ N −1/2
(12)
On the other hand, for every ϕ ∈ H 1 (Y ), we have:
M ε ∇ ψwξε ∇ϕ dy = M ε ∇wξε ∇ ψ ϕ − ϕ dz dy − M ε ∇wξε ∇ψ ϕ − ϕ dz dy Y
Y
+ n
Using (8), and ψ = 1 in n (A − Bi )ξ ∇ϕ dy.
Y
Y
M ε ∇ψ∇ϕ wξε dy. Y
ε i=1 ωi ,
i=1
Y
the first term on the right-hand side is
ωiε
The second and third terms on the right-hand side can be estimated by using (10), (11) and Poincaré–Wirtinger’s inequality. We then get:
n
ε
M ε ∇ ψw ε ∇ϕ dy − (A − Bi )ξ ∇ϕ dy
C N +1/2 ∇ϕ L2 (Y )N , ∀ϕ ∈ H 1 (Y ), ξ
δ i=1
Y
and then, from (6), we get:
M ε ∇ ψw ε − v ε ∇ϕ dy C ξ ξ
ωiε
ε δ N +1/2
∇ϕ L2 (Y )N ,
∀ϕ ∈ H1 (Y ).
Y
Taking here ϕ = ψwξε − vξε , we deduce ∇(ψwξε − vξε ) L2 (Y )N Cεδ −N −1/2 , which joining to (12) shows
2 1
ε ∇ wξ − vξε dy = 0, ∀ξ ∈ RN . lim ε→0 ε Y
From (7) and (9) we then conclude that (H ε − D ε )ξ tends to zero, for every ξ ∈ RN . Step 3. Let us now consider the general case.
(13)
J. Casado-Díaz et al. / C. R. Acad. Sci. Paris, Ser. I 342 (2006) 353–358
357
We fix δ ∈ (0, 1). Since in (5) and (6), ωiε can be replaced by ωiε + a, for every a ∈ RN , then, choosing a appropriately, we can assume that there exists C > 0 which does not depend on ε nor on δ, such that
n
ε (14) ωi ∩ (Y \ Yδ ) Cδε.
i=1
We take ωiε,δ
= ωiε
∩ Yδ ,
M
ε,δ
= AχRN \
n
ε,δ i=1 ωi
+
n
Bi χωε,δ .
i=1
i
ε,δ , the extension by As above, we also denote by ωiε,δ , the extension by Y -periodicity of ωiε,δ ⊂ Y to RN and by M ε,δ ε,δ Y -periodicity of M ε,δ χY to RN . We define wξ , vξ as the solutions of − div M ε,δ ∇wξε,δ = ε,δ ∇v ε,δ = − div M ξ
n
div(Bi − A)χωε,δ ξ
in D RN ,
N ∇wξε,δ ∈ L2 RN ,
(15)
div(Bi − A)χ ξ ωε,δ
in D RN ,
vξε,δ ∈ H1 (Y ).
(16)
i
i=1 n
i
i=1
From Step 2 (see (13)), we know
2 1
ε,δ ∇ wξ − vξε,δ dy = 0, lim ε→0 ε
∀ξ ∈ RN .
(17)
Y
Let us now estimate wξε − wξε,δ . Taking it as test function in the difference of (8) and (15) we get: n
M ε ∇ wξε − wξε,δ ∇ wξε − wξε,δ dx = (A − Bi )∇ wξε + ξ ∇ wξε − wξε,δ dx, i=1 ε ε,δ ωi \ωi
RN
which gives
ε
∇ w − w ε,δ 2 dx C ξ ξ RN
n
2 1 + ∇wξε dx.
ε i=1 (ωi ∩(Y \Yδ ))
From Meyers’ theorem [6] applied to (8), we know that there exists p > 2 such that n ε ε ∇w p N C div(Bi − A)χωε,δ ξ W −1,p (2Y ) + ∇wξ L2 (2Y )N , ξ L (Y ) i i=1 √ √ and as wξε satisfies ∇wξε L2 (RN )N C ε, we deduce ∇wξε Lp (Y )N C p ε. Thus, estimate (18) shows ε
∇ w − w ε,δ 2 N N Cδ (p−2)/(2p) ε 1/2 . ξ ξ L (R ) A similar reasoning proves ε
∇ v − v ε,δ 2 N Cδ (p−2)/(2p) ε 1/2 . ξ ξ L (Y ) These estimates and (17), then prove
lim sup ∇ vξε − vξε,δ L2 (Y )N Cδ (p−2)/(2p) ε 1/2 . ε→0
So, from (7) and (9), we deduce that
lim sup H ε − D ε ξ Cδ (p−2)/(2p) ,
∀δ ∈ (0, 1)
ε→0
and then (H ε − D ε )ξ tends to zero, for every ξ ∈ RN .
(18)
358
J. Casado-Díaz et al. / C. R. Acad. Sci. Paris, Ser. I 342 (2006) 353–358
References [1] G. Allaire, Shape Optimization by the Homogenization Method, Appl. Math. Sci., vol. 146, Springer-Verlag, New York, 2002. [2] J. Casado-Díaz, J. Couce-Calvo, J.D. Martín-Gómez, Optimality conditions for nonconvex multistate control problems in the coefficients, SIAM J. Control Optim. 43 (1) (2004) 216–239. [3] A.V. Cherkaev, Variational Methods for Structural Optimization, Appl. Math. Sci., vol. 140, Springer-Verlag, New York, 2000. [4] G. Dal Maso, R.V. Kohn, The local character of G-closure, Unpublished work. [5] V.V. Jikov, S.M. Kozlov, O.A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994. [6] N.G. Meyers, An Lp -estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 3 (17) (1963) 189–206. [7] F. Murat, L. Tartar, H -convergence, in: A. Cherkaev, R. Kohn (Eds.), Topics in the Mathematical Modelling of Composite Materials, in: Progr. Nonlinear Differential Equations Appl., Birkhäuser, Boston, 1997, pp. 21–44. [8] F. Murat, L. Tartar, Calculus of variations, in: A. Cherkaev, R. Kohn (Eds.), Topics in the Mathematical Modelling of Composite Materials, in: Progr. Nonlinear Differential Equations Appl., Birkhäuser, Boston, 1997, pp. 139–173. [9] O. Pironneau, Optimal Shape Design for Elliptic Systems, Springer-Verlag, New York, 1984. [10] L. Tartar, An Introduction to the homogenization method in optimal design, in: A. Cellina, A. Ornelas (Eds.), Optimal Shape Design, CIM/CIME, Summer School, Trôia, 1–6 June 1998, in: Lecture Notes in Math., vol. 1740, Springer-Verlag, Berlin, 2000, pp. 47–156.