An asymptotic formula for the voltage potential in a perturbed ε-periodic composite medium containing misplaced inclusions of size ε

We consider composite media made of homogeneous inclusions with C1,α,boundaries. Our goal is to compare the potential uε in a perfectly periodic composite with the potential uε,d, of a perturbed periodic medium, where the periodicity defects consist of misplaced inclusions. We give an asymptotic expansion of the difference uε,d − uε away from the defects and show that, to first order, a misplaced inclusion manifests itself via a polarization tensor, which is characterized.


Introduction
In this work, we consider a composite medium made of an array of inclusions embedded in a homogeneous background material. We assume that the inclusions are centred on an ε-periodic lattice, except for a small number of them that may have been misplaced: the centres of these 'defects' are at a distance of order ε from the lattice points. Our goal is to compare, sufficiently far from the defects, the potential u ε,d of the perturbed medium with the potential u ε of a perfectly periodic medium.
Fengya et al . [10] (see also [2] and the references therein) studied the perturbations of the potential caused by the presence of small inhomogeneities when the reference or background medium is homogeneous (or sufficiently smooth), and derived an asymptotic expansion for the difference between the perturbed and background potentials. The first correction term in their asymptotic expansion is of the order of the volume ε n of the inhomogeneities and has the following structure: (1.1) where u is the background potential and M j is a polarization tensor, which characterizes how the presence of the jth inhomogeneity, centred at z j , is felt in the far field. The above expression also involves the gradient of the background Green function G(x, z), which makes the expansion particularly interesting for numerical detection of inhomogeneities: linear sampling or Music algorithms detect the singularities of the Green functions, and have proven quite efficient in both impedance imaging of inhomogeneities of low volume fraction [7] and inverse scattering by small inclusions [3]. In this paper we derive a similar asymptotic formula when the background medium is periodic. Our main result, theorem 4.1, shows that inhomogeneities or defects similar in size to the period affect the perturbed potential in a manner similar to the case of a homogeneous background. Indeed, the first term in our asymptotic formula (4.4) has the same structure as (1.1). It involves the gradient of the homogenized potential at the centre of the inclusions, the gradient of the homogenized Green function, and a polarization tensor that combines the influence of the defect at infinity and the interaction of the defect with the periodic structure. Thus, numerical detection of such periodicity defects should be possible using Music algorithms, provided that one has accurate knowledge of the background potential (which might be expensive in practice).
A possible application of our analysis concerns photonic crystals, periodic composite arrays of dielectric materials. In these structures, propagation of waves may be prohibited in certain intervals of frequencies, as a result of destructive interferences between the waves and the structure of the composite [13]. For a mathematical perspective, see the enthusiastic review by Kuchment [14].
Photonic crystals are an example of structures where periodicity or near-periodicity seems to play an important role. As the current manufacturing processes may not guarantee perfect periodicity, it is interesting to study the influence of periodicity defects in these structures, in view of developing methods for non-destructive control.
Our analysis relies on fine regularity results on the potential gradients [15,16], which require that the inclusions be somewhat smooth: their boundaries have regularity C 1,α for some 0 < α 1. Under this hypothesis, ∇u ε,d ∞ can be shown to be bounded independently of ε and of the distance between inclusions, misplaced or not.
The paper is organized as follows. In § 2, we recall classical results about Green functions for uniformly elliptic operators in divergence form with merely bounded and measurable coefficients. We are particularly interested in their behaviour at infinity, and throughout the paper we work in dimension 3 (although some of our results are valid and given in any dimension). Section 3 is concerned only with the background potential u ε . In § 3.1, we recall a few classical results of periodic homogenization, while § 3.2 contains several estimates on the potential u ε and on its gradient. We recall such interior estimates, which were derived by Avellaneda and Lin [4], when the coefficients of the medium have regularity C 0,µ . We then give interior estimates on the gradient ∇u ε , in the case of a composite medium made of a homogeneous background conductor containing homogeneous inclusions with C 1,α boundaries (in which case the conductivity is only globally L ∞ ). These estimates are similar to those given in [15] (and so is their proof) but here we allow a non-zero source term.
We chose to study the case of Neumann boundary conditions for the periodic and perturbed media. The analysis also led us to compare the potentials u ε,d and u ε with the potential u 0 of the homogenized medium. In particular, we give an L 2 error estimate on u ε −u 0 . This kind of estimate is well known in the case of Dirichlet boundary conditions. In the case of Neumann boundary conditions, our result generalizes to dimension 3, a two-dimensional estimate obtained by Moskow and Vogelius [20]. The proofs of all the estimates in this section are given in the appendix. Section 4 contains the main result. We derive there the asymptotic expansion of the potential in the domain with defects. We give the expression of the polarization tensor associated with a periodicity defect and compare it with the formula of [10] that describes the effect at infinity of an inhomogeneity embedded in a smooth matrix.
Throughout the paper, C denotes a generic positive constant, independent of ε.

Properties of the Green function
In this section, we present some known results and properties of the Green function for the elliptic operator when the conductivity a(x) is merely a bounded measurable function in R n . The detailed proofs of the following results can be found in [18,21,22] in the symmetric case and are extended to the case of non-symmetric coefficients in [12].
Let Ω be a smooth bounded domain in R n . We consider a medium with conductivity a ∈ L ∞ (Ω) that is uniformly elliptic Given a Radon measure µ defined on Ω, a function u ∈ L 1 (Ω) is called a weak solution vanishing at the boundary ∂Ω of the equation Lu = µ, if it satisfies  Stampacchia [21] extends the De Giorgi-Nash theorem on C 0,α regularity of solutions to elliptic equations and shows that, when Ω is sufficiently smooth and f ∈ W −1,p (Ω) with p > n, the solution u to (2.2) lies in C 0 (Ω). Moreover, one has where C depends only on p. Consequently, given a Radon measure µ, a function u is a weak solution vanishing on ∂Ω of the equation Lu = µ if and only if ∀f ∈ C 0 (Ω), There is at most one solution to this problem. By (2.3), this solution satisfies ∀f ∈ C 0 (Ω), (Ω), 1/p + 1/p = 1/n, and u W 1,p 0 (Ω) The transformation µ → u is thus the adjoint operator G * of G: as G(W −1,p (Ω)) ⊂ C 0 (Ω), the image by G * of the space of Radon measures on Ω is contained in (Ω). This proves the following theorem. As a consequence, one can define a Green function for L in Ω, as follows.
Definition 2.2. The Green function G(x, y), associated with the operator L on Ω, is defined as the weak solution vanishing on ∂Ω of the equation where δ y is the Dirac mass at y.
The Green function provides a representation formula (see [18, theorem 6.1]): for every Radon measure µ, the integral is finite almost everywhere (a.e.) and is the weak solution vanishing on ∂Ω of the equation Lu = µ. The Green function has the following properties.
Theorem 2.3 (Littman et al . [18]; Grüter and Widman [12]). For each y ∈ Ω, Further, let G andḠ be the Green functions of two uniformly elliptic operators L andL, with ellipticity constants λ, Λ andλ,Λ, respectively. Then for any compact subset K ⊂⊂ Ω, there exist positive constants c and C, which depend only on K, Ω, n and on the ellipticity constants, such that The Lorentz spaces L * p (Ω) involved in these estimates are defined by and are related to the classical L p spaces via the estimates When n 3, as the radius of Ω tends to infinity, the Green function converges to a function G(·, y), Hölder continuous in R n \ {y}. Moreover, G(·, y) ∈ W 1,q loc (R n ) ∩ W 1,2 loc (R n \ y), q < n/(n − 1), and the representation formula (2.5) is valid. In particular, given f ∈ W −1,2 (R n ), the solution u ∈ W 1,2 (R n ) of Lu = f can be represented by The estimates (2.7) hold uniformly in R n with constants that depend only on the ellipticity constants and on n. Comparing the Green functions of L and of the Laplace operator in R n , we see from (2.7) that where C depends only on λ, Λ and n.
In the rest of this paper, we will be concerned with families of operators of the form L ε = div(a(x/ε)∇) defined over a smooth domain Ω ⊂ R 3 , where a is a [0, 1] 3 -periodic piecewise constant function.

Asymptotic behaviour of the background potential and of the associated Green function in periodic composite materials
Let Ω be a smooth bounded domain in R 3 that contains a periodic composite medium composed of cells of size ε. These cells are deduced from the unit cell Y = (0, 1) 3 by translation and rescaling, and are of the form εp + εY , p ∈ Z 3 . The unit cell Y contains an inclusion D 0 ⊂⊂ Y with boundary of class C 1,α , 0 < α 1. We assume that Let 0 < λ Λ and 0 < µ < 1. A(λ, Λ, µ, α) denotes the class of Y -periodic functions a such that a is C 0,µ inD 0 and in Y \ D 0 and such that 0 < λ a(x) Λ in Y . We also denote by L(λ, Λ, µ, α) the class of elliptic operators with coefficients in A(λ, Λ, µ, α) of the form where a ε (x) = a(x/ε). We call these media 'composites with sufficiently smooth inclusions'.

Homogenization
As ε tends to zero, we consider the sequence of elliptic problems with the normalization The effective behaviour of the composite and the asymptotic behaviour of u ε are described in terms of solutions ). The function u ε can be sought formally with the ansatz [6,23] where each function u i (x, y) is Y -periodic with respect to the fast variable y = x/ε. The function u 0 (x, y) = u 0 (x) is independent from y and is the unique solution in W 1,2 (Ω) to the homogenized equation The effective properties of the medium are expressed by the constant, symmetric, positive definite, homogenized matrix A defined by The functions u 1 and u 2 can be written in terms of derivatives of u 0 , up to arbitrary functionsũ 1 ,ũ 2 of the variable x only: If we approximate u ε to first order by (3.7), we may chooseũ 1 = 0. If we seek an approximation up to second order away from the boundary (neglecting boundary layers)ũ 2 may be chosen to be 0, butũ 1 must satisfy

Error estimates
In this section, we give W 1,∞ -interior estimates for solutions u ε to i.e. we are concerned only with perfectly periodic media. We are particularly interested in pointwise estimates on the gradients of u ε , which will be used in the proofs in § 4. When the conductivity a has global Hölder regularity on Y , a C 0,µ (Y ) M , Avellaneda and Lin [4] proved that the potentials u ε are uniformly Lipschitz.
where f ∈ L n+δ for some δ > 0 and g ∈ C 1,ν (∂Ω), 0 < ν 1. There exists a constant C that depends only on λ, Λ, µ M, Ω, ν and δ, such that The regularity hypothesis on a can be relaxed to cover the case of composite media that contain inclusions with sufficiently smooth boundaries. For such media, one can show that the gradient of the potential is uniformly bounded, independently of the inter-inclusion distance. Results of this nature were first obtained by Li and Vogelius [16], and then generalized to strongly elliptic systems by Li and Nirenberg [15]. We state here the version of [15] in the scalar case.
Let D be a bounded domain in R 3 containing L disjoint subdomains D 1 , . . . , D L , of class C 1,α , 0 < α 1, with D = ( L l=1D l ) \ ∂D. We assume that any point x ∈ D belongs to at most two of the boundaries of the D l . For η > 0, we set . Let 0 < µ < 1 and assume that the conductivity a is uniformly elliptic in D and belongs to C µ (D l ), 1 l L.
For any η > 0, there exists a constant C such that, for any Here C depends only on λ, Λ, µ, L, α, η, a C α (D l ) and on the C 1,α norm of the D l . In particular, (3.14) In the following, for each r, 0 < r < 1, and x ∈ R 3 , we set The constant C in the above theorem may, however, grow with the number of inclusions. However, in the case of periodic media, uniform pointwise estimates on the gradients do hold as in theorem 3.1. This is established in the following result, due to Li and Nirenberg (see also the remark in § 5.3 of [4]). Its proof relies on theorem 3.2 and on the three-step compactness method of [4]. Then where C is independent of ε (and thus is independent of the number of inclusions and of the distance between their boundaries).
We will need a slightly different version of theorem 3.3 for solutions of elliptic equations in divergence form, with a source term of the following particular form.
Theorem 3.4 (interior gradient estimates). Assume that . Then where C is independent of ε (and thus of the number of inclusions and of the distances between their boundaries).
On the basis of theorem 3.4, one can proceed as in [5], and generalize to composite media with sufficiently smooth inclusions, error estimates between the potential u ε and the homogenized potential u 0 , and between the ε-periodic Green function and the Green function for the homogenized medium.
However, as we intend to apply such results to Neumann problems, we first state the following L 2 error estimate.
normalized with the condition that Ω u ε = 0. Let u 0 denote the solution to the corresponding homogenized problem also normalized by Ω u 0 = 0. Then the following estimate holds Estimates of this sort are well known for Dirichlet boundary conditions [1,6,19]. For Neumann boundary conditions, a similar error estimate was derived in two dimensions by Moskow and Vogelius [20] in the case of a convex polygon using harmonic conjugates. We show in § A.2 how this estimate generalizes to three dimensions.
We now state uniform error estimates between u ε and u 0 .
for some 0 < σ 1. There then exists a constant C that depends only on λ, Λ, µ, α, Ω and ω such that We remark that, by lemma 3.5, this theorem applies to solutions of (3.16), (3.17). Let G ε and G 0 denote the respective Green functions, vanishing on ∂Ω, of the operators L ε and L 0 . From theorem 3.6, we derive an estimate on the convergence rate of G ε to G 0 . This result is applied, in § 4, when the source is far from the defect. For this reason, we consider below the case G ε (x, y) when x ∈ ω ⊂⊂ Ω and y ∈ Ω \ ω with dist(y, ω) > 0. In [5], when the coefficients have Hölder regularity, similar estimates are established that are valid on the whole of Ω (away from the source). Their derivation requires uniform boundary estimates on L ε -harmonic functions. It would be interesting to study whether such estimates also hold in our context.
Theorem 3.7. Assume that ω ⊂⊂ Ω is a smooth domain. Let G ε and G 0 be the Green functions, vanishing on ∂Ω, for the operators L ε and L 0 (see § 2). There exists a positive constant C, independent of ε, such that, for y ∈ Ω \ ω with dist(y, ω) > 0,

Main result: asymptotics of the perturbed potential
Let Y denotes the unit cell (0, 1) 3 in R 3 . We assume that Y contains an inclusion D 0 , the boundary of which has regularity C 1,α for some 0 < α < 1. We also assume that Let a be a measurable Y -periodic function equal to a constant k in D 0 , 0 < k < ∞, k = 1, and equal to 1 in Y \D 0 . As in § 3, we consider a bounded domain Ω ⊂ R 3 formed by the union of cells, translated and rescaled by ε from the elementary cell Y . The conductivity in Ω is denoted by a ε (x) = a(x/ε). We consider the elliptic operator L ε = − div(a ε (x)∇·). We call the solution u ε to background electrostatic potential in Ω. We study the influence of a particular perturbation of such a medium which consists in misplacing one inclusion. More precisely, let p ∈ Z 3 such that Y p ε := ε(p + Y ) ⊂ Ω, and so that dist(Y p ε , ∂Ω) ε. If the medium were completely periodic, the inclusion contained in the cell Y p ε would occupy the subset ω ε,1 := ε(p + D 0 ). Instead, the inclusion lies in a subset ω ε,2 = ε(p + δ + D 0 ), for some 0 < |δ| < 1. For simplicity, we assume that ω ε,2 does not intersect any of the remaining inclusions. Let ω ε denote the symmetric difference of the sets ω ε,1 and ω ε,2 .
The conductivity a ε,d of the perturbed medium is thus given by The associated potential u ε,d solves div(a ε,d (x)∇u ε,d ) = 0 in Ω, Without loss of generality, we assume that x 0 = 0 belongs to the convex hull of ω ε . LetΩ (note thatω is independent of ε), and define the function a d in R 3 by a d (y) = a ε,d (εy). Throughout this section, we denote by a + (x) and a − (x) the outward and inward limits, respectively, of the discontinuous function a through an interface. Let G ε be the Green function associated with the operator L ε , the solution to vanishing on ∂Ω (see § 2), and G 0 be the Green function of the homogenized operator L 0 , defined by (3.8).
We now state the main result of this paper: an asymptotic expansion for u ε,d −u ε . This expansion has the same structure as that derived in the case of a homogeneous background medium [10], although it involves the homogenized potential u 0 and the homogenized Green function G 0 . As mentioned in § 1, the presence of the Green function (and its singularity) should make this expansion useful for numerical detection purposes.
We note that our analysis extends easily to the case of several misplaced inclusions (or to the case of O(ε) defects with different constant conductivities), provided that they are at distances greater than O(ε) apart.
Theorem 4.1. Assume that Ω and the Neumann data g are sufficiently regular that the homogenized potential u 0 is smooth inside Ω. For any z ∈ Ω at a distance d 0 > 0 away from ω ε , we have The term O(ε 3+(1/4) ) is uniformly bounded by Cε 3+(1/4) , where the constant C depends on d 0 , k and α. The polarization tensor M is given by for 1 i, j 3, where the cell function χ = (χ i ) 1 i 3 is defined by (3.4) and where the auxiliary functions ϕ j,d are defined by (4.14) below.
To prove the theorem, we first establish three lemmas. We introduce two auxiliary functions v ε,d and v d , respective solutions to with µ = div y ((a d − a)(I + ∇ y χ)∇ x u 0 (0)).
Proof. Since the support of µ is included inω, the function v d can be represented in terms of the Green function G associated with (see [18, theorem 6.1], where it is shown that the integral on the right-hand side of (4.9) exists a.e. as a consequence of Fubini's theorem). It follows from (2.8) that Noting that G( · , z) is C 1,µ away from y = z as a consequence of theorem 3.2, we can differentiate (4.9) with respect to y, and we conclude from (4.10) that which proves the lemma.

Thus, integrating by parts yields
Lemma 3.5, theorem 3.6 and the smoothness of the homogenized potential u 0 show that Since a ε is bounded, we conclude that Lemma 4.4. There exists a constant C, independent of ε, such that Proof. Lemma 4.3 shows that it is sufficient to prove that In view of (4.6) and (4.7), φ ε,d solves div(a d (y)∇φ ε,d ) = 0 inΩ ε , By integrating by parts and changing variables back to the fixed domain Ω, we see that The trace theorem and the Poincaré-Wirtinger inequality imply that Since v d decays at infinity (see (4.8)), we have and therefore and the lemma is proved.
Proof of theorem 4.1. Let u ε,d and u ε be the electrostatic-potential solutions to (4.2) and (4.1), respectively. Let z be a point in Ω, at a distance d > 0 away from ω ε . We apply the Green formula in Ω to get Using the continuity of u ε,d and the jump conditions satisfied by its normal derivative across ∂ω ε , the difference between these two equations yields Combining the W 1,∞ -error estimate (3.21) for the Green function G ε and the fact that ∇ x u ε is bounded on every compact subset of Ω that contains ω ε (theorem 3.4), we show that Hence, Thus, by a Taylor expansion of G ε about the origin, Since u ε,d and v d are harmonic inω, the first term in the right-hand side of the above expression vanishes and I 1 reduces to Invoking theorem 3.6 in a fixed subset ω ⊂⊂ Ω that contains ω ε , we see that for some constant C independent of ε, and thus Thus, up to O(ε 3+(1/4) ), the term Turning to I 2 , integration by parts and the change of variables y = x/ε give Lemma 4.4 shows that ∇ y r ε,d (εy) L 2 (Ωε) cε 3/2 . Moreover, theorem 3.4 implies that ∇ x G ε (x, z) is uniformly bounded in ω ε . Consequently, Thus, (4.13) yields To clarify the structure of this expression, we introduce the functions ϕ j,d , 1 j 3, which are solutions of the following: allows us to rewrite where the tensor M is defined by Using the jump condition satisfied by ϕ j,d across ∂ω, one sees that M ij can be expressed as which proves theorem 4.1.
This formula defines a polarization tensor in the spirit of [9,10]. It describes the influence on the far field of a localized defect within the periodic medium. One easily checks that the expression of M ij reduces to that given in [10] when, instead of a misplaced inclusion, one considers a defect (a d = 1 in ω ε,2 ) in a homogeneous background medium (a constant and χ = 0). Also, adapting the proof of [10, lemma 5] shows that M is symmetric.

Appendix A. Proofs of the estimates
The proofs of theorems 3.4-3.7 and of lemma 3.5 are collected in this appendix.

A.1. Proof of theorem 3.4
The proof of this result is based on two main ingredients. The first is the 'threestep compactness method' of Avellaneda and Lin [4,5], who proved Hölder and Lipschitz estimates on u ε , when the coefficients of L ε are smooth (Hölder continuous). The second is the Hölder regularity results for the gradients in composite media containing inclusions with C 1,α -regular boundaries [15,16]. Theorem 3.4 generalizes theorem 3.3 to non-zero right-hand side. Its proof closely follows that of theorem 3.3 (in [15, theorem 0.2]), which itself is based on the arguments of [4,5].
In the sequel, for each r, 0 < r < 1, and x ∈ R 3 , we set We recall the classical characterization of Hölder spaces [8] in terms of the seminorm For each 0 < α < 1, there exist positive constants c 1 , c 2 , which depend only on Ω and α such that, for all u ∈ C 0,α (Ω), We assume that the coefficient a is piecewise smooth and that the boundaries of the inclusions have regularity C 1,α for some 0 < α < 1. Let µ = α/2(α + 1). We begin by proving interior Hölder estimates on u ε (see [15, theorem 5.1]).
Theorem A.1 (interior Hölder estimates). Let f ∈ L ∞ (B 1 ) and h ∈ C 0,µ (B 1 ) 3 . We then assume that u ε satisfies There exists a constant C, which depends only on µ, λ, Λ and α, but which is independent of ε and of the distances between the inclusions, such that The theorem results from the following three lemmas. The difference with [4] mainly lies in the proof of the third lemma, where the regularity hypotheses on the conductivity are determinant.
Lemma A.2 (one-step improvement). There exist θ > 0 and 0 < ε 0 < 1, which depend only on µ, α, λ and Λ, such that if u ε , f and h satisfy Proof. Let µ < µ < 1. As the homogenized operator L 0 is elliptic with constant coefficients, solutions to − div(A∇u 0 ) = 0 in B 1 are smooth. In particular, there exists 0 < θ < 1 such that We fix a value of θ for which (A 4) holds. We prove (A 3) by contradiction, as follows. Assume that there is a sequence L j εj , u εj , f εj , h εj that satisfies 1 and lim f εj L ∞ (B1) = lim h εj C 0,µ (B1) = lim ε j = 0, and such that − Extracting a subsequence, we find an operator L 0 , limit of the operators L j εj in the sense of homogenization, and a function u 0 ∈ H 1 loc (B 1 ), such that u εj u 0 weakly in L 2 (B 1 ), As f εj + ε j div(b εj h εj ) converges to 0 strongly in H −1 (B 1 ), we see that L 0 (u 0 ) = 0 in B 1 . Taking limits in (A 5) we get which is a contradiction. Hence, (A 3) holds for some ε 0 > 0.
Lemma A.3 (iteration). Let θ and ε 0 be as in lemma A.2. Then, for all u ε ∈ L 2 (B 1 ), f ∈ L ∞ (B 1 ) and h ∈ C 0,µ (B 1 ) that satisfy and for all k 1 such that ε/θ k ε 0 , Proof. The proof is by induction on k. Lemma A.2 shows that (A 6) holds for k = 1. Let and, for k satisfying ε/θ k ε 0 and x ∈ B 1 , let Then w ε solves By the induction hypothesis, we see that Thus, we can apply lemma A.2: w ε satisfies (A 3) which, expressed in terms of u ε , yields (A 6).
Thus, for all s 1/ε 0 , we have (A 13) Setting r = sε and combining this last identity with (A 12), (A 10) and (A 13), we finally obtain which is (A 9) at x = 0. By translation, this estimate remains true for all x ∈ B 1/2 . The lemma (and theorem A.1) is proved.
Let χ be the cell function defined in (3.4). To prove theorem 3.4, we again apply the three-step method, this time to estimate the quantity .
Lemma A.5 (one-step improvement). There exist 0 < θ, ε 0 < 1 which depend only on λ, Λ, µ and α, such that, if u ε , f and h satisfy Proof. Let µ < µ < 1. Recalling (3.8), let u 0 and f 0 satisfy Classical regularity estimates [11] show that u 0 ∈ C 1,µ (Ω). Thus, there exists 0 < θ < 1, which depends only on λ and Λ, such that By fixing this value of θ, we prove (A 14) by contradiction. Suppose on the contrary that there is a sequence ε j → 0 and sequences L j εj , u εj , f εj and h εj , such that and for which Passing to a subsequence (not renamed) and using theorem A.1, we find an operator L 0 and functions u 0 ∈ H 1 loc (B 1 ) and f 0 ∈ L ∞ (B 1 ), such that We also note that ε j div(b εj h εj ) → 0 strongly in H −1 (B 1 ), such that L 0 u 0 = f 0 in B 1 . Before passing to the limit in (A 16), we show that |(∇u εj ) 0,θ | is uniformly bounded by a constant that depends only on θ. Indeed, let v ∈ C ∞ 0 (B (1+θ)/2 ) satisfy 0 v 1 and v ≡ 1 on B θ . We have One can easily check that Therefore, given the uniform bounds on u εj and f εj , we conclude that Returning to (A 16), and passing to the limit ε j → 0 yields which contradicts the fact that θ < 1.
The same estimate can be established in B(x, ε/2ε 0 ) for any x ∈ B 1/2 . The proof of theorem 3.4 follows by combining this estimate with (A 2).

A.2. Proof of lemma 3.5
Error estimates in L 2 between u ε and u 0 are well known for Dirichlet problems. For Neumann boundary conditions, Moskow and Vogelius [20] derived twodimensional estimates, using the fact that the harmonic conjugate of the potential satisfies Dirichlet boundary conditions. Our proof does not use this property, although we follow the structure of Moskow and Vogelius's argument.
Step 1. We transform the equation into a first-order system a ε ∇u ε − v ε = 0, − div(v ε ) = 0. and seek an asymptotic expansion for both u ε and v ε . Such an expansion is given explicitly in three dimensions, in [6, pp. 58-65]. Recalling the notation of § 3.1, one may easily check that the first term in the expansion of u ε must be the potential u 0 of (3.8) and that − div y v 0 = 0, Denoting by e p , 1 p 3, the canonical basis of R 3 , we set u 1 (x, y) = −χ j (y) ∂u 0 ∂x j and we define the functionsχ p ∈ H 1 # (Y ) 3 by curl y (a −1 (y) curl y (χ p )) = curl y (a −1 (y)e p ), div y (χ p ) = 0, where curl yχ and ∇ y χ denote the matrices whose columns are the vectors curl yχp and ∇χ j , respectively (see [6]). In particular, according to theorem 3.2, the above relation shows that, under our hypothesis on conductivity,χ ∈ W 1,∞ (Y ) 3 . It also shows thatχ(x/ε) has a trace on ∂Ω, which is uniformly bounded in L ∞ (∂Ω). Following [6], we set As L ε G ε (·, y) = 0 in ω 2 , theorem 3.4 shows that there is a positive constant C independent of ε and y, such that ∇G ε (·, y) L ∞ (ω2) + ∇G 0 (·, y) L ∞ (ω2) C.
Theorem 2.3 in [1] gives us an interior L 2 estimate for the convergence rate of w ε to w 0 : for some constant C independent of ε, w ε − w 0 L 2 (ω2) Cε.