Univalent Solutions of an Elliptic System of Partial ...

Univalent Solutions of an Elliptic System of Partial Differential Equations Arising in Homogenization PATRICIA B AUMAN , A NTONELLA M ARINI & V INCENZO N ESI A BSTRACT. In this paper we prove that solutions to an elliptic system of partial differential equations in divergence form whose boundary values are the restriction of a diffeomorphism of degree one onto a convex domain in two dimensions are mappings whose differential has a nonnegative determinant. Under appropriate regularity assumptions on the domain, the boundary values, and the coefficients of the elliptic system, it is shown that solutions are diffeomorphisms whose differential has a strictly positive determinant. We also describe applications of our results to problems arising in homogenization.

1. I NTRODUCTION In this paper we derive some results with applications to homogenization on the existence of univalent solutions to a linear elliptic system of partial differential equations in two dimensions. The results obtained here were inspired by some observations made by Bauman and Phillips in [5] concerning solutions of certain nonlinear elliptic systems arising in two-dimensional elasticity. To describe our results in more detail, let Ω be a simply connected, bounded open set in R2 with Lipschitz boundary and let (1.1) (1.2)

σij = σji ∈ L∞ (Ω) m|ξ|2 ≤

2 X

for 1 ≤ i, j ≤ 2, with

σij (x)ξi ξj ≤ M|ξ|2

i,j=1

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for all ξ in R2 and almost every x in Ω, where 0 < m ≤ M < ∞. Let Φ ≡ (Φ1 , Φ2 ) ¯ into R2 and let ψ ≡ (ψ1 , ψ2 ) be a C 1 diffeomorphism of some neighborhood of Ω denote the restriction of the diffeomorphism Φ to the boundary of Ω. Assume that ψ maps ∂Ω onto the boundary of a convex domain ∆ ⊂ R2 and that the degree of Φ in ∆ is one. ¯ ; R2 ), where uk Let us define u(x) = (u1 (x), u2 (x)) in W 1,2 (Ω; R2 ) ∩ C(Ω satisfies: (1.3)

2 X

Dj (σij (x) · Di uk (x)) = 0

in Ω,

i,j=1

(1.4)

uk = ψ k

on ∂Ω,

for k = 1, 2. Our main result (see also Theorem 2.5) is the following theorem. Theorem 3.1. For Ω, ∆, and ϕ as above, we have det Du(x) ≥ 0 a.e. in Ω. ¯ ; R2 ), and σij is in Moreover, if in addition, Ω is a C 1,α domain, Φ is in C 1,α (Ω α ¯ 1 ,α ¯ C (Ω) for 1 ≤ i, j ≤ 2 for some α > 0, then u is a C -homeomorphism from Ω ¯ onto ∆ and det Du > 0 in Ω. The above result generalizes a classical theorem due to Rad´o [20] for harmonic maps in two dimensions. Motivated by inverse conductivity problems, the case of a single equation in (1.3) has been treated by Alesandrini [1] and by Alessandrini and Magnanini [2] and [3]. In [1], the author treats the case of Dirichlet boundary conditions under the assumptions that σ is Lipschitz continuous and isotropic. In [2] the isotropy assumption is dropped and oblique derivative problems are considered. Finally, in [3] the authors deal with the case when σ is only measurable, and establish various results concerning Dirichlet and Neumann boundary conditions, as well as results for the Stekloff eigenvalue problem. A consequence of Theorem 3.1 (with applications to homogenization) is the following result. Corollary 3.2. Let A be a real 2 × 2 constant matrix and let Ω be a bounded, simply connected, convex domain in R2 with Lipschitz boundary. Let σ be the symmetric 2 × 2 matrix-valued function whose ij th element is σij , and let w0 = 1,2 (w01 , w02 ) ∈ W0 (Ω; R2 ) satisfy 



div σ (x) · (Dw0 (x) + A)t = 0 in Ω, where for a 2 × 2 matrix B = [bij ], B t is the transpose of B and div B is the vector in R2 defined by (div B)j =

2 X ∂bij , ∂xi i=1

Univalent Solutions of an Elliptic System of PDE Arising in Homogenization 749 for j = 1, 2. Then, det A · det(Dw0 (x) + A) ≥ 0, for almost every x in Ω. Corollary 3.2 can be used to obtain new results in some so-called G-closure problems, which are problems that arise in homogenization as well as in optimal design. In particular, new results concerning bounds on the effective conductivity for certain composite materials are obtained in [16] as a direct application of Corollary 3.2. Other applications can be found in [17, 18]. In Section 4 we give a brief description of these connections. 2. T HE SMOOTH CASE Throughout this paper, we assume that Ω, σ = [σij ], Φ, ψ, ∆, and u = (u1 , u2 ) satisfy the hypotheses stated in paragraphs two and three of Section 1. In this section, we assume (in addition) that Ω is a bounded domain of class ¯ ; R2 ), and σij is in C α (Ω) ¯ for 1 ≤ i, j ≤ 2, for some C 1,α in R2 , Φ is in C 1,α (Ω α > 0. Our main result is Theorem 2.4, which asserts that under these further ¯ onto ∆ ¯ and det Du > 0 in Ω. hypotheses, u is a C 1,α -homeomorphism from Ω We also prove (cf. Theorem 2.5) that, under suitable additional regularity assump¯ onto ∆ ¯ such that det Du > 0 tions, the mapping u is a diffeomorphism from Ω ¯ in Ω. ¯ ; R2 ), y0 ∈ v(Ω) ¯ \ v(∂Ω), and Recall that, if v ≡ (v 1 , v 2 ) is in C 1 (Ω −1 det Dv(x) 6= 0 for all x in v (y0 ), then the Brouwer degree of v at y0 is given by X

deg(v, Ω, y0 ) =

sgn[det Dv(x)],

x∈v −1 (y0 )

¯ ; R2 ) where Dv(x) denotes the differential of v at x . More generally, if v ∈ C(Ω 2 and y0 ∈ R \ v(∂Ω), then Z

(2.1)

deg(v, Ω, y0 ) =

Ω

f (v(x)) · det Dv(x) dx,

where f is any smooth real-valued function on R2 with compact support in the R connected component of R2 \ v(∂Ω) containing y0 , and R2 f (y) dy = 1. It is well known that deg(v, Ω, ·) is integer-valued and constant on connected components of R2 \ v(∂Ω), and within such domains it depends on v only through v|∂Ω . See [19]. The following result follows easily from the degree theory, the maximum principle, and the convexity of the domain ∆. Lemma 2.1. The mapping u satisfies u(Ω) = ∆.

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¯ ; R2 ). Proof. Note that from elliptic regularity theory, the mapping u is in C 1,α (Ω ¯ (See [10, Theorem 8.34].) Since Φ is a homeomorphism of degree one from Ω ¯ onto ∆ and u = Φ on ∂Ω, we have (2.2)

deg(u, Ω, y) = deg(Φ, Ω, y) = 1,

for all y in ∆.

❐

To show that ∆ ⊂ u(Ω), consider the set E ≡ {y ∈ ∆ | u−1 (y) 6= ∅}. If E is not dense in ∆, there exists anR open ball Br (y0 ) in ∆ with Br (y0 ) ∩ E = ∅. Hence, if f ∈ C0∞R(Br (y0 )) and Br (y0 ) f (y) dy = 1, we have f ◦ u = 0 in Ω and deg(u, Ω, y0 ) ≡ Ω f (u(x)) det Du(x) dx = 0, which contradicts (2.2). Thus, E is dense in ∆. Now from the continuity of u and the fact that u(∂Ω) = ∂∆, we deduce that ∆ ⊂ u(Ω). To prove the converse, we note that since ∆ is convex, it is sufficient to show that u(Ω) ⊂ H for any open half plane H ⊃ ∆. Since uniform rotations and translations of u (in either the domain or range variables) satisfy uniformly elliptic boundary value problems of the form (1.3) and (1.4), we may assume without loss of generality that H = {(y 1 , y 2 ) ∈ R2 | y 2 > 0}. Since H ⊃ ∆, we have H¯ ⊃ ∂∆ = u(∂Ω) and thus u2 (x) ≥ 0 on ∂Ω. It follows from the strong maximum principle that u2 (x) > 0 for all x in Ω. Thus, u(Ω) ⊂ H . This proves that u(Ω) ⊂ ∆. Remark 2.2. The above proof is valid assuming only the hypotheses of paragraphs two and three of Section 1. In this case, the mapping u is in W 1,2 (Ω; R2 ) ∩ ¯ ; R2 ). (See [10, Theorems 8.3 and 8.29].) Thus, the conclusion of Lemma C(Ω 2.1 holds under these weaker hypotheses. We shall need the following lemma to prove Theorem 2.4. Lemma 2.3. Let y0 = (y01 , y02 ) be any point in ∆, and fix any ϑ ∈ [0, π ]. ¯ and let Define uϑ (x) = (u1 (x) − y01 ) cos ϑ + (u2 (x) − y02 ) sin ϑ for all x in Ω 1 ¯ | uϑ (x) = 0}. Then Γ is a connected imbedded C -curve in Ω ¯ , and Γ = {x ∈ Ω ∇uϑ 6= 0 on Γ .

Proof. Note that since ∆ is a bounded convex domain in R2 , the set γ ≡ ¯ | (y 1 − y01 ) cos ϑ + (y 2 − y02 ) sin ϑ = 0} is a closed line {y = (y 1 , y 2 ) ∈ ∆ ¯ segment in ∆, and Γ = u−1 (γ). Thus, γ ∩ ∂∆ consists of two distinct points (namely the endpoints of γ ) in ∂∆. Since u|∂Ω = Φ|∂Ω = ψ is a homeomorphism of ∂Ω onto ∂∆, it follows that Γ ∩ ∂Ω = u−1 (γ) ∩ ∂Ω also consists of two distinct points, say p and q, in ∂Ω. Let ∂ψ/∂s denote the derivative of ψ ≡ Φ|∂Ω with respect to arclength of ¯ , we ∂Ω and consider ∂ψ/∂s evaluated at p . Since Φ is a diffeomorphism on Ω have ∂ψ/∂s 6= 0 on ∂Ω. As ∆ is convex and of class C 1 , the tangent line to ∂∆ at ψ(p) cannot intersect y0 ∈ ∆, and, in particular, it cannot be parallel to γ . Since (cos ϑ, sin ϑ) is a vector perpendicular to γ (by the definition of γ), it follows that (∂ψ/∂s)(p) is not perpendicular to (cos ϑ, sin ϑ). Hence,

Univalent Solutions of an Elliptic System of PDE Arising in Homogenization 751

❐

(∂uϑ /∂s)(p) = h(cos ϑ, sin ϑ), (∂ψ/∂s)(p)i 6= 0. Similarly, (∂uϑ /∂s)(q) 6= 0. ¯ | uϑ (x) = 0} is a C 1 curve in neighborhoods of p and q, Thus, Γ ≡ {x ∈ Ω respectively, intersecting ∂Ω at these points nontangentially. Since u1 (x) and u2 (x) satisfy the linear elliptic partial differential equation (1.3) in Ω, the same is true of uϑ (x). It follows from F. Schulz’s generalization of Hartman and Wintner’s theorem (see [21, Theorem 7.2.4] and [12]) that uϑ has at most isolated critical points in Γ ∩ Ω = {x ∈ Ω | uϑ (x) = 0}. Moreover, in a neighborhood of any such point, Γ is made up of 2K (K ≥ 2) arcs meeting at this point. If a critical point exists, then due to the fact that Γ ∩ ∂Ω consists of just two points, it follows that Γ must contain a closed loop; that is, some connected component D ⊂ {x ∈ Ω | uϑ (x) 6= 0} has ∂D ⊂ Γ . Since uϑ = 0 on Γ , it follows from the maximum principle that uϑ ≡ 0 on D. But this is impossible by unique continuation for solutions of (1.3) (cf. [7, Part II, Chapter 6]). Thus, ∇uϑ 6= 0 on Γ , and Γ is an imbedded C 1 curve. It must be connected, since otherwise it would contain a closed loop.

We can now prove the following theorem. ¯ onto ∆ ¯ , and Theorem 2.4. The mapping u is a C 1,α -homeomorphism from Ω det Du > 0 in Ω.

❐

Proof. If det Du is not positive in Ω, then {x ∈ Ω | det Du(x) ≤ 0} is ¯ ; R2 ) and deg(u, Ω, ·) a nonempty set. On the other hand, since u ∈ C 1,α (Ω is one in ∆, the set {x ∈ Ω | det Du(x) > 0} is also nonempty. It follows that det Du(x0 ) = 0 for some x0 in Ω. Thus, there exists ϑ in [0, π ] such that cos ϑ∇u1 (x0 ) + sin ϑ∇u2 (x0 ) = 0. For ϑ as above, y0 = u(x0 ), and uϑ (x) defined as in Lemma 2.3, we have uϑ (x0 ) = 0 and ∇uϑ (x0 ) = 0, a contradiction. Thus, det Du > 0 in Ω. Now it follows from (2.1) and Lemma 2.1 that u is a one-to-one and onto ¯ onto ∆ ¯ , we deduce that u−1 is ¯ . Since u is continuous on Ω mapping from Ω ¯. continuous on ∆ For the sake of completeness, we note that if Ω and σ have higher regularity, ¯ . In particular, then det Du > 0 on ∂Ω and hence u is a diffeomorphism on all of Ω we have the following result. Theorem 2.5. Assume, in addition to the hypotheses of this section, that Ω is a ¯ for 1 ≤ i, ¯ , i.e., σij is Lipschitz continuous on Ω C 2 domain and σij is in C 0,1 (Ω) ¯ ¯ ¯ j ≤ 2. Then det Du > 0 in Ω, and u is a diffeomorphism from Ω onto ∆. 2,2 (Ω; R2 ) ∩ Proof. Note that from elliptic regularity theory, u ∈ Wloc ¯ ; R2 ). By Theorem 2.4, it is sufficient to show that det Du 6= 0 on ∂Ω. C 1,α (Ω If not, there exists x0 ∈ ∂Ω such that det Du(x0 ) = 0. Since rotations and translations of u = (u1 , u2 ) (in the range variables) are solutions of system (1.3) and since det Du is invariant with respect to such rotations and translations, we may assume without loss of generality that u(x0 ) = (0, 0), det Du(x0 ) = 0, and

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∆ = u(Ω) ⊂ H = {(y 1 , y 2 ) ∈ R2 | y 2 > 0}. Thus, the tangent line to ∂∆ at u(x0 ) = (0, 0) is perpendicular to e2 = (0, 1). Now det Du(x0 ) = 0 implies that for some ϑ in [0, π ] we have

(2.3)

cos ϑ∇u1 (x0 ) + sin ϑ∇u2 (x0 ) = 0.

❐

If ϑ 6= π /2, the vector νϑ = (cos ϑ, sin ϑ) is not equal to e2 . Thus, the line through (0, 0) perpendicular to νϑ is nontangential to ∂∆ at (0, 0), and it must contain a point y0 in ∆. Now defining uϑ (x), γ , and Γ as in Lemma 2.3, we have (0, 0) ∈ γ , x0 ∈ Γ , and thus ∇uϑ (x0 ) 6= 0 by Lemma 2.3. But this contradicts (2.3). Hence, we must have ϑ = π /2 and ∇u2 (x0 ) = 0. To show that this is impossible, we note that u2 satisfies the nondivergence form elliptic equation obtained by applying the product rule in (1.3). On the other hand, u(Ω) = ∆ ⊂ {(y 1 , y 2 ) | y 2 > 0} and so u2 (x) > 0 in Ω. Since u2 (x0 ) = 0, the Hopf maximum principle (see [10, Lemma 3.4 and Theorem 9.1]) ensures that (∂u2 /∂n)(x0 ) > 0, where n is the inward unit normal to ∂Ω at x0 . But this is impossible, since ∇u2 (x0 ) = 0. Thus, det Du 6= 0 in ∂Ω. 3. T HE GENERAL CASE From this point on we assume only the hypotheses of paragraphs two and three in ¯ ; R2 ). By approximation Section 1. Recall that in this case, u ∈ W 1,2 (Ω; R2 )∩C(Ω and Theorem 2.4, we can prove the following theorem. Theorem 3.1. The mapping u satisfies det Du(x) ≥ 0 a.e. in Ω. ¯; Moreover, if in addition, Ω is a C 1,α domain, Φ is a diffeomorphism in C 1,α (Ω 2 α ¯ 1 ,α R ), and σij is in C (Ω) for 1 ≤ i, j ≤ 2 for some α > 0, then u is a C ¯ onto ∆ ¯ and det Du > 0 in Ω. homeomorphism for Ω Proof. By Theorem 2.4 we need only show that det Du(x) ≥ 0 a.e. in Ω. ¯ , it follows Since Φ is a C 1 -diffeomorphism of degree one on a neighborhood of Ω 1 ,α ¯ 2 that there exists a sequence {Φm } ⊂ C (Ω; R ) such that {Φm } and {DΦm } are ¯ , and Φm → Φ ¯ , det DΦm > 0 in Ω uniformly bounded and equicontinuous in Ω ¯ ; R2 ). Let {Ωm } be a sequence of simply connected smooth domains in C 1 (Ω ¯ m ⊂ Ω, Φ(Ωm ) is a convex subdomain of ∆, and ∂Ωm is uniformly such that Ω Lipschitz continuous. Let um = (u1m , u2m ), where uim satisfies (1.3) in Ωm for ¯ by i = 1, 2, and um = Φm on ∂Ωm . Define vm in Ω vm (x) =

( um (x), Φm (x),

for x in Ωm , ¯ | Ωm . for x in Ω

Univalent Solutions of an Elliptic System of PDE Arising in Homogenization 753

❐

From existence, uniqueness, and a priori estimates for solutions of (1.3) and (1.4), ¯ ; R2 ), and it follows that {vm } is uniformly bounded and equicontinuous in C α (Ω 1 ,2 0 0 is uniformly bounded in W (Ω ) for each subdomain Ω b Ω. Thus, there exists ¯ ; R2 ) and a subsequence, {vmk }, such that vmk converges to u strongly in C α (Ω 1 ,2 2 0 weakly in W (Ω; R ) for each subdomain Ω b Ω. From this and a result of J. M. Ball ([4, Theorem T3.4]), det Dvmk converges in the sense of distributions to det Du in each subdomain Ω0 b Ω. Since det Dvmk > 0 in Ωmk by Theorem 2.4, we have det Du ≥ 0 a.e. in Ω. A consequence of Theorem 3.1 with applications to homogenization is the following result. Corollary 3.2. Let A be a real 2 × 2 constant matrix and let Ω be a bounded, simply connected, convex domain in R2 with Lipschitz boundary. Let σ = [σij ] be a symmetric 2 × 2 matrix-valued function on Ω which satisfies (1.1) and (1.2). Let 1,2 w0 = (w01 , w02 ) be in W0 (Ω; R2 ) such that (3.1)

div[σ (x) · (Dw0 (x) + A)t ] = 0 in Ω,

where for a 2 × 2 matrix B = [bij ], B t is the transpose of B and div B is the vector in P2 R2 defined by (div B)j = i=1 ∂bij /∂xi for j = 1, 2. Then det A · det(Dw0 (x) + A) ≥ 0 for almost every x in Ω. Proof. The result is trivial if det A = 0, so let us assume that det A > 0. Let a1 ≡ (a11 , a12 ) be the first row of A and a2 ≡ (a21 , a22 ) be the second row of A. Define Φ = (Φ1 , Φ2 ) ∈ C ∞ (R2 ; R2 ) by Φ1 (x) = ha1 , xi, Φ2 (x) = ha2 , xi for all x in R2 . Note that Φ is a diffeomorphism of degree 1 from R2 to R2 and, from our assumptions on Ω, the domain ∆ ≡ Φ(Ω) is a bounded convex domain in R2 . Define v0 = (v01 , v02 ), where (3.2)

v01 (x) = w01 (x) + Φ1 (x),

v02 (x) = w02 (x) + Φ2 (x),

for almost every x in Ω. Then v0 ∈ W 1,2 (Ω; R2 ), v0 = Φ on ∂Ω, and ∇v0k = ∇w0k + ak for k = 1, 2 almost everywhere in Ω. From (3.1) it follows that v0 satisfies the hypotheses of Theorem 3.1. Thus, det Dv0 = det(Dw0 + A) ≥ 0 almost everywhere in Ω. This proves the theorem if det A > 0. e and v˜ 0 , Now if det A < 0, we replace Φ and v0 in the above argument by Φ respectively, where

 e 1 (x) = a2 , x , Φ

 e 2 (x) = a1 , x , Φ

and

 v˜ 01 (x) = w02 (x) + a2 , x ,

 v˜ 02 (x) = w01 (x) + a2 , x .

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PATRICIA B AUMAN , A NTONELLA M ARINI & V INCENZO N ESI 



❐

Then the above argument implies that det Dv˜ = det ∇w02 + a2 , ∇w01 + a1 ≥ 0 almost everywhere in Ω, and thus det(Dw + A) ≤ 0 a.e. in Ω. 4. A PPLICATIONS TO G - CLOSURE PROBLEMS

In this section we give a brief account of some of the applications of our results. Let Ω be a simply connected, bounded open set in R2 with Lipschitz boundary, and let Q = [0, 1] × [0, 1] be the unit cube in R2 . Let σ (x) = [σij (x)] satisfy (1.1), (1.2), and in addition assume that σ is Q-periodic on R2 . Definition 4.1. The effective conductivity (or homogenized coefficient) H is the 2 × 2 matrix defined via the following relation (4.1) hHξ, ξi := lim+

inf

ε→0 v∈W 1,2 (Ω) 0

1 |Ω|

Z

Ω

∇v(x)+ξ, 

σ

  x (∇v(x) + ξ) dx, ε

∀ξ ∈ R2 .

Let us recall some well-known facts. The definition above is well-posed: the matrix H depends neither on Ω nor on the particular boundary conditions imposed, nor on the vector ξ . Moreover, H is constant and symmetric. In fact, the inequalities (2.2) are satisfied when σ is replaced by H (with the same constants m and M ). The reader is referred to the books by [6], [9], and [13] for a comprehensive treatment of the theory of asymptotic expansions, Γ -convergence, and homogenization, respectively. A self-contained, shorter presentation, including the results stated in the present section, can be found in [15]. An important problem arising in homogenization theory is to determine the set of possible values for H , when only partial information on the matrix σ is provided. Perhaps the problem which has been most thoroughly examined among this type of problems, is the following. Consider the class of matrices satisfying the hypotheses stated in the present section. Furthermore, assume that the following data are given: an integer N ≥ 2 which represents the number of phases, N regions assigned via the characteristic functions χi (x), with corresponding (positive) conσi , i = 1, . . . , N , and N positive numbers pi , i = 1, . . . , N with ductivities P p = 1, called the volume fractions of the corresponding phases. Then, take i i PN σ of the form σ (x) = ( i=1 σi χi (x))I , where the characteristic functions {χi } satisfy (4.2)

1 |Ω|

Z Ω

χi (x) dx = pi ,

i = 1, 2, . . . N.

The particular family {χi } of characteristic functions is called the microgeometry of the problem. The so called G-closure problem in this context entails finding the range of the effective conductivity H defined by (4.1), with σ given

Univalent Solutions of an Elliptic System of PDE Arising in Homogenization 755 above, under the constraint (4.2). This set is usually denoted by G. In this case, G = G(σ1 , σ2 , . . . , σN , p1 , p2 , . . . , pN ). The name G-closure is motivated by the notion of G-convergence introduced by the pioneering work of Spagnolo [22]. For a review see [25]. The first progress towards solving G-closure problems, was made in [11]. However, the complete solution to this problem for the case N = 2 was found much later. See [23] and [8]. The case N > 2 is still open, although partial results can be found in [11] and [14]. The usual strategy to determine a G-closure is twofold. The first step consists of proving bounds on the set G, depending on the conductivities and their volume fractions, as the microgeometry varies in the range of all admissible ones. These bounds read as inequalities to be satisfied for any choice of H ∈ G. The second step consists of the explicit calculation of optimal microgeometries. In other words, one aims to find microgeometries for which the corresponding homogenized coefficient H belongs to G and it saturates the bounds proved in step one. Clearly, one is able to implement the second step only if the bounds are optimal, i.e., if they cannot be improved. It can be shown that the bounds proved in [14] or in [11] are not optimal when N ≥ 3. Indeed in [16], the author improves upon them making use of Corollary 3.2 of the present paper in an essential way. The details of that work are beyond the scope of the present work. Here, we limit ourselves to rephrasing the G-closure problem in order to make more transparent the relevance of Corollary 3.2. We start from Definition 4.1. For ε > 0 fixed, (4.1) gives a family of variational principles indexed by the vector ξ ∈ R2 . For each such vector, and for ε fixed, the (unique) minimizer is a solution of a linear partial differential equation. It is straightforward to see that Definition 4.1 is equivalent to the following definition in vector notation (one uses the ellipticity of σ and simply sums two contributions). Definition 4.2. The effective conductivity (or homogenized coefficient) H is the 2 × 2 (constant and symmetric) matrix defined by

(4.3)

t

tr[AHA ] := lim+

inf

ε→0 w∈W 1,2 (Ω;R2 ) 0

1 |Ω|

Z

 Ω

tr (Dw(x) + A)  ×σ

  x (Dw(x) + A)t dx, ε

for all A ∈ M2×2 , where M2×2 denotes the set of two by two real matrices.

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One of the classical methods to bound the effective conductivity, exploits the differential constraints on the vector fields involved in Definition 4.2. In particular, one has the following “null-lagrangian property:” (4.4) ∀A ∈ M2×2 , ∀u ∈ W01,2 (Ω; R2 ),

1

Z

|Ω|

Ω

det(Du(x) + A) dx = det A.

The family of equalities above describes a very special case of quasiconvexity. Note that for fixed ε > 0, the Euler-Lagrange equations relative to the variational principle (4.3) are precisely the equations studied in Corollary 3.2, namely equations (3.1). Therefore, it is rather natural that Corollary 3.2 should be instrumental to establish improved bounds on the effective conductivity. We remark that, in order to study a G-closure problem, one is allowed to use a variety of quasiconvex functions. The technique of compensated compactness [24] gives a great deal of flexibility. However, Corollary 3.2 suggests that the determinant is particularly interesting in dimension two, since it captures a simple pointwise information. The interested reader is referred for more details to [16]. Some other applications along the same lines can be found in [17] and [18]. We remark that in the latter work the authors consider a G-closure problem relative to a nonlinear operator. Also in this case one can improve the known bounds using Corollary 3.2. Acknowledgements. P. Bauman was partially supported by NSF Grant #DMS9623438. P. Bauman and A. Marini were partially supported by C.N.R. (GNFM) and MURST. V. Nesi was partially supported by C.N.R. (short term mobility program).

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PATRICIA B AUMAN : Department of Mathematics Purdue University West Lafayette, IN 47907, U. S. A. A NTONELLA M ARINI : Department of Pure and Applied Mathematics University of L’Aquila 67010 L’Aquila, ITALY. V INCENZO N ESI : Department of Mathematics University of Rome I 00185 Rome, ITALY. Received : June 1st, 1999; revised: October 12th, 2000.