Relaxation through Homogenization for Optimal Design Problems with Gradient Constraints 1

Relaxation through Homogenization for Optimal Design Problems with Gradient Constraints

¹

R. L^IPTON2

Communicated by K. A. Lurie

Abstract. The problem of the relaxation of optimal design problems for multiphase composite structures in the presence of constraints on the gradient of the state variable is addressed. A relaxed formulation for the problem is given in the presence of either a finite or infinite number of constraints. The relaxed formulation is used to identify mini- mizing sequences of configurations of phases.

Key Words. Gradient constraints, relaxation, optimal design.

1. Introduction

In the study of the optimal design of composite structures, much effort has focused on the problems of optimization of design criteria in the absence of constraints on the gradients of the field variables. In this article, a theory is developed for the relaxation of optimal design problems for multiphase composite structures in the presence of constraints on the gradient. An example of where such a theory would be useful is the design of a multiphase insulator with a prescribed overall dielectric constant subject to the con- straint that, inside the composite, the electric field does not exceed a critical value. This type of problem is important in the applications (Ref. 1), where large values of the local electric field are attributed to material breakdown (Ref. 2).

To fix the ideas, the paper is written in the physical context of multi- phase dielectric materials. The design domain is a bounded open set Ω in R^m and the multiphase dielectric is made up of N anisotropic dielectric materials with dielectric constants given by the mBm symmetric positive

1This work was supported by NSF Grant DMS-0072469 and by Air Force Office of Scientific Research Grant F49620-99-1-0009.

2Professor, Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana.

27

0022-3239兾02兾0700-0027兾02002 Plenum Publishing Corporation

matrices A1, A2, A3, . . . , AN. A particular choice of the component dielec- tric constants is specified by the array

Aq G(A1, A2, A3, . . . , AN)

and the dielectric constant at each point in the design domain is given by C (Aq, x)G∑

i G1

N χi(x)Ai.

Here, χiis the indicator function of the set occupied by the ith dielectric phase, with χiG1 for x in the ith phase and χiG0 elsewhere. The design space consists of all partitions of Ω into N Lebesgue measurable subsets Ω1,Ω2, . . . ,ΩN occupied by the different dielectrics subject to the resource constraints

meas(Ωi)⁄γi. Here,

∑

i

γi¤meas(Ω),

and the vector of resource constraints is written as γq G(γ1,γ2, . . . ,γN).

The electric potential in the composite is denoted byφ and, for a prescribed distributed charge density, f satisfies

−div(C(Aq, x)∇φ)Gf. (1) Here, the potentialφ is zero on the boundary ofΩ. From the theory of the Poisson equation, it is well known that the potential lies in the Sobolev space H0¹(Ω) and f can be chosen from the dual space H⁻¹(Ω). The design criteria is a function of the potential and is denoted by F(φ). The types of criteria considered here are those associated with overall energy dissipation, given by

F (φ)G冮Ωφf dx,

or the distance of the potential from a desired target potentialφˆ , given by F (φ^)G冮Ω兩φ^Aφˆ 兩²dx.

Both of these objective functions are continuous with respect to weak con- vergence in H0¹(Ω). The results given here are not restricted to this situation

and apply to any design criteria that is continuous with respect to G-conver- gent sequences of designs (Ref. 3).

For a given tolerance KH0 and a prescribed nonnegative function p(x), the basic gradient constraint is given by

冮Ω

p(x)兩∇φ兩²dx ⁄ K². (2)

For simplicity, p(x) is chosen to be in the space of infinitely differentiable functions continuous on the closure ofΩ, i.e., p(x) in C^S(Ω¯ ). For a finite collection of functions p1, p2, . . . , pj, one has the set of constraints given by

冮Ω

pk(x)兩∇φ兩²dx ⁄ K², pk(x) ¤ 0, k G1, 2, . . . , j. (3) Last, one may consider the infinite number of constraints given by

冮Ω

p(x)兩∇φ兩²dx ⁄ K²冮Ω

p(x) dx, p(x) ¤ 0 in C^S(Ω¯ ), (4) which is equivalent to the local constraint

兩∇φ兩⁄K, almost everywhere inΩ. (5)

The basic gradient-constrained optimal design problem is given by P G inf

configurations, meas(Ωi)⁄γi

F (φ), (6)

where φ is a solution of the equation of state (1) and is subject to the gradient constraint given by (2). Optimal design problems subject to either a finite or an infinite number of gradient constraints are formulated in the same way.

The goal of this analysis is to present a theory of relaxation for these problems that provides the theoretical underpinnings for a numerical approach to the solution of the design problem. The problem as stated above is not readily amenable to numerical solution. The fundamental rea- son for this is that problems of this type do not possess configurations for which the infimum in (6) is obtained; see Refs. 4–6. Thus, any approach that seeks to identify optimal configurations is most likely to fail. Instead, one must seek a methodology for systematically identifying minimizing sequences of configurations that approach the infimum of (6).

In the absence of gradient constraints, problems of this kind have been the object of intense and fruitful study. One approach to these problems is the relaxation of the design problem through the extension of the design

space. This is accomplished using the theory of homogenization. This approach can be found in the seminal work of Ref. 7 and Ref. 8. The approach taken here follows the philosophy given in Ref. 7 and Ref. 8 and provides a relaxation of the design problem (6) through an extension of the design space. However, the extended design space for the gradient-con- strained design problem is seen to be a generalization of the one developed for the unconstrained problem. Loosely speaking, in the absence of gradient constraints, the design space is extended to include all effective dielectric tensors made from composites formed from the constituent dielectric mater- ials. However, for gradient-constrained optimal design problems, it is found that the relaxed problem requires the extension of the design space to include all the effective dielectric tensors viewed as functions of the dielectric tensors of the constituent materials.

In order to proceed, the optimal design problem over configurations is reformulated in an equivalent way. For a given array Aq, a neighborhood

N

(Aq) of Aq is introduced. Arrays in this neighborhood are denoted by Pq G(P1, P2, . . . , PN).

The neighborhood is chosen such that all matrices Piin the array Pq satisfy the constraint

0Fλ^FPiFΛ.

The associated set of controls C (Pq, x)G∑

i G1

N χi(x)Pi,

for Pq in the neighborhood

N

(Aq) for which 兰Ωχidx ⁄γi, is denoted by

C

(Aq,γq ), and the design problem is given by P G inf

C (Pq, x) in C (Aq,γq )

F (φ^{), (7)}

where φ is a solution of the equation of state (1) and is subject to the gradient constraint given by (2). As before, optimal design problems subject to either a finite or an infinite number of gradient constraints are formulated in the same way.

The notion of homogenization is given in the theories of G-convergence and H-convergence; see Ref. 9 and Ref. 10. Both notions of convergence coincide for the problem addressed here. A comprehensive introduction to the theory of G-convergence together with extensive references to the litera- ture is given in Ref. 11.

Definition 1.1. G-Convergence. The sequence {Cⁿ(Pq, x)}^Sn G1 G-con- verges to C^S(Pq, x) if and only if, for any open set ω⊂⊂ Ω and any f in H⁻¹(ω), the solutions wⁿin H0¹(ω) of

−div(Cⁿ(Pq, x)∇wⁿ)Gf (8) satisfy

wⁿ%w , weakly in H0¹(ω),

Cⁿ(Pq, x)∇wⁿ%C^S(Pq, x)∇w , weakly in L²(ω)^N, where w is the H0¹(ω) solution of

−div(C^S(Pq, x)∇w ) Gf. (9) The resource constraints for a G-convergent sequence of controls is handled in the usual way by assuming that the associated sequence of characteristic functions {χⁿi}^S_{n G1} satisfy

χⁿi %θi, in L^S(Ω) weak*. (10)

The design space is extended to include the set of all functions C^S(Pq, x) for which there exists a sequence of controls {Cⁿ(Pq, x)}^Sn G1 G-converging to C^S(Pq, x) for all Pq in

N

(Aq) and兰Ωθidx ⁄γi. The set of all such functions C^S(Pq, x) is the G-closure of the set

C

(Aq,γq) and is denoted by

G C

(Aq,γq).

We point out that

C

(Aq,γq ) is contained in

G C

(Aq,γq ). Its evident that the extension of the design space introduced here is more general than the usual one in that it involves the notion of tensor-valued effective dielectric func- tions as opposed to effective dielectric tensors.

The next step in the relaxation procedure is the correct description of the gradient constraints (2)–(5) in the extended design space. Given a sequence {Cⁿ(Pq, x)}^Sn G1 that G-converges to C^S(Pq, x) for every Pq in

N

(Aq), the local corrector function associated with the sequence {Cⁿ(Aq, x)}^Sn G1 is introduced. Consider a cube Q(x, r) of side length r centered at some point x inΩ. For r sufficiently small, Q(x, r) is contained withinΩ. Given a vector E in R^N, the local corrector function is written as ϕ^n,rE , where ϕ^n,rE is the H0¹(Q(x, r)) solution of

−div(Cⁿ(Aq, y)(∇ϕ^n,rE CE )) G−div(C^S(Aq, y)E ), for y in Q(x, r). (11) The local corrector functions are used to define the directional derivatives of the G-limit C^S(Pq, x) in the following theorem.

Theorem 1.1. Given that {Cⁿ(Pq, x)}^Sn G1 G-converges to C^S(Pq, x) for every Pq in

N

(Aq), the directional derivative of C^S(Pq, x) at Aq with respect

to the ith component conductivity in the direction specified by the mBm symmetric matrix Miis given by

∂C^S兾∂MiE · E

G ∑

k G1

m ∑

l G1 m

[Mi]kl冤^{limr→0n→limS}(1兾兩Q(x, r)兩) B冮Q(x,r)

χⁿi(∇kϕ^n,rE CEk)(∇lϕ^n,rE CEl) dy冥 ⁽¹²⁾

and exists almost everywhere inΩand for every E in R^m.

From Theorem 1.1, it follows that the definition of the ith phase gradi- ent, denoted by (∇ⁱklC^S(Aq, x))E · E, is given by the local formula

(∇ⁱklC^S(Aq, x))E · E

Glim

r→0 lim

n→S

(1兾兩Q(x, r)兩)冮Q(x,r)

χⁿi(∇kϕ^n,rE CEk) (∇lϕ^n,rE CEl) dy (13)

and

(∇ⁱkkC^S(Aq, x))E · E

Glim

r→0 lim

n→S(1兾兩Q(x, r)兩)冮Q(x,r)

χⁿi兩∇ϕ^n,rE CE )兩²dy, (14)

where repeated indices indicate summation. Theorem 1.1 provides explicit formulas for the derivatives of the G-limit in terms of the microscopic prob- lems (11). The identification of the suitable microscopic problems together with the local formulas (12)–(14) are necessary for numerical solution schemes based on relaxation through homogenization; see Ref. 12. General differentiability properties of the G-limits obtained without the use of local formulas are presented in Ref. 13. Differentiability properties for the the effective dielectric tensor in the contexts of periodic homogenization and statistically homogeneous random media were presented in Ref. 14.

The relaxation scheme developed here follows from the homogen- ization result given below. In order to state the result, one considers a sequence of configurations and a function C^S(Pq, x) for which the sequence of controls {Cⁿ(Pq, x)}^Sn G1 G-converges to C^S(Pq, x) for every Pq in

N

(Aq).

From homogenization theory, there exists a potential φ^S such that the sequence of potentials {φ^n}Sn G1 in H0¹(Ω) satisfying

−div(Cⁿ(Aq, x)∇φⁿ)Gf (15)

converges weakly toφ^Sin H0¹(Ω), where

−div(C^S(Aq, x)∇φ^S)Gf. (16) The following homogenization theorem allows one to pass to the homogenization limit in the gradient constraints of the form (2)–(5).

Theorem 1.2. Homogenization Limit of Weakly Converging Gradients. For any function p(x) in C^S(Ω¯ ),

n→limS冮Ω

p(x)兩∇φⁿ兩²dx G冮Ω

p(x)(σ(x)CI )∇φ^S·∇φ^Sdx, (17) where the covarianceσ(x) is a nonnegative matrix defined by

σ(x)E · EG∑

i G1

N (∇ⁱkkC^S(Aq, x))E · EA兩E兩², (18)

for all vectors E in R^m.

The homogenization limit of the gradient constraints is given in the following corollary.

Corollary 1.1. Homogenization Limit of Gradient Constraints. For some nonnegative p in C^S(Ω¯ ), if it is known that

冮Ω

p(x)兩∇φⁿ兩²dx ⁄ K² (19)

for the sequence {φⁿ}^Sn G1 given above, then∇φ^Ssatisfies

冮^{Ωp(x)(σ(x)CI )∇φS·∇φSdx ⁄ K2. (20)}

Similarly, if one has the local constraint兩∇φⁿ兩⁄K almost everywhere inΩ, then∇φ^Ssatisfies

1₍σ(x)CI )∇φ^S·∇φ^S⁄K, (21)

almost everywhere inΩ.

The covariance can be thought of as a new effective property charac- terizing the interaction between the local microstructure and the macro- scopic electric field A∇φ^S. It has been introduced in the contexts of periodic and statistically homogeneous random composites in Refs. 15–16.

In the general context of G-convergent sequences of matrices, the local for- mula forσ is given in Theorem 1.1 of Ref. 17.

The relaxed formulation for the basic gradient-constrained optimal design problem is introduced. For p(x) ¤ 0 and p(x) in C^S(Ω¯ ), the relaxed formulation of (7) is

QP G inf

C^S(Pq, x) in G C (Aq,γq)

F (φ), (22)

subject to the constraint

冮Ω

p(x)(σ(x)CI )∇φ·∇φdx ⁄ K², (23) whereφ ^{is the H}0¹(Ω) solution to the state equation

−div(C^S(Aq, x)∇φ)Gf, (24) andσ is defined by (18).

The first important feature of the relaxed problem is that the infimum of the relaxed design problem is attained by a control in

G C

(Aq,γq ).

Theorem 1.3. There exists a control Cˆ^S(Pq, x) in

G C

(Aq,γq ) and a state variableφˆ in H0¹(Ω) for which

−div(Cˆ^S(Aq, x)∇φˆ )Gf (25) and

QP GF (φˆ ), (26)

where

冮Ω

p(x)(σˆ (x)CI )∇φˆ ·∇φˆ dx⁄K² (27) and

σˆ (x)E · EG∑

i G1

N (∇ⁱkkCˆ^S(Aq, x))E · EA兩E兩², (28)

for every E in R^m.

The second important feature is the connection between the minimizer Cˆ^S(Pq, x) of the relaxed problem and minimizing sequences of configur- ations. To make the connection, the following optimal design problems are introduced. For p(x) ¤ 0 and p(x) in C^S(Ω¯ ),

PjG inf

C (Pq, x) in C (Aq,γq)

F (φ), (29)

subject to the constraint

冮Ω

p(x)兩∇φ兩²dx ⁄ K²(1C1兾j), (30)

whereφ is the H0¹(Ω) solution to the state equation

−div(C(Aq, x)∇φ^{)Gf. (31)} Its clear that, as j tends to infinity, the constraints given in the design prob- lems Pjapproach the constraint associated with (7). The problems Pjshare the same feature as (7) in that they are optimal design problems over con- figurations of N dielectrics. The connection between the minimizer of the relaxed problem and minimizing sequences of configurations is given in the following theorem.

Theorem 1.4. Given a minimizer Cˆ^S(Pq, x) of QP, there is a sequence of configurations and associated controls {C^j(Pq, x)}^Sj G1 in

C

(Aq,γq) such that, for all Pq in

N

(Aq), the sequence {C^j(Pq, x)}^Sj G1 G-converges to Cˆ^S(Pq, x) and the state variablesφ^jsatisfy the constraints (30). For this case, one has

j→limS

F (φ^j)GQP. (32)

Moreover, given any (H0, there exists an index JH0 such that, for all jHJ,

Pj⁄F (φ^j)⁄PjC( (33)

and

j→limS

PjGQP. (34)

Thus, one can use the minimizer for the relaxed problem to recover nearly- optimal constrained-gradient designs of N dielectrics.

The locally-constrained optimal design problem is given by inf

C (Pq, x)inC (Aq,γq)F (φ^{), (35)}

whereφ is a H0¹(Ω) solution of the equation of state (1) and is subject to the gradient constraint given by

兩∇φ兩⁄K, almost everywhere inΩ. (36)

The local constraint can be given also in the equivalent form (4). Moreover, choosing a countable dense subset {pk}^Sk G1 of the set of nonnegative func- tions in C^S(Ω¯ ), the local constraint is equivalent to

冮Ω

pk(x)兩∇φ兩²dx ⁄ K²冮Ω

pk(x) dx, for 1⁄kFS. (37) The relaxed formulation of (35) is

QR G inf

C^S(Pq, x) in G C (Aq,γq )

F (φ^{), (38)}

subject to the local constraint

1(σ(x)CI )∇φ·∇φ^⁄K, almost everywhere inΩ, (39) whereφ is the H0¹(Ω) solution to the state equation

−div(C^S(Aq, x)∇φ)Gf (40) andσ is defined by (18). The first important feature of the relaxed problem is that the infimum of the design problem is attained by a control in

G C

(Aq,γq).

Theorem 1.5. There exists a control Cˆ^S(Pq, x) in

G C

(Aq,γq) and a state variableφ^{ˆ in H}0¹(Ω) for which

−div(Cˆ^S(Aq, x)∇φˆ )Gf (41) and

QR GF (φˆ ), (42)

where

1₍σˆ (x)CI )∇φˆ ·∇φˆ ⁄ K, almost everywhere inΩ, (43) and

σˆ (x)E · EG∑

i G1

N (∇ⁱkkCˆ^S(Aq, x))E · EA兩E兩², (44)

for every E in R^m.

The second important feature is the connection between the minimizer Cˆ^S(Pq, x) of the relaxed problem and minimizing sequences of configur- ations. To make the connection, consider a countable dense subset {pk}^Sk G1 of the nonnegative functions in C^S(Ω¯ ). For a finite collection of these functions SjG{pk}^jk G1, the following optimal design problems are

introduced:

P˜_jG inf

C (Pq,x) in C (Aq,γq)F (φ^{), (45)}

subject to the constraints

冮Ω

pk(x)兩∇φ兩²dx ⁄ K²冮Ω

pk(x)(1C1兾j) dx, for pkin Sj, (46) whereφ is the H0¹(Ω) solution to the state equation

−div(C(Aq, x)∇φ)Gf. (47) It is clear that, as j tends to infinity, the constraints given in the design problems P˜j approach the constraint given by (37). The problems P˜j share the same feature as (35) in that they are optimal design problems over all configurations of N dielectrics. The connection between the minimizer of the relaxed problem and minimizing sequences of configurations is given in the following theorem.

Theorem 1.6. Given a minimizer Cˆ^S(Pq, x) of QR, there is a sequence of configurations and associated controls {C^j(Pq, x)}^S_{j G1} in

C

(Aq,γq) such that, for all Pq in

N

(Aq), the sequence {C^j(Pq, x)}^S_{j G1} G-converges to Cˆ^S(Pq, x) and the state variablesφ^jsatisfy the constraints (46). For this case, one has

j→limS

F (φ^j)GQR. (48)

Moreover, given any (H0, there exists an index JH0 such that, for all jHJ,

P˜_j⁄F (φ^j)⁄P˜jC( (49)

and

j→limS

P˜_jGQR. (50)

Thus, one can use the minimizer for the relaxed problem to recover nearly- optimal constrained-gradient designs of N dielectrics.

The paper is organized as follows. In Section 2, the local formula for the derivative of the effective dielectric tensor viewed as a function of its constituent dielectric constants is derived together with a local homogen- ization theorem. In Section 3, Theorem 1.2 is derived; the relaxation the- orems are proved in Section 4. In Section 5, an overview of the results given

here and recent results for problems where the design criteria are given in terms of the L²norm of the gradient are discussed in the conclusion.

2. Local Formulas for the Derivative of the G-Limit and Local Homogenization

In this section, we prove Theorem 1.1. We consider a sequence {Cⁿ(Pq, x)}^Sn G1 that G-converges to C^S(Pq, x) for every Pq in

N

(Aq). Given the sequence {Cⁿ(Aq, x)}^Sn G1, a vector E in R^m, and an open set ω⊂ Ω, we introduce the local corrector functionsϕⁿE, whereϕⁿE is the H0¹(ω) solution of

−div(Cⁿ(Aq, x))(∇ϕⁿECE )) G−div(C^S(Aq, x)E ), inω. (51) Equation (51) is written in the variational form given by

冮ω

Cⁿ(Aq, x)(∇ϕⁿECE ) ·∇ψdxA冮ω

(C^S(Aq, x)E ) ·∇ψdx G0, (52) for allψ in H0¹(ω). In the sequel, a generic open subset ofΩis denoted by ω. The measure of ω is denoted by 兩ω兩, and standard a priori estimates using the Cauchy inequality in (52) give

冮ω

兩∇ϕⁿE兩²dx ⁄ (4Λ²兾λ²)兩ω兩兩E兩², (53)

冮ω

兩∇ϕⁿECE兩²dx ⁄ (4Λ²兾λ^2C1)兩ω兩兩E兩². (54) We now list the convergence properties for the sequence of corrector functions.

Lemma 2.1. The sequence {ϕⁿE}^Sn G1 converges weakly to zero in H0¹(ω). Thus,

∇ϕⁿE%0, weakly in L²(ω^{)N, (55)}

and

Cⁿ(Aq, x)(∇ϕⁿECE ) % C^S(Aq, x)E, weakly in L²(ω)^N. (56) The proof is standard; see Ref. 10.

Forδβ and Mq such that Aq CδβMq lies in

N

(Aq), consider the sequence {Cⁿ(Aq CδβMq, x)}^Sn G1. Here, Mq is taken to be an array of dielectric tensors

that are identically zero except for the ith component dielectric tensor Mi. Here,

兩Mi兩G

1

^∑k,l

[M_i]²_kl.

The sequence of coefficients {Cⁿ(Aq CδβMq, x)}^Sn G1 differs from {Cⁿ(Aq, x)}^Sn G1 by the increment δβMiχⁿi. The G-limit for the sequence {Cⁿ(Aq CδβMq, x)}^Sn G1 is written as C^S(Aq CδβMq, x). We set

δC GC^S(Aq CδβMq, x)AC^S(Aq, x,)

and examine the dependence ofδC with respect toδβ.

The first step is to examine the dependence of (1兾兩ω兩)兰ωδC E · E dx with respect to the increment δβ. The correctors associated with the sequence {Cⁿ((Aq CδβMq, x)}^Sn G1 are denoted byϕˆⁿECE · x, whereϕˆⁿEare the H0¹(ω) solutions of

冮ω

Cⁿ((Aq CδβMq, x)(∇ϕ^ˆnECE ) ·∇ψdx

A冮ω

(C^S(Aq CδβMq, x)E ) ·∇ψdx G0, (57) for allψ ^{in H}0¹(ω^{), and}

Cⁿ(Aq CδβMq, x)(∇ϕ^ˆnECE ) % C^S(Aq CδβMq, x)E, weakly in L²(ω)^N, (58)

∇ϕ^ˆnE%0, weakly in L²(ω^{)N. (59)}

For any choice of p(x) in C^S(ω¯), it follows from (56) and (58) that (1兾兩ω兩)冮ω

p(x)δC E · E dx Glim

n→S

(1兾兩ω兩)冮ω

p(x)(Cⁿ(Aq CδβMq, x)(∇ϕ^ˆnECE )

ACⁿ(Aq, x)(∇ϕⁿECE )) · E dx. (60) Writing

Cⁿ(Aq CδβMq, x)GCⁿ(Aq, x)CδβMiχⁿi, ϕˆⁿEGϕⁿECδϕⁿ,

where

δϕ^nGϕˆⁿEAϕⁿE,

and substitution into (60) gives

(1兾兩ω兩)冮ω

pδC E · E dx Glim

n→S(1兾兩ω兩)冮ω

pδβMiχⁿi(∇ϕⁿECE ) · E

CpδβMiχⁿi∇δϕⁿ· ECpCⁿ(Aq, x)∇δϕⁿ· E dx. (61) To proceed further, we subtract (52) from (57) to obtain, for every ψ in H0¹(ω), the equation

0G冮ωδβMiχⁿi(∇ϕⁿECE ) ·∇ψdxC冮ω

Cⁿ(Aq, x)∇δϕⁿ·∇ψdx

C冮ωδβMiχⁿi∇δϕⁿ·∇ψdxA冮ωδCE ·∇ψdx. (62) Choosingψ^G( pϕⁿE) in (62), substitution into (61), and taking limits give

(1兾兩ω兩)冮ω

pδC E · E dx Gδβ^lim

n→S

[(1兾兩ω兩)冮ω

pMiχⁿi(∇ϕⁿECE ) · (∇ϕⁿECE ) dx

C(1兾兩ω兩)冮ω

pMiχⁿi(∇ϕⁿECE ) ·∇δϕⁿdx]. (63)

Writing

S

¯ Glim sup

n→S (1兾兩ω兩)冮ω

pδβMiχⁿi(∇ϕⁿECE ) ·∇δϕⁿdx, (64a)

S

^{q G}lim inf

n→S

(1兾兩ω兩)冮ω

pδβMiχⁿi(∇ϕⁿECE ) ·∇δϕⁿdx, (64b)

standard a priori estimates are used to obtain

兩

S

^¯_{兩⁄兩兩p兩兩}SK兩E兩²1兩δβ兩, 兩

S

^q _{兩⁄兩兩p兩兩}SK兩E兩²1兩δβ兩, (65) where K is a constant independent of δβ and the choice of ω⊂ Ω and 兩兩p兩兩^S is the maximum value of p onω¯ . We establish the following lemma.

Lemma 2.2. For everyω⊂ Ωand function p in C^S(ω¯), the limit

n→limS

[(1兾兩ω兩)冮ω

pMiχⁿi(∇ϕⁿECE ) · (∇ϕⁿECE ) dx] (66) exists and

(1兾兩ω兩)冮ω

pδC E · E dx Gδβlim

n→S冤^{(1兾兩ω兩)}冮ω

pMiχⁿi(∇ϕⁿECE ) · (∇ϕⁿECE ) dx冥^{CδβR(δβ,ω, p), (67)}

where

兩R(δβ,ω, p)兩⁄兩兩p兩兩^SK兩E兩²1兩δβ兩. (68) Proof. We put

HnG(1兾兩ω兩)冮ω

pMiχⁿi(∇ϕⁿECE ) · (∇ϕⁿECE ) dx, (69a)

RnG(1兾兩ω兩)冮ω

pMiχⁿi(∇ϕⁿECE ) ·∇δϕⁿdx, (69b) and from (54) we see that

H¯ Glim sup

n→S

Hn and Hq Glim inf

n→S

Hn

are finite. Passing to subsequences as necessary, we deduce that (1兾兩ω兩)冮ω

pδC E · E dx GδβH¯ CδβR¯ (δβ,ω, p), (70) where for each δβ, R¯ (δβ,ω, p) is a cluster point of the sequence {Rn}n G1^S

and from (65) it is evident that

R¯ (δβ,ω, p)⁄兩兩p兩兩^SK兩E兩²1兩δβ兩.

Similar considerations give (1兾兩ω兩)冮ω

pδC E · E dx GδβHq CδβRq(δβ^,ω^{, p), (71)} where Rq(δβ^,ω, p) is a cluster point of the sequence {Rn}^Sn G1 and

Rq(δβ,ω, p)⁄兩兩p兩兩^SK兩E兩²1兩δβ兩.

It is evident from (70) and (71) that Hq GH¯ . From this, we deduce that Rq(δβ^,ω^{, p)GR¯ (}δβ^,ω^{, p). R(}δβ^,ω^{, p),}

and the lemma is established. 䊐

With Lemma 2.2 in hand, we establish Theorem 1.1. We consider sub- setsω ofΩ given by cubes Q(x, r) and the associated local corrector func- tions are denoted byϕ^n,rE . We put

H(x, r) G lim

n→S(1兾兩Q(x, r)兩)冮Q(x,r)

Miχⁿi(∇ϕ^n,rE CE ) · (∇ϕ^n,rE CE ) dy, (72) and we introduce the functions f¯(x) and fq(x) defined everywhere inΩ by

f¯(x).lim sup

r→0 H(x, r), (73)

_(x).lim inff

r→0 H(x, r). (74)

The estimate (54) shows that these functions are bounded onΩ. Given the increment δβ, we consider the intersection of the Lebesgue points of C^S(Aq, x) and C^S(Aq CδβMq, x). We choose pG1, and from Lemma 2.2 we have the following identity:

δC E · EGlim

r→0(1兾兩Q(x, r)兩)冮Q(x,r)

δC E · E dy Gδβlim

r→0(H(x, r)CR(δβ, Q(x, r), 1)), (75) almost everywhere on Ω and for any E in R^m. Passing to subsequences if necessary, one has

δC E · E Gδβ( f¯(x)CR¯ (δβ, x)), a.e. onΩ, (76) where R¯ (δβ, x) is a cluster point of the sequence {R(δβ, Q(x, r), 1)}rH0 and

兩R¯(δβ^{, x)}兩⁄K兩E兩²兩1兩δβ兩.

A similar consideration shows that

δC E · E Gδβ( f_(x)CRq(δβ, x)), a.e. onΩ, (77) where Rq(δβ, x) is a cluster point of the sequence {R(δβ, B(x, r), 1)}rH0 and

兩Rq(δβ, x)兩⁄K兩E兩²1兩δβ兩.

We subtract (77) from (76) to obtain

f¯(x)Af_(x) GRq(δβ, x)AR¯ (δβ, x), a.e. onΩ, (78)

to see that

兩 f¯(x)Af_(x)兩F2 K兩E兩²1兩δβ兩, a.e. onΩ. (79) Since f¯(x) and f_(x) are independent of the incrementδβ, we deduce that

limr→0 n→limS

(1兾兩Q(x, r)兩)冮Q(x,r)

Miχⁿi(∇ϕ^n,rE CE ) · (∇ϕ^n,rE CE ) dy (80) exists almost everywhere onΩ. In this way, it is seen that

Rq(δβ, x)GR¯ (δβ, x).R(δβ, x) and that

δC E · E Gδβlim

r→0 lim

n→S

(1兾兩Q(x, r)兩)

B冮Q(x,r)

Miχⁿi(∇ϕ^n,rE CE ) · (∇ϕ^n,rE CE ) dy

CδβR(δβ, x), a.e. onΩ, (81)

where

R(δβ, x)FK兩E兩²1兩δβ兩.

It is evident from (81) that (∂C^S兾∂Mi)E · E exists and is given by (∂C^S兾∂Mi)E · EGlim

r→0 lim

n→S(1兾兩Q(x, r)兩)

冮Q(x,r)

Miχⁿi(∇ϕ^n,rE CE ) · (∇ϕ^n,rE CE ) dy, (82) and Theorem 1.1 follows.

Collecting results, we substitute (81) into (67) of Lemma 2.3 to obtain the local homogenization theorem.

Theorem 2.1. Local Homogenization Theorem. Under the hypoth- eses of Theorem 1.1, given anyω⊂ Ωand p in C^S(ω¯), the sequence of local correctors {ϕⁿECE · x}^Sn G1 defined by (51) satisfy

n→limS(1兾兩ω兩)冮ω

pMiχⁿi(∇ϕⁿECE ) · (∇ϕⁿECE ) dx

G(1兾兩ω兩)冮ω

p(∂C^S兾∂Mi)E · Edx. (83)

For future reference, it is noted that the estimate (54) gives the bound

σ(x)E · E ⁄ (4Λ²兾λ²)兩E兩², (84)

almost everywhere inΩfor E in R^m.

3. Homogenization of the Products of Weakly Converging Sequences of Gradients

In this section, Theorem 1.1 and Theorem 2.1 are applied together with a localization argument to establish Theorem 1.2. Recall that the sequences of solutions of (15), denoted by {φⁿ}^Sn G1, converge weakly to the solu- tion φ^S of (16) and the associated sequence of coefficient matrices {Cⁿ(Pq, x)}^Sn G1 G-converges to C^S(Pq, x) for every Pq in

N

(Aq).

The first step in the proof is to approximate φ^S by piecewise-affine functions. Given (H0, there exists a function w⁽ in H0¹(Ω) which is piece- wise affine onΩand

冮Ω

兩∇w⁽A∇φ^S兩²dxF(²; (85)

see for example Ref. 18. The gradient of w⁽ is constant on the open sets ω^{i( and}

Ω¯ G *^{κ(( )}

i G1ω¯ ⁱ⁽.

In each open setω^{i(, one has} w⁽GEⁱ· xCcⁱ,

where Eⁱis a constant vector in R^mand cⁱis a constant.

The global corrector {φ^n,(}Sn G1 is introduced. The right-hand side f⁽G−div(C^S(Aq, x)∇w⁽)

is chosen andφ^n,( is defined to be the H0¹(Ω) solution of

−div(Cⁿ(Aq, x)∇φ^{n,()Gf(. (86)} One obtains easily the following convergence properties.

Lemma 3.1. For ( fixed,

φ^n,(%w⁽, as n→^S, in H0¹(Ω), (87a) Cⁿ(Aq, x)∇φ^n,(%C^S(Aq, x)∇w⁽, in L²(Ω)^m. (87b)

The error between the weakly converging sequences {φⁿ}^Sn G1 and the sequence of global correctors {φ^n,(}^Sn G1 can be controlled uniformly with respect to n. This is made precise in the following lemma.

Lemma 3.2. There exists a constant C0 independent of n and ( such that

冮Ω

兩∇φ^nA∇φ^n,(兩²dx ⁄ C²0(Λ兾λ)²(², for all nH0. (88)

Proof. We start by showing that 兩兩 f⁽Af兩兩H⁻¹(Ω)⁄ (Λ.

Givenψ in H0¹(Ω), we have

兩FH⁻¹( f⁽Af ),ψ^HH0¹兩G冨冮Ω

C^S(Aq, x) (∇w⁽A∇φ) ·∇ψdx冨

⁄Λ冮Ω

兩∇w⁽A∇φ兩兩∇ψ兩, dx

⁄ (Γ兩兩ψ兩兩H0¹(Ω), (89) and the claim follows. Next, one has the estimate given by

λ冮Ω

兩∇φ^nA∇φ^n,(兩²dx ⁄冮Ω

Cⁿ(Aq, x)(∇φ^nA∇φ^n,() · (∇φ^nA∇φ^n,() dx

⁄兩兩 f⁽Af兩兩H⁻¹C0

1

_冮Ω^{兩∇φnA∇φn,(兩2dx, (90)}

where C0depends only onΩand m and comes from the Poincare´–Friedrichs inequality. The inequality (88) follows by noting that

兩兩 f⁽Af兩兩H⁻¹⁄Λ(. 䊐

Collecting results, we write φ^Gw⁽Cr⁽, φ^nGφ^n,(Cz^n,(. Here, r⁽ and z^n,( are in H0¹(Ω) and satisfy

冮Ω

兩∇r⁽兩²dxF(², 冮Ω

兩∇z^n,(兩²dxF(², for all nH0. (91)

Given p(x) in C^S(Ω¯ ), one observes that 1G∑

j G1 N χⁿj(x)

and writes

冮Ω

p兩∇φⁿ兩²dxG冮Ω

p冢^{j G1∑N χnj(x)}冣^{兩∇(φn,(Czn,()兩2dx}

G∑

i G1 κ(( )

∑

j G1 N 冮ωⁱ(

pχⁿj兩∇φ^n,(兩²dxCO(( ). (92) The lemma that gives the connection between global and local correc- tors is provided by the following localization result.

Lemma 3.3. Localization Lemma. Letϕ^i,n,(Eⁱ be the H0¹(ωⁱ⁽) solution of

−div(Cⁿ(Aq, x))(∇ϕ^i,n,(Eⁱ CEⁱ))G−div(C^S(Aq, x)Eⁱ). (93) Then,

lim

n→S冮ωⁱ(

兩∇ϕ^i,n,(Eⁱ CEⁱA∇φ^n,(兩²dx G0. (94)

Proof. From the properties of the local corrector introduced in Sec- tion 2, one has

ϕ^i,n,(Eⁱ %0, weakly in H0¹(ωⁱ⁽), (95a)

Cⁿ(Aq, x)(∇ϕ^i,n,(Eⁱ CEⁱ) % C^S(Aq, x)Eⁱ, weakly in L²(ωⁱ⁽)^m. (95b) The properties of the global correctorφ^n,( imply that it is a solution of

−div(Cⁿ(Aq, x)∇φ^n,()G−div(C^S(Aq, x)Eⁱ), onωⁱ⁽; (96) i.e., for allψ in H0¹(ωⁱ⁽),

冮ωⁱ⁽

Cⁿ(Aq, x)∇φ^n,(·∇ψdx G冮ωⁱ⁽

C^S(Aq, x)Eⁱ·∇ψdx, (97) and

φ^n,(%w⁽GEⁱ· xCcⁱ, weakly in H¹(ω^{i(), (98a)} Cⁿ(Aq, x)∇φ^n,(%C^S(Aq, x)Eⁱ, weakly in L²(ω^{i()m. (98b)} The lemma now follows immediately from Remark 1 of Section 10 in Ref.

10. 䊐

The local homogenization Theorem 2.1 and the localization Lemma 3.3 are applied and we work up from small to intermediate scales to find that

∑

i G1 κ(( )

冮ωⁱ⁽

p(σ(x)CI )Eⁱ· Eⁱdx

G∑

i G1 κ(( )

∑

j G1 N 冮ωⁱ⁽

p∇^jkkC^S(Aq, (x))Eⁱ· Eⁱdx

G∑

i G1 κ(( )

∑

i Gj N

n→limS冮ωⁱ(

pχⁿj兩∇ϕ^i,n,(Eⁱ CEⁱ兩²dx

Glim

n→S ∑

i G1 κ(( )

∑

j G1 N 冮ωⁱ⁽

pχⁿj兩∇φ^n,(兩²dx. (99)

Proceeding from large scales down to intermediate scales, we see that

n→limS冮Ω

p兩∇φⁿ兩²dxG lim

n→S ∑

i G1 κ(( )

冮ωⁱ⁽

p兩∇φ^n,(兩²dxCO(( )

G∑

i G1 κ(( )

冮ωⁱ(

p(σ(x)CI ) Eⁱ· EⁱdxCO(( )

G冮Ω

p(σ(x)CI )∇w⁽·∇w⁽dxCO(( ). (100) Since ( is arbitrary and recalling (84) and (91), we obtain the desired result given by

n→limS冮Ω

p兩∇φⁿ兩²dx G冮Ω

p(σ(x)CI )∇φ^S·∇φ^Sdx, (101) and Theorem 1.2 follows.

4. Relaxation

In this section, the properties of the relaxed problem (38) given by Theorems 1.5 and 1.6 are established. The proofs of Theorems 1.3 and 1.4 follow identical lines. In establishing the properties of the relaxed problem, the following compactness result will be used.

Theorem 4.1. Compactness Property. Given any sequence {Cⁿ(Pq, x)}^Sn G1 in

C

(Aq,γq ), there exists a subsequence {C^n′(Pq, x)}^Sn′G1 and a

function C^S(Pq, x) in

G C

(Aq,γ^q) such that C^n′(Pq, x) G-converges to C^S(Pq, x) for all Pq in

N

(Aq).

This property follows immediately from Ref. 13.

A special sequence of controls {Cⁿ(Pq, x)}^Sn G1 in

C

(Aq,γq) is constructed.

This sequence will be used to establish Theorems 1.5 and 1.6. To construct this sequence, consider a minimizing sequence {C^S,n(Pq, x)}^Sn G1 for QR. The associated set of electric potentials for the sequence {C^S,n(Aq, x)}^Sn G1 is denoted by {φⁿ}^Sn G1 and

QR G lim

n→SF (φⁿ).

For each n, there exists a sequence of controls {C^n,k(Pq, x)}^Sk G1 in

C

(Aq,γq) such that C^n,k(Pq, x) G-converges to C^S,n(Pq, x) for every Pq in

N

(Aq).

The associated potentialsφ^n,kin H0¹(Ω) satisfy the state equation

−div(C^n,k(Aq, x)∇φ^{n,k)Gf, (102)} and for every nonnegative p in C^S(Ω¯ ), the homogenization Theorem 1.2 gives

lim

k→S冮Ω

p(x)兩∇φ^n,k兩²dx G冮Ω

p(x)(σⁿ(x)CI )∇φⁿ·∇φⁿdx

⁄K²冮Ω

p(x) dx, (103)

where the covariance matrixσⁿ(x) is σⁿ(x)E · EG∑

i G1

N (∇ⁱkkC^S,n(Aq, x))E · EA兩E兩², (104)

for all vectors E in R^m. A countable dense subset {pl}^Sl G1 of the set of nonnegative functions in C^S(Ω¯ ) is introduced. Given n, put

SnG{p_l}ⁿ_{l G1};

then, there exists an index knfor which

冮Ω

pl兩∇φ^n,kn兩²dx ⁄ K²冮Ω

pl(1C1兾n) dx, for every plin Sn, (105)

兩F(φ^n,kn)AF(φⁿ)兩F1兾n, (106)

and

−div(C^n,kn(Aq, x)∇φ^n,kn)Gf. (107)

Passing to a subsequence if necessary and appealing to the compactness property, there exists a function C˜^S(Pq, x) in

G C

(Aq,γq ) for which

{C^n,kn(Pq, x)}^Sn G1G-converges to C˜^S(Pq, x) (108) for all Pq in

N

(Aq).

Definition 4.1. The sequence {(C^n,kn(Pq, x)}^Sn G1 constructed above satisfying (105)–(108) is called a configuration minimizing sequence.

The potential associated with the control C˜^S(Pq, x) is denoted byφ˜ and

−div(C˜^S(Aq, x)∇φ˜ )Gf. (109) We proceed to establish Theorem 1.5. For each plin {pl}^Sl G1, it follows that

冮Ω

pl(x)(σ˜ (x)CI )∇φ˜ ·∇φ˜ dxG lim

n→S冮Ω

pl兩∇φ^n,kn兩²dx

⁄K²冮Ω

pldx, (110)

where

σ˜(x)E · EG∑

i G1

N (∇ⁱkkC˜^S(Aq, x))E · EA兩E兩², (111)

for every E in R^m. Thus, by density, for any nonnegative p in C^S(Ω¯ ), one has

冮Ω

p(x)(σ˜(x)CI )∇φ˜ ·∇φ˜ dx⁄K²冮Ω

p(x) dx, (112)

and C˜^S(Pq, x) is an admissible control for QR. It is evident from the conti- nuity of the objective function and (106) that F(φ˜ )GQR and Theorem 1.5 follows.

To prove Theorem 1.6, one considers the design problems P˜j given by (45) and starts by showing that P˜j⁄QR. For the configuration minimizing sequence {(C^n,kn(Pq, x)}^Sn G1 of Definition 4.1, it is evident that, given the index j, one has that, for all nHj, C^n,kn(Pq, x) is admissible for P˜j and

P˜_j⁄F (φ^n,kn).

Sending n to infinity shows that P˜_j⁄QR.