Xavier Blanc
email: blanc@ann.jussieu.fr

Marc Josien
email: marc.josien@enpc.fr

Claude Le Bris
email: claude.le-bris@enpc.fr

& C Le Bris

Precised approximations in

come

Introduction

The present paper follows up on the articles [START_REF] Vions | A possible homogenization approach for the numerical simulation of periodic microstructures with defects[END_REF][START_REF]Local proles for elliptic problems at dierent scales: defects in, and interfaces between periodic structures[END_REF][START_REF]On correctors for linear elliptic homogenization in the presence of local defects[END_REF][START_REF]On correctors for linear elliptic homogenization in the presence of local defects: the case of advection-diusion[END_REF]. In these works, we studied homogenization theory for linear elliptic equations, for which the coecients are assumed to be periodic and perturbed by local defects, that is, L r (R d ) functions, r ∈]1, +∞[. As expected, the macroscopic behavior, in the homogenization limit, is dened by the periodic background only. However, if one is interested in ner convergence properties, possibly with a convergence rate, then the defect may have an impact. In such a case, a corrector taking the defect into account is necessary. Its existence has been proved in [START_REF] Vions | A possible homogenization approach for the numerical simulation of periodic microstructures with defects[END_REF] in the case r = 2, and in [START_REF]Local proles for elliptic problems at dierent scales: defects in, and interfaces between periodic structures[END_REF][START_REF]On correctors for linear elliptic homogenization in the presence of local defects[END_REF] in the general case. Formal arguments in [START_REF] Vions | A possible homogenization approach for the numerical simulation of periodic microstructures with defects[END_REF][START_REF]Local proles for elliptic problems at dierent scales: defects in, and interfaces between periodic structures[END_REF] indicate that this adapted corrector is important for having a good convergence rate and/or convergence in a ner topology.

The aim of the present paper is to prove that the corrector constructed in [START_REF] Vions | A possible homogenization approach for the numerical simulation of periodic microstructures with defects[END_REF][START_REF]Local proles for elliptic problems at dierent scales: defects in, and interfaces between periodic structures[END_REF][START_REF]On correctors for linear elliptic homogenization in the presence of local defects[END_REF][START_REF]On correctors for linear elliptic homogenization in the presence of local defects: the case of advection-diusion[END_REF] indeed allows for such convergence results. The work [START_REF] Le Bris | Compactness methods in the theory of homogenization[END_REF] and, more recently, [START_REF] Hen | Periodic homogenization of Green and Neumann functions[END_REF], are the two major reference works on these issues. They both address the periodic setting, and we will briey summarize the important results they established in Section 1.1 below.

Our proofs, in the setting of a periodic geometry perturbed by a local defect, closely follow the general pattern of the proofs exposed in [START_REF] Hen | Periodic homogenization of Green and Neumann functions[END_REF] and reproduce many key ingredients and details of both [START_REF] Le Bris | Compactness methods in the theory of homogenization[END_REF] and [START_REF] Hen | Periodic homogenization of Green and Neumann functions[END_REF]. For the sake of clarity and brevity, and also with a specic pedagogic purpose because the arguments may become very rapidly technical, we have however decided to present our proofs in the particular case of equations, as opposed to systems. Some simplications of the proofs of [START_REF] Le Bris | Compactness methods in the theory of homogenization[END_REF][START_REF] Hen | Periodic homogenization of Green and Neumann functions[END_REF], which all apply to systems as well as to equations, are then possible. The reader might better, then, appreciate the string of key arguments, in the absence of some unnecessary technicalities. Similarly, we have also provided some additional internal details of the proofs which can be useful to non experts for a better comprehension. Our results carry over to elliptic systems (satisfying the Legendre condition, as is the case for [START_REF] Le Bris | Compactness methods in the theory of homogenization[END_REF][START_REF] Hen | Periodic homogenization of Green and Neumann functions[END_REF]), provided some of the arguments are adjusted, and then follow those of [START_REF] Le Bris | Compactness methods in the theory of homogenization[END_REF][START_REF] Hen | Periodic homogenization of Green and Neumann functions[END_REF] even more closely. We did not check all the details in this direction.

One interesting feature we emphasize in the present contribution is that the results of [START_REF] Le Bris | Compactness methods in the theory of homogenization[END_REF][START_REF] Hen | Periodic homogenization of Green and Neumann functions[END_REF] of the periodic setting indeed carry over not only to the perturbed periodic setting, but also to a quite general abstract setting, which we make precise in Section 1.2 below. The latter observation about the generalization of the results of [START_REF] Le Bris | Compactness methods in the theory of homogenization[END_REF] and related works to non periodic setting is corroborated by the recent works [START_REF]Quantitative stochastic homogenization : local control of homoge-nization error through corrector[END_REF][START_REF]A regularity theory for random elliptic operators[END_REF]. Some of the necessary assumptions presented there (in the context of random homogenization) are quite close in spirit to our own formalization. We consider the following problem:

   -div a x ε ∇u ε = f in Ω,
u ε = 0 on ∂Ω.

(1.1)

Here, Ω is a domain of R d , the regularity of which will be made precise below. The right-hand side f is in L q (Ω) for some q ∈]1, +∞[, and the matrix-valued coecient a satises the following assumptions:

a = a per + a (1.2)
where a per denotes a periodic unperturbed background, and a the perturbation, with

          
a per (x) + a(x) and a per (x) are both uniformly elliptic, in

x ∈ R d , a per ∈ L ∞ (R d ) d×d , a ∈ L ∞ (R d ) ∩ L r (R d ) d×d , for some 1 ≤ r < +∞ a per , a ∈ C 0,α unif R d d×d for some α > 0, (1.3)
where C 0,α unif (R d ) denotes the space functions that are, uniformly on R d , Hölder continuous with coefcient α.

From now on, we will not make the distinction between the spaces L q (Ω), (L q (Ω))

d and (L q (Ω)) d×d , denoting z L q (Ω) the norm of z even if z is a vector-valued or a matrix-valued function. The same convention is adopted for Hölder spaces C 0,α .

We also note that we assume d ≥ 3. All our proofs and results can be adapted to the dimension d = 2. Of course, dimension 1 is specic and can be addressed by (mostly explicit) analytic arguments that we omit here.

All the results we present here have been announced in [START_REF]Approximation locale précisée dans des problèmes multi-échelles avec défauts localisés[END_REF], and are part of the PhD thesis [START_REF] Wf Tosien | Etude mathématique et numérique de quelques modèles multi-échelles issus de la mécanique des matériaux [Mathematical and numerical study of some multiscale models for the materials sciences[END_REF].

The periodic case

In the periodic case, that is, a ≡ 0, it is well-known (see for instance [START_REF]eF fensoussnD tFEvF vionsD nd qF pniolou, Asymptotic Analysis for Periodic Structures[END_REF]) that problem (1.1) converges to the following homogenized problem

-div [a * ∇u * ] = f in Ω, u * = 0 on ∂Ω, (1.4)
where a * is a constant matrix. It is classical that u ε -→ u * in L 2 (Ω), and that ∇u ε -∇u * in L 2 (Ω). In order to have strong convergence of the gradient, correctors need to be introduced, that is, the solutions to the following problem -div (a per (x) (p + ∇w p,per (x))) = 0, w p,per is periodic, (1.5) posed for each xed vector p ∈ R d . It is well-known (see, here again, [START_REF]eF fensoussnD tFEvF vionsD nd qF pniolou, Asymptotic Analysis for Periodic Structures[END_REF]), that problem (1.5) has a unique solution (up to the addition of a constant), for any p ∈ R d . Given (1.3), elliptic regularity implies that ∇w p,per ∈ C 0,α unif (R d ). Introducing the remainder

R ε per (x) := u ε (x) -u * (x) -ε d j=1 w per,ej x ε ∂ j u * (x), (1.6)
the results of [START_REF]eF fensoussnD tFEvF vionsD nd qF pniolou, Asymptotic Analysis for Periodic Structures[END_REF] imply that ∇R ε per -→ 0 in L 2 (Ω), with the following convergence rate:

∇R ε per L 2 (Ω) ≤ C f L 2 (Ω) √ ε, (1.7)
for some constant independent of f . The convergence rate O( √ ε) is mainly due to the existence of a boundary layer, and an O(ε) convergence can actually be proved for interior domains.

In [START_REF]Estimation d'erreur et éclatement en homogénéisation périodique[END_REF][START_REF]Error estimate and unfolding for periodic homogenization[END_REF], the generalization of the above results (both (1.7) and interior convergence of order ε) are proved under more general assumptions (Ω of class C 1,1 , a ∈ L ∞ and the corrector is not assumed to have its gradient in L ∞ ). Also in [START_REF]Error estimate and unfolding for periodic homogenization[END_REF], in the case of Lipschitz domains, a convergence up to the boundary of order ε γ , for some 0 < γ ≤ 1/3, is established.

In order to have a O(ε) convergence rate up to the boundary, an adaptation of the corrector is needed. This question was studied in [START_REF]Error estimates for periodic homogenization with non-smooth coecients[END_REF] in the case of non-Hölder coecients. For the case of systems (as opposed to equations) it was studied in [START_REF] Hen | Convergence rates in L 2 for elliptic homogenization problems[END_REF] (actually also with non-homogeneous Dirichlet conditions).

Issues regarding the convergence of the remainder are also addressed in [START_REF] Le Bris | Compactness methods in the theory of homogenization[END_REF], where Avellaneda and Lin proved uniform (with respect to ε) continuity for the operator L ε which, to the couple (f, g), associates the solution u ε of (1.1) with Dirichlet condition u ε = g. This continuity holds from L q (Ω) × C 0,γ (∂Ω) to C 0,µ (Ω) if q > d, with µ = min(γ, d/q). If q ≤ d, with homogeneous Dirichlet conditions, the continuity holds from L q (Ω) to W 1,r (Ω), with 1/r + 1/q = 1/d. These results also hold for systems, and actually improve an earlier and more restricted work [START_REF]Homogenization of elliptic problems with L p boundary data[END_REF]. In [START_REF]Compactness methods in the theory of homogenization. II. Equations in nondivergence form[END_REF], the same kind of results were extended to equations in non-divergence form. In [START_REF] Hen | Periodic homogenization of Green and Neumann functions[END_REF], estimates were proved for the convergence of the Green functions associated to (1.1), both for Dirichlet and Neumann conditions. These estimates allow to prove the convergence rate of R ε in W 1,p .

All the above results are valid only for periodic coecients. In the preprint [START_REF]A regularity theory for random elliptic operators[END_REF], some important results of [START_REF] Le Bris | Compactness methods in the theory of homogenization[END_REF] were extended to the stochastic case, using the idea that, in [START_REF] Le Bris | Compactness methods in the theory of homogenization[END_REF], periodicity was only used to ensure some uniform H-convergence. This is also a key idea of the present work.

The periodic case with a local defect

In order to develop the approximation estimates for (1.1)-(1.2)-(1.3) for a ≡ 0, we dene the corrector problem

     -div (a (p + ∇w p )) = 0 in R d , |w p (x)| 1 + |x| -→ |x|→+∞ 0. (1.8)
In the special case a ≡ 0, a Liouville-type theorem was proven in [4], showing that (1.8) reduces to (1.5): up to the addition of a constant, the only solution that is strictly sublinear at innity is the periodic solution. In the case a ≡ 0, it has been proven in [START_REF] Vions | A possible homogenization approach for the numerical simulation of periodic microstructures with defects[END_REF][START_REF]Local proles for elliptic problems at dierent scales: defects in, and interfaces between periodic structures[END_REF][START_REF]On correctors for linear elliptic homogenization in the presence of local defects[END_REF] (see also the recent work [START_REF]Riesz transform under perturbations via heat kernel regularity[END_REF], that brings a dierent perspective) that Problem (1.8) has a solution, that reads as

w p = w p,per + w p , (1.9)
where w p,per is the solution to (1.5), w p is the solution to

-div (a ∇ w p ) = div ( a (p + ∇w p,per )) in R d , (1.10) and, if a ∈ L r (R d ), ∇ w p ∈ L r (R d ), for any r ∈]1, +∞[.
Even if a ≡ 0, the proofs of [START_REF]eF fensoussnD tFEvF vionsD nd qF pniolou, Asymptotic Analysis for Periodic Structures[END_REF] still imply in this case that u ε -→ u * in L 2 (Ω) and ∇u ε -∇u * in L 2 (Ω), as ε → 0, where u * solves (1.4), and the matrix a * is equal to the periodic homogenized matrix. However, in order to improve and quantify this convergence, [START_REF] Vions | A possible homogenization approach for the numerical simulation of periodic microstructures with defects[END_REF][START_REF]Local proles for elliptic problems at dierent scales: defects in, and interfaces between periodic structures[END_REF][START_REF]On correctors for linear elliptic homogenization in the presence of local defects[END_REF] show that we need to replace the periodic corrector (1.5) by the solution to (1.8), and dene

R ε (x) := u ε (x) -u * (x) -ε d j=1 w ej x ε ∂ j u * (x), (1.11)
instead of (1.6). Then we have:

Theorem 1.1 (Local defects in periodic backgrounds). Assume d ≥ 3. Consider (1.1), where the matrix-valued coecient a satises (1.2), and a per and a satisfy (1.3). Assume that Ω is a C 2,1 domain, that Ω 1 ⊂⊂ Ω, that r = d and dene

ν r = min 1, d r ∈]0, 1]. (1.12)
Let f ∈ L 2 (Ω), and let u ε , u * be the solutions to (1.1) and (1.4), respectively. Dene R ε by (1.11), where the corrector w p with p = e j , 1 ≤ j ≤ d, is dened by (1.9)-(1.10)-(1.5) (thus in particular solves (1.8)). Then R ε satises the following:

1. R ε ∈ H 1 (Ω), and R ε L 2 (Ω) ≤ Cε νr f L 2 (Ω) , (1.13)
∇R ε L 2 (Ω1) ≤ Cε νr f L 2 (Ω) . (1.14) 2. If f ∈ L q (Ω) for some q ≥ 2, then R ε ∈ W 1,q (Ω) and ∇R ε L q (Ω1) ≤ Cε νr f L q (Ω) . (1.15) 3. If f ∈ C 0,β (Ω) for some β ∈]0, 1[, then R ε ∈ W 1,∞ (Ω) and ∇R ε L ∞ (Ω1) ≤ Cε νr ln 2 + ε -1 f C 0,β (Ω) , (1.16)
where, in (1.13) through (1.16), the various constants C > 0 do not depend on f nor on ε.

Given (1.12), this result gives two dierent behaviors of the remainder R ε according to r < d or r > d. In the rst case, the defect is so localized that the estimates are exactly those of the periodic case [START_REF] Hen | Periodic homogenization of Green and Neumann functions[END_REF]. On the contrary, if r > d, the defect is spread out, and the quality of approximation deteriorates as r grows. In the critical case r = d, we can apply the results of the case r > d in order to have the above estimates, in which ε νr is replaced by ε ν , for any ν < ν r = 1.

As already pointed out in [START_REF]Local proles for elliptic problems at dierent scales: defects in, and interfaces between periodic structures[END_REF], the case r = d is a critical case for the existence of a corrector. Indeed, even if a per ≡ 1, hence ∇w per = 0, the corrector equation reads as

-∆ w p = div ( ap) . Hence, as |x| → +∞, w p (x) ≈ C (x -y) • ( a(y) p)
|x -y| d dy, for some constant C = 0. This makes clear the fact that a(y) ∼ |y| -1 is reminiscent of the criticality of the space L d (R d ).

Remark 1.2. In Theorem 1.1, the domain Ω is assumed to be C

prove that R ε ∈ W 1,∞ (Ω).

Abstract general assumptions

As we shall see below, Theorem 1.1 is a consequence of a more general, abstract, result that we state in the present subsection. The point is that, in the theory of [START_REF] Le Bris | Compactness methods in the theory of homogenization[END_REF], the periodicity of the matrix-valued coecient a is essentially useful in order to have a bounded corrector. This assumption may be replaced by uniform H-convergence (a notion which is made precise below in Denition 1.3).

Let us now emphasize that (1.1) considers a rescaled coecient a x ε , which is a strong assumption of our setting. This implies, since a * is dened as some weak limit of functions of x/ε, that a * is homogeneous of degree 0. Hence, if it is continuous, it must be a constant. This is why we hereafter assume that a * is a constant matrix.

(1.17)

We now introduce a set of assumptions that formalize our mathematical setting. We consider a matrix-valued coecient a that satises the following conditions Assumption (A1). There exists µ > 0 such that

∀x ∈ R d , ∀ξ ∈ R d , µ|ξ| 2 ≤ (a(x)ξ) • ξ ≤ 1 µ |ξ| 2 .

Assumption (A2

). There exists α ∈]0, 1[ such that a ∈ C 0,α unif (R d ). Assumptions (A1) and (A2) are standard, and were made already in [START_REF] Le Bris | Compactness methods in the theory of homogenization[END_REF]. We now give more specic assumptions that aim at generalizing periodicity. The rst one is the existence of a corrector: Assumption (A3). For any p ∈ R d , there exists w p ∈ H 1 loc (R d ) solution to the corrector equation (1.8).

As in the periodic case, we assume that the gradient of the corrector is bounded uniformly:

Assumption (A4). For any p ∈ R d , the gradient of w p is in L 2 unif (R d ), that is:

∇w p L 2 unif (R d ) := sup x∈R d ∇w p L 2 (B(x,1)) < +∞,
where B(x, 1) denotes the unit ball of center x.

In the periodic case, we have ∇w p

• ε -0 as ε → 0. Moreover, this property is uniform with respect to translation. This is a property we will impose here:

Assumption (A5). For any sequence (y n ) n∈N of vectors in R d and any sequence ε n → 0, and for

any p ∈ R d , Q ∇w p x ε n + y n dx -→ n→+∞ 0,
where Q is the unit cube of R d .

With a view to addressing non-symmetric matrix-valued coecients, note that, in contrast to (1.2), the fact that a satises Assumption (A3)-(A4)-(A5) does not imply that a T does. We will in some situations need to assume that a T also satises Assumption (A3)-(A4)-(A5), and likewise other assumptions that follow below. In such a case, we denote by w T p the corrector associated to the coecient a T . We will assume that the convergence to the homogenized matrix a * is uniform in the following sense:

Assumption (A6). There exists a constant matrix a * such that, for any sequence (y n ) n∈N of vectors in R d , any sequence ε n → 0 and for any

p ∈ R d , Q a x ε n + y n p + ∇w p x ε n + y n dx -→ n→+∞ a * p,
where the matrix a * is the homogenized matrix in (1.4).

It is stated in Proposition 2.4 below that this implies uniform H-convergence, in the sense of the following denition:

Denition 1.3. We say that the matrix-valued coecient a x ε uniformly H-converges to a * if for any sequence ε n → 0 and any sequence

(y n ) n∈N , a x ε n + y n H-converges to a * .
For the denition of H-convergence itself, we refer to [27, Denition 1] or [START_REF]The general theory of homogenization[END_REF]Denition 6.4].

As we will see below, an important quantity in order to analyze the behaviour of the remainder R ε dened by (1.11) is the potential associated with a. In order to dene it, we rst introduce the vector eld M k dened by

M i k (x) = a * ik - d j=1 a ij (x) (δ jk + ∂ j w e k (x)) , 1 ≤ i ≤ d, (1.18)
which is divergence-free, according to (1.8). Hence, formally, there exists B ij k (x), which is skewsymmetric with respect to the indices i, j, and is solution to

div(B k ) = M k , that is, ∀i, j, k ∈ {1, . . . , d}, B ij k = -B ji k . (1.19) ∀j, k ∈ {1, . . . , d} , d i=1 ∂ i B ij k = M j k . (1.20)
A simple way to build this potential B is to solve the following equation

∀ i, j, k ∈ {1, . . . , d} , -∆B ij k = ∂ j M i k -∂ i M j k . ( 1
∂ i B ij k = -∆M j k .
Hence, up to the addition of a harmonic function, we nd (1.20). In most cases, this harmonic function is necessarily a constant (think for instance of the periodic case).

The above construction can be made precise in the periodic case (see [START_REF]Homogenization of dierential operators and integral functionals[END_REF], pp 26-27). We will see below how and why the construction also makes sense in our setting (1.2)- (1.3).

The link between B and R ε will be clear below when we write the equation satised by R ε (see (2.6)-(2.7)). In order to apply a method close to that of [START_REF] Le Bris | Compactness methods in the theory of homogenization[END_REF], we are going to assume that, in some sense, εw p (x/ε) and εB(x/ε) vanish as ε → 0. This is the meaning of the following two assumptions Assumption (A7). There exists C > 0 and ν ∈ [0, 1[ such that, for any x ∈ R d , any y ∈ R d , and any k ∈ {1, . . . , d},

|x -y| ≥ 1 ⇒ |w e k (x) -w e k (y)| ≤ C|x -y| 1-ν .

Assumption (A8

). There exists a potential B ∈ H 1 loc (R d ) dened by (1.21), and there exists C > 0 such that, for any x ∈ R d and any

y ∈ R d , |x -y| ≥ 1 ⇒ |B(x) -B(y)| ≤ C|x -y| 1-ν .
Here, the constant ν ∈ [0, 1[ is assumed to be the same as in Assumption (A7). Proposition 5.5 below will establish that, in the case of a coecient a satisfying (1.2) and (1.3), the above assumptions are satised with ν = ν r dened by (1.12).

Our main result in this general abstract setting is Theorem 1.4 (Abstract general setting). Assume d ≥ 3 and that the coecients a and a T (and their respective correctors w p and w T p ) satisfy Assumptions (A1) through (A6), and (A7)-(A8) for some ν > 0. Assume that Ω is a C 2,1 domain, and that Ω 1 ⊂⊂ Ω. Let ε ∈]0, 1[ and let u ε , u * , R ε be dened by (1.1), (1.4), (1.11), respectively, where f ∈ L 2 (Ω). Then we have

1. R ε ∈ H 1 (Ω), and R ε L 2 (Ω) ≤ Cε ν f L 2 (Ω) , (1.22)
∇R ε L 2 (Ω1) ≤ Cε ν f L 2 (Ω) . (1.23) 2. If f ∈ L p (Ω) for some p ≥ 2, then R ε ∈ W 1,p (Ω)
and

∇R ε L p (Ω1) ≤ Cε ν f L p (Ω) . (1.24) 3. If f ∈ C 0,β (Ω) for some β ∈]0, 1[, then R ε ∈ W 1,∞ (Ω) and ∇R ε L ∞ (Ω1) ≤ Cε ν ln 2 + ε -1 f C 0,β (Ω) , (1.25)
where in (1.22) through (1.25), the various constants C > 0 do not depend on f nor on ε.

The proof of Theorem 1.4 will consist in applying the strategy of proof of [START_REF] Le Bris | Compactness methods in the theory of homogenization[END_REF] and [START_REF] Hen | Periodic homogenization of Green and Neumann functions[END_REF], which were originally restricted to the periodic case. Here, periodicity is replaced by Assumptions (A3) through (A8). The proofs follow those of [START_REF] Le Bris | Compactness methods in the theory of homogenization[END_REF][START_REF] Hen | Periodic homogenization of Green and Neumann functions[END_REF], but we need to everywhere keep track of the use of assumptions (A3) through (A8), and check that these properties are sucient to proceed at each step of the arguments. Remark 1.5. As we already pointed out in Remark 1.2 for the specic case of localized defects, in Theorem 1.4, the assumption that Ω is of class C 2,1 is, here again, only needed for the estimate

∇R ε L ∞ (Ω) .
Given this result, it is clear that proving Theorem 1.1 amounts to proving that, in the case of a defect, Assumptions (A1) through (A8) are satised with ν = ν r dened by (1.12).

Our article is organized as follows. In Section 2, we start with some comments on Assumptions (A1) through (A8). Then we study the existence and uniqueness of the potential B, and we relate it to the remainder R ε , using (2.6)-(2.7), that is,

-div a x ε ∇R ε = div (H ε ) , with
H ε i (x) = ε d j,k=1 a ij x ε w e k x ε ∂ j ∂ k u * (x) -ε d j,k=1 B ij k x ε ∂ j ∂ k u * (x),
Our method to prove estimates on R ε relies on some regularity properties of the operator -div(a(x/ε)∇•) on the one hand, and bounds on the right-hand side H ε on the other hand. In Section 3, we prove such regularity estimates in the homogeneous case (that is, if the right-hand side is 0). In Section 4, we extend these results to the inhomogeneous case. Finally, in Section 5, we conclude the proof of Theorem 1.4 (abstract setting) and that of Theorem 1.1 (local defects).

2 Preliminaries

Some remarks on our assumptions

Alternative formulations of our Assumptions. Assume (A1), (A2) and (A3). Then, it is clear that Assumptions (A4) and (A5) are equivalent to

∇w p x ε n + y n - n→+∞ 0 in L 2 (D),
for any bounded Lipschitz domain D, any p ∈ R d , and for any sequences (y n ) n∈N and ε n → 0.

Similarly, if Assumptions (A1), (A2) and (A3) are satised, Assumptions (A4) and (A6) are equivalent to

a x ε n + y n p + ∇w p x ε n + y n - n→+∞ a * p in L 2 (D),
for any bounded Lipschitz domain Ω, any p ∈ R d , and for any sequences (y n ) n∈N and ε n → 0.

Another important point is that Assumptions (A4) and (A5) are in fact equivalent to some strict sublinearity condition at innity for the corrector: Lemma 2.1. Assume that the matrix-valued coecient a satises Assumptions (A1) and (A3). Then, it satises Assumptions (A4) and (A5) if and only if

∀p ∈ R d , lim |x|→+∞ sup y∈R d |w p (x + y) -w p (y)| 1 + |x| = 0. (2.1)
Proof. We rst assume that Assumptions (A4) and (A5) are satised and prove (2.1) using a contradiction argument. If (2.1) does not hold, then there exists two sequences (y n ) n∈N and (x n ) n∈N such that

|x n | -→ n→+∞ +∞, and |w p (x n + y n ) -w p (y n )| 1 + |x n | ≥ γ > 0,
where γ does not depend on n. Dening

ε n = |x n | -1 and x n = x n /|x n |, this inequality implies ε n → 0, |x n | = 1, ε n w p x n ε n + y n -w p (y n ) ≥ γ > 0. Hence, dening v n (x) := ε n w p x ε n + y n -w p (y n ) , we have v n (0) = 0, |v n (x n )| ≥ γ > 0, |x n | = 1. (2.2)
Moreover, Assumption (A5) implies [START_REF]Elliptic partial dierential equations of second order[END_REF]Theorem 8.24] imply that v n is bounded C 0,β (B(0, 2)) for some β > 0. Hence, up to extracting a subsequence, it converges in C 0 (B(0, 1)) to some v ∈ C 0 (B(0, 1)). Now, extracting a subsequence once again, we have

∇v n = ∇w p • ε n + y n - n→+∞ 0 in L 2 loc (R d ). (2.3) Since -div a x εn + y n ∇ (v n (x) + p • x) = 0, Nash-Moser estimates
x n → x, with |x| = 1. Hence, (2.2) implies v(0) = 0, |v(x)| ≥ γ > 0, |x| = 1.
Since (2.3) implies ∇v = 0, we have reached a contradiction.

Conversely, if (2.1) is satised, then there exists A > 0 such that

∀x ∈ B(0, A) C , ∀y ∈ R d , |w p (x + y) -w p (y)| ≤ 1 + |x|.
If necessary, we can take A large enough to have A ≥ 2. In particular, we have

|w p (x+y)-w p (y)+p•x| ≤ 1 + A + A|p| on ∂B(0, A). Recalling that -div x [a(x + y)∇ x (w p (x + y) -w p (y) + p • x)] = 0, in B(0, A), this implies that ∀x ∈ B(0, A), |w p (x + y) -w p (y) + px| ≤ 1 + A + A|p|.
Then, we apply the Caccioppoli inequality, which gives a constant C depending only on the coecient a such that B(y,1)

|∇w p (z) + p| 2 dz ≤ C B(y,2) |w p (z) -w p (y) + p • (z -y)| 2 dz = C B(0,2) |w p (x + y) -w p (y) + p • x| 2 dx ≤ C (1 + A + A|p|) |B(0, 2)|.
This implies Assumption (A4). In order to prove Assumption (A5), we integrate by parts, nding

Q ∇w p x ε n + y n dx = ∂Q ε n w p x ε n + y n n(x)dx = d j=1 ∂Q + j ε n w p x ε n + y n -w p - x ε n + y n n(x)dx.
Here, ∂Q ± j denotes the faces of the cube Q, namely the set of equations {|x k | < 1/2 k = j, x j = ±1/2}, and n(x) is the outer normal to Q at point x. Applying (2.1), we nd (A5).

Logical links between our assumptions. We have the following logical links between the assumptions Lemma 2.2. Assume that the matrix-valued coecient a satises Assumptions (A1) and (A3).

1. If it satises Assumption (A7), then it satises Assumptions (A4) and (A5).

If it satises Assumption (A8), then it satises Assumption (A6).

Proof. We rst prove Assertion 1: if (A7) holds, then clearly (2.1) is satised. Hence, applying Lemma 2.1, we have (A4) and (A5).

As for Assertion 2, B satises (1.20), hence

Q   d j=1 a ij x ε n + y n δ jk + ∂ j w e k x ε n + y n -a * jk   dx = Q d j=1 ∂ j B ij k x ε n + y n dx = d j=1 ∂Q ε n B ij k x ε n + y n -B ij k (y n ) e j • n(x)dx, (2.4)
where n(x) is the outer normal to Q at point x. Applying Assumption (A8), we have, for any x ∈ ∂Q,

ε n B ij k x ε n + y n -B ij k (y n ) ≤ Cε n |x| 1-ν ε ν-1 n ≤ C|x|ε ν n .
Inserting this estimate into (2.4), we prove Assumption (A6).

Remark 2.3. The above proof implies that, if B satises (2.1), that is,

lim |x|→+∞ sup y∈R d |B(x + y) -B(y)| 1 + |x| = 0, (2.5)
then it satises Assumption (A6). Indeed, (2.5) is sucient, with (2.4), to prove (A6).

Uniform H-convergence. First, we prove that under Assumptions (A1) through (A6), we have a uniform H-convergence property, in the sense of Denition 1.3:

Proposition 2.4. Assume that the matrix-valued coecient a satises Assumptions (A1) through (A6). Then, for any sequence (y n ) n∈N of R d and any sequence (ε n ) n∈N of positive numbers such that ε n → 0, and any bounded domain Ω, the coecient a x εn + y n H-converges to a * on Ω, where a * is dened by Assumption (A6).

Proof. This is a standard application of homogenization tools (div-curl lemma in particular, see [22, Lemma 1.1]), so we skip it. The only important point is that all the estimates, hence the convergences, are uniform with respect to y n .

The following example proves that (A6) is not satised in general: in dimension 1, dene

a(x) = 2 if 2 n ≤ x ≤ 2 n + 2 n log(1+|n|) , n ∈ Z, 1 otherwise.
Then it is clear that a * = 1, and that the corrector is equal to w = 1-a a . Hence, using y n = 2 n and ε n = log(1 + |n|)2 -n , we have

1 0 a y n + x ε n w y n + x ε n dx = -1.
Hence, Assumption (A6) is not satised.

The matrix-valued coecients a and a T . If the matrix-valued coecient a is not symmetric, we will in the sequel need to assume that both a and a T satisfy assumptions (A3) through (A8) (note that (A1) and (A2) are stable under transposition of a).

In full generality, the existence of strictly sublinear correctors satisfying Assumptions (A4) and (A5) for the coecient a does not imply the existence of correctors for the adjoint coecient a T satisfying the same properties, as the following two-dimensional counter-example shows it. Note that it extends mutatis mutandis to any dimension d ≥ 3.

Consider

a(x 1 , x 2 ) = 1 γ(x 2 ) 0 1 ,
where γ ∈ L ∞ (R), and |γ| ≤ 1, so that a is indeed uniformly elliptic. Then div(ae 1 ) = div(ae 2 ) = 0, hence the correctors associated with a are all equal to 0. We also compute

div a T e 1 = γ (x 2 ).
Assume that a T admits a corrector for the vector e 1 , and that it sastises (A4) and (A5). We denote it by w T e1 . It is solution to

∂ 2 1 w T e1 + ∂ 2 γ(x 2 ) ∂ 1 w T e1 + 1 + ∂ 2 2 w T e1 = 0. Hence, v = ∂ 1 w T e1 is a solution to ∆v+∂ 2 (γ(x 2 )∂ 1 v) = 0.
This is an elliptic equation, and v ∈ L 2 unif (R d ) according to (A4). Hence, applying the Liouville theorem, v is a constant. If this constant is not 0, then w T e1 cannot be sublinear at innity. Hence v = 0, which means that w T e1 depends only on x 2 . Hence ∂ 2 2 w T e1 = -γ (x 2 ). This implies

w T e1 (x 1 , x 2 ) = C 2 - x2 0 (γ(z) + C 1 ) dz.
We choose for γ the function

γ = χ * γ 0 , with γ 0 (z) = 1 2 n∈N 1 [2 2n+1 ,2 2n+2 ] (z) + 1 [-2 2n+2 ,-2 2n+1 ] (z) ,
where χ is a smooth compactly-supported function such that 0 ≤ χ ≤ 1 and χ = 1. For this γ, it is easily seen that w T e1 cannot be strictly sublinear at innity.

On the value of ν r . Let us point out that the value (1.12) of ν r is optimal in the following sense: rst, in the periodic case, we recover the results of [START_REF] Hen | Periodic homogenization of Green and Neumann functions[END_REF] (with ν r = 1, that is, both the correctors and the potential are bounded). Second, we have the following example, in dimension one, in which (R ε ) is bounded from below, up to a logarithmic term, by ε νr . It is unclear to us whether a similar example can, or not, be constructed in higher dimensions. It however strongly suggests that the convergence rate stated is sharp.

Consider a ∈ L r (R), 0 ≤ a ≤ 1, a(x) = a(-x)
, and a per = 1.

Then a * = 1, and the corrector is easily seen to be equal to

w(x) = w(x) = - x 0 a(z) 1 + a(z)
dz.

In the special case f = 1, if we solve (1.1) and (1.4) with Ω =] -1, 1[, one easily computes

(u ε ) (x) = - x 1 + a x ε , (u * ) (x) = -x.
Hence, computing (R ε ) , we have

(R ε ) (x) = (u ε ) (x) -(u * ) (x) -εw x ε (u * ) (x) -w x ε (u * ) (x) = - x 1 + a x ε + x -ε x/ε 0 a(z) 1 + a(z) dz -x a x ε 1 + a x ε = -ε x/ε 0 a(z) 1 + a(z) dz. Hence, since 0 ≤ a ≤ 1, (R ε ) (x) ≥ ε 2 x/ε 0 a(z)dz.
Using

a(z) = 1 (1 + |z|) 1/r (1 + log(1 + |z|) 1+δ ) 1/r , δ > 0, we nd that, if x > 0, (R ε ) (x) ≥ ε 2 x/ε 0 dz (1 + |z|) 1/r (1 + log(1 + |z|) 1+δ ) 1/r ≥ ε 2 x/ε 0 dz (1 + |x|/ε) 1/r (1 + log(1 + |x|/ε) 1+δ ) 1/r = ε 2 |x| ε 1 (1 + |x|/ε) 1/r (1 + log(1 + |x|/ε) 1+δ ) 1/r ≥ Cε 1/r log ε -1 -(1+δ)/r .
Hence, estimate (1.14) is optimal, up to logarithmic terms.

Equation satised by the remainder

We now prove Proposition 2.5. Assume (A1), (A3), (A4), and that there exists

B ∈ W 1,1 loc R d solution to (1.19)- (1.20). Then R ε dened by (1.11) solves -div a x ε ∇R ε = div (H ε ) , (2.6)
H ε i (x) = ε d j,k=1 a ij x ε w e k x ε ∂ j ∂ k u * (x) -ε d j,k=1 B ij k x ε ∂ j ∂ k u * (x), (2.7)
where w e k is the corrector dened by (1.8)

with p = e k , 1 ≤ k ≤ d.
Proof. By denition of R ε , that is, (1.11),

∇R ε (x) = ∇u ε (x) -∇u * (x) - d j=1 ∂ j u * (x)∇w ej x ε -ε d j=1 w ej x ε ∇∂ j u * (x).
We have, using

-div a x ε ∇u ε = -div (a * ∇u * ), -div a x ε ∇R ε = div   a x ε -a * ∇u * + a x ε d j=1 ∂ j u * (x)∇w ej x ε +ε a x ε d j=1 w ej x ε ∇∂ j u * (x)   ,
in the sense of distributions. Using the denition (1.18

) of M k = (M 1 k , . . . , M d k ) T , this reads as -div a x ε ∇R ε = div   - d k=1 ∂ k u * (x)M k x ε + ε a x ε d j=1 w ej x ε ∇∂ j u * (x)  
We concentrate on the rst term of the right-hand side, and use div(M k ) = 0:

div d k=1 ∂ k u * (x)M k x ε = d j=1 d k=1 M j k x ε ∂ j ∂ k u * (x)
.

We now use the potential B dened by (1.19)-(1.20), and write

div d k=1 ∂ k u * (x)M k x ε = d i=1 d j=1 d k=1 ∂ i B ij k x ε ∂ j ∂ k u * (x) = d i=1 ∂ i   d j=1 d k=1 εB ij k x ε ∂ j ∂ k u * (x)   - d i=1 d j=1 d k=1 B ij k x ε ∂ i ∂ j ∂ k u * (x).
The right-most term vanishes because, for each k, B k is skew-symmetric and

D 2 (∂ k u * ) is symmetric. Considering (2.6)-(2.7
), a natural strategy to prove bounds on R ε is the following: rst prove bounds on H ε , then prove elliptic regularity estimates for the operator -div(a(x/ε)∇•) that are uniform with respect to ε.

The following two Lemmas achieve the rst step of this strategy, establishing bounds on H ε .

Lemma 2.6. Assume (A1) through (A4). Then, the correctors dened by Assumption (A3) satisfy

∀p ∈ R d , ∇w p ∈ C 0,α unif (R d ). (2.8)
If in addition Assumption (A8) holds, the potential B dened by (1.21) satises

∇B ∈ C 0,α unif (R d ).
(2.9)

Proof. Estimate (2.8) is a direct consequence of elliptic estimates [START_REF]Elliptic partial dierential equations of second order[END_REF]Theorem 8.32]. Similarly, (1.21)

reads -∆B ij k = ∂ j M i k -∂ i M j k = div(M i k e i -M j k e j )
, where M j k is dened by (1.18). Using (2.8), M j k ∈ C 0,α unif (R d ). Thus, applying [START_REF]Elliptic partial dierential equations of second order[END_REF]Theorem 8.32] again, we have (2.9).

Lemma 2.7. Assume (A1)-(A2)-(A3), and (A7)-(A8) for some ν > 0, and let H ε be dened by (2.7).

Then, for any R > 0 and any q ∈ [1, +∞], if D 2 u * ∈ L q (Ω), we have

H ε L q (Ω∩B(0,R)) ≤ Cε ν R 1-ν D 2 u * L q (Ω∩B(0,R)) , (2.10)
where the constant C does not depend on D 2 u * , R, ε.

Moreover, α being dened by Assumption (A2), for any

β ∈ [0, α], if u * ∈ C 2,β (Ω), we have [H ε ] C 0,β (Ω∩B(0,R)) ≤ Cε ν R 1-ν D 2 u * C 0,β (Ω∩B(0,R)) + Cε ν-β R 1-ν D 2 u * L ∞ (Ω∩B(0,R)) , (2.11)
where C does not depend on D 2 u * , R, ε.

We recall here that the Hölder semi-norm

[•] C 0,β (Ω ) is dened by [v] C 0,β (Ω ) = sup x =y∈Ω |v(x) -v(y)| |x -y| β .
(2.12)

Remark 2.8. Lemma 2.7 is proved under Assumptions (A1), (A2), (A3), (A7), (A8) only. However, applying Lemma 2.2, this in fact implies that Assumptions (A4), (A5), (A6) are satised.

Proof. First, it is clear that

H ε L q (Ω∩B(0,R)) ≤ ε a L ∞ (R d ) d k=1 w e k • ε L ∞ (Ω∩B(0,R)) + B • ε L ∞ (Ω∩B(0,R))
× D 2 u * L q (Ω∩B(0,R)) . (2.13) Note that, w e k and B being dened up to the addition of a constant, we can always assume that w e k (0) = 0 and B(0) = 0. Hence, if |x|/ε > 1, Assumptions (A7) and (A8) imply

w e k x ε ≤ C |x| ε 1-ν , B x ε ≤ C |x| ε 1-ν . (2.14) If |x|/ε ≤ 1, we use Lemma 2.6, which implies that ∇w e k ∈ L ∞ (R d ) and ∇B ∈ L ∞ (R d ),
whence

w e k x ε ≤ C |x| ε ≤ C |x| ε 1-ν , B x ε ≤ C |x| ε ≤ C |x| ε 1-ν . ( 2
[H ε ] C 0,β (Ω∩B(0,R)) ≤ ε a L ∞ (R d ) d k=1 w e k • ε L ∞ (Ω∩B(0,R)) D 2 u * C 0,β (Ω∩B(0,R)) + ε a L ∞ (R d ) d k=1 w e k • ε C 0,β (Ω∩B(0,R)) D 2 u * L ∞ (Ω∩B(0,R)) + ε a • ε C 0,β (Ω∩B(0,R)) d k=1 w e k • ε L ∞ (Ω∩B(0,R)) D 2 u * L ∞ (Ω∩B(0,R)) + ε B • ε L ∞ (Ω∩B(0,R)) D 2 u * C 0,β (Ω∩B(0,R)) + ε B • ε C 0,β (Ω∩B(0,R)) D 2 u * L ∞ (Ω∩B(0,R)) (2.16)
Here again, we use (2.14)-(2.15), which imply

d k=1 w e k • ε L ∞ (Ω∩B(0,R)) + B • ε L ∞ (Ω∩B(0,R)) ≤ Cε ν-1 R 1-ν .
(2.17)

Using Assumption (A2), we also have, since

β ≤ α, a • ε C 0,β (Ω∩B(0,R)) ≤ Cε -β . (2.18)
Using (2.8), we have, for |x -y| < ε,

εw e k x ε -εw e k y ε |x -y| β ≤ C ∇w e k L ∞ (R d ) |x -y| 1-β = C ∇w e k L ∞ (R d ) |x -y| ν-β |x -y| 1-ν ≤ C ∇w e k L ∞ (R d ) ε ν-β R 1-ν .
If |x -y| > ε, we use Assumption (A7), which implies

εw e k x ε -εw e k y ε |x -y| β ≤ Cε ν |x -y| 1-β-ν = Cε ν |x -y| -β |x -y| 1-ν ≤ Cε ν-β R 1-ν .
Collecting the above estimates, we obtain

εw e k x ε C 0,β (Ω∩B(0,R)) ≤ Cε ν-β R 1-ν . ( 2

.19)

A similar argument allows to prove that Next, we are going to prove elliptic regularity estimates for the operator -div(a(x/ε)∇•) that are uniform in ε. This will in turn allow to prove estimates on R ε using (2.6).

εB x ε C 0,β (Ω∩B(0,R)) ≤ Cε ν-β R 1-ν . ( 2

Estimates in the homogeneous case

Our aim is now to prove, as a rst step, that, if the coecient a satises (A1) through (A6), then a solution v ε to

-div a x ε + y ∇v ε = 0, (3.1)
satises Lipschitz bounds uniformly in ε > 0 and y ∈ R d . To this end, we apply the compactness method of Avellaneda and Lin [START_REF] Le Bris | Compactness methods in the theory of homogenization[END_REF]. Loosely speaking, since as ε vanishes, the equation homogenizes into div (a * ∇v * ) = 0, for which Lipschitz bounds hold, thus, for ε suciently small, such bounds should survive. On the other hand, for ε "large", bounded away from zero, they also hold, uniformly, by standard elliptic regularity results, thus, intuitively, the result.

Hölder estimates

The main result of this Section is a generalization of [1, Lemma 24] to the present setting: Theorem 3.1. Assume that the matrix-valued coecient a satises (A1) through (A6). Assume that Ω is a C 2,1 bounded domain, that β ∈]0, 1[, y ∈ R d , and g ∈ C 0,β B(0, 1) . Assume that v ε is a solution to

   -div a x ε + y ∇v ε = 0
in Ω ∩ B(0, 1),

v ε = g
in (∂Ω) ∩ B(0, 1). Then there exists a constant depending only on a, β and Ω such that

v ε C 0,β (Ω∩B(0,1/2)) ≤ C g C 0,β ((∂Ω)∩B(0,1)) + v ε L 2 (Ω∩B(0,1)) (3.2)
In order to prove Theorem 3.1, we rst assume that B(0, 1) ⊂ Ω. In such a case, (3.2) becomes an interior estimate. Its proof is the matter of Lemma 3.2 and Lemma 3.3 below. In a second step, we allow for B(0, 1) to intersect ∂Ω and prove the same type of estimates (Lemma 3.4 and 3.5 below).

We rst prove a result that generalizes [1, Lemma 7] (with f = 0 there) to the present setting. Lemma 3.2. Assume (A1) through (A6), and let β ∈]0, 1[. There exists θ ∈]0, 1/4[ depending only on µ (see Assumption (A1)) and β, there exists a ε 0 > 0 depending only on a, β and θ, such that,

∀y ∈ R d , if v ε is a solution to -div a x ε + y ∇v ε = 0 in B(0, 1), (3.3)
then

- B(0,θ) v ε -- B(0,θ) v ε 2 dx ≤ θ 2β - B(0,1) |v ε (x)| 2 dx. (3.4)
Proof. We reproduce the proof of [1, Lemma 7], and use, instead of periodicity, uniform H-convergence.

Consider v * a solution to -div(a * ∇v * ) = 0 in B(0, 1/2). The matrix a * being constant, we have

- B(0,θ) v * -- B(0,θ) v * 2 dx ≤ θ 2 ∇v * 2 L ∞ (B(0,θ)) ≤ θ 2 ∇v * 2 L ∞ (B(0,1/4)) ≤ Cθ 2 - B(0,1/2) |v * (x)| 2 dx.
The right-most inequality is a consequence of elliptic regularity results. It may be proved by successively applying [START_REF]Elliptic partial dierential equations of second order[END_REF]Theorem 8.32], and [START_REF]Elliptic partial dierential equations of second order[END_REF]Theorem 8.24]. Hence, for θ suciently small, -

B(0,θ) v * -- B(0,θ) v * 2 dx ≤ θ 2β 2 d+1 - B(0,1/2) |v * (x)| 2 dx (3.5)
We then x such a θ and argue by contradiction to prove that v ε satises -

B(0,θ) v ε -- B(0,θ) v ε 2 dx ≤ θ 2β 2 d - B(0,1/2) |v ε (x)| 2 dx. (3.6)
If it does not hold, then we can build sequences ε n → 0 and

y n ∈ R d such that - B(0,θ) v εn -- B(0,θ) v εn 2 dx > θ 2β 2 d - B(0,1/2) |v εn (x)| 2 dx, (3.7)
where v εn solves (3.3) (with ε = ε n and y = y n ). Normalizing v εn if necessary, we may assume that -B(0,1)

|v εn | 2 = 1.
Applying the Caccioppoli inequality [15, page 76], the sequence (v εn ) n∈N is bounded in H 1 (B(0, 1/2)). Hence we can extract a subsequence converging strongly in L 2 (B(0, 1/2)) and weakly in H 1 (B(0, 1/2)), to some limit v * ∈ H 1 (B(0, 1/2)). Applying Proposition 2.4 (this where we use assumptions (A1) through (A6)), we see that v * is a solution to -div(a * ∇v * ) = 0 in B(0, 1/2). Hence it satises (3.5). On the other hand, strong convergence in L 2 (B(0, 1/2) allows to pass to the limit in (3.7), reaching a contradiction. We have proved (3.6), which clearly implies (3.4).

Exactly as in [1, Lemma 8] (with f = 0 there), a proof by induction (which we therefore do not include here) from Lemma 3.2 allows to prove the following Lemma 3.3. Under the assumptions of Lemma 3.2, let θ ∈]0, 1/4[ and ε 0 be given by Lemma 3.2. If ε ∈]0, θ k ε 0 [, and if v ε satises (3.3), then

- B(0,θ k ) v ε -- B(0,θ k ) v ε 2 dx ≤ θ 2kβ - B(0,1) |v ε (x)| 2 dx.
Following the sketch of the proof of [1, Lemma 10] (with f = 0 and g = 0 there), and using uniform H-convergence where periodicity was used in [START_REF] Le Bris | Compactness methods in the theory of homogenization[END_REF], we obtain Lemma 3.4. Assume (A1) through (A6) with β ∈]0, 1[, and that Ω is a C 1,α bounded domain such that, say, 0 ∈ ∂Ω. There exists θ ∈]0, 1/4[ and ε 0 > 0 depending only on a, β and Ω, such that, for any ε < ε 0 , any y ∈ R d , and any solution v ε of

   -div a x ε + y ∇v ε = 0 in Ω ∩ B(0, 1), v ε = 0 on (∂Ω) ∩ B(0, 1), (3.8)
we have

- Ω∩B(0,θ) |v ε (x)| 2 dx ≤ θ 2β - Ω∩B(0,1) |v ε (x)| 2 dx. (3.9)
Proof. Assume temporarily that v * is a solution to

-div (a * ∇v * ) = 0 in Ω ∩ B(0, 1), v * = 0 in (∂Ω) ∩ B(0, 1). (3.10)
In particular, we have v * (0) = 0, hence, for any θ ∈]0, 1/4[,

- Ω∩B(0,θ) |v * | 2 ≤ Cθ 2 ∇v * 2 L ∞ (Ω∩B(0,1/4)) .
Applying the boundary gradient estimate [START_REF]Elliptic partial dierential equations of second order[END_REF]Corollary 8.36], we have

∇v * L ∞ (Ω∩B(0,1/4)) ≤ C v * L ∞ (Ω∩B(0,1/2)) , hence - Ω∩B(0,θ) |v * | 2 ≤ Cθ 2 v * 2 L ∞ (Ω∩B(0,1/2)) (3.11)
We apply [START_REF]Elliptic partial dierential equations of second order[END_REF]Theorem 8.25]. This gives v * L ∞ (Ω∩B(0,1/2)) ≤ C-

Ω∩B(0,1)
|v * | 2 . Hence, inserting this estimate into (3.11), we nd

- Ω∩B(0,θ) |v * | 2 ≤ Cθ 2 - Ω∩B(0,1) |v * | 2
Thus, for θ > 0 suciently small,

- Ω∩B(0,θ) |v * | 2 ≤ θ 2β 2 - Ω∩B(0,1) |v * | 2 . (3.12)
We now x θ > 0 to this value, and argue by contradiction: if (3.9) does not hold, then one can nd a sequence ε n → 0 and a sequence y n such that, for each n the solution v εn of (3.8) (with ε = ε n , y = y n ) satises

- Ω∩B(0,θ) |v εn | 2 > θ 2β - Ω∩B(0,1) |v εn | 2 . (3.13)
Multiplying v εn by a normalizing constant if necessary, we may assume that -Ω∩B(0,1)

|v εn | 2 = 1. (3.14)
The sequence (v εn ) n∈N is bounded in H 1 (Ω ∩ B(0, 1/2)) according to Caccioppoli's inequality [15, Proposition 2.1, p 76]. Hence, we can extract weak convergence in L 2 (Ω ∩ B(0, 1) ∩ H 1 (Ω ∩ B(0, 1/2)) and strong convergence in L 2 (Ω ∩ B(0, 1/2)). We Denote by v * its limit. Inequality (3.13) implies

θ 2β - Ω∩B(0,1) |v * | 2 ≤ θ 2β lim inf n→+∞ - Ω∩B(0,1) |v εn | 2 ≤ lim inf n→+∞ - Ω∩B(0,θ) |v εn | 2 = - Ω∩B(0,θ) |v * | 2 .
In addition, Proposition 2.4, which is valid since we assumed (A1) through (A6), allows to prove that v * is a solution to (3.10), hence satises (3.12). We therefore reach a contradiction, concluding the proof.

Here again, using an induction argument as in the proof of [1, Lemma 11] (with f = 0 there), we have Lemma 3.5. Under the same assumptions as those of Lemma 3.4, with θ > 0 and ε 0 > 0 dened by the conclusion of Lemma 3.4, we have, for any integer

k ≥ 0, if ε < θ k ε 0 , - Ω∩B(0,θ k ) |v ε | 2 ≤ θ 2kβ - Ω∩B(0,1) |v ε | 2 .
The four above Lemmas allow us to proceed with the proof of Theorem 3.1. We rst deal with the case of interior estimate, that is, ∂Ω ∩ B(0, 1) = ∅, then we prove the general case.

Proof of Theorem 3.1. Assume rst that ∂Ω ∩ B(0, 1) = ∅. Then the proof is exactly that of [1, Lemma 9] with f = 0, in which periodicity is not used. Next, if ∂Ω ∩ B(0, 1) = ∅, we follow the proof of [1, Lemma 24].

Lipschitz estimates

In this Section, we prove the following result, which is the generalization of [1, Lemma 16] (with f = 0 there) to he present setting: Theorem 3.6. Assume (A1) through (A6). Let y ∈ R d , R > 0, and assume that

v ε ∈ H 1 (B(0, 2R)) is a solution to -div a x ε + y ∇v ε (x) = 0 in B(0, 2R).
Then, there exists a constant C depending only on the coecient a such that

sup x∈B(0,R) |∇v ε (x)| ≤ C R - B(0,2R) |v ε | 2 1/2 . (3.15)
As we did for the proof of Hölder estimates above, we are going to apply the proof of [START_REF] Le Bris | Compactness methods in the theory of homogenization[END_REF], replacing, when necessary, periodicity by assumptions (A3) through (A6).

We rst prove a result that is the generalization of [1, Lemma 14] (with f = 0 there) to our setting.

Lemma 3.7. Assume that the matrix-valued coecient a satises Assumptions (A1) through (A6), and let γ ∈]0, 1[. Then there exists ε 0 > 0 and θ ∈]0, 1/4[ depending only on a and γ such that, if

ε < ε 0 and if v ε satises -div a x ε + y ∇v ε (x) = 0 in B(0, 1), (3.16)
then

sup x∈B(0,θ) v ε (x) -v ε (0) - d j=1 x j + ε w ej x ε + y -w ej (y) - B(0,θ) ∂ j v ε ≤ θ 1+γ - B(0,1) |v ε | 2 1/2 (3.17)
Proof. As in the proof of Lemma 3.4, we argue by contradiction. Let v * ∈ H 1 (B(0, 1/2)) be a solution to

-div (a * ∇v * ) = 0 in B 0, 1 2 . (3.18)
Since a * is constant, ∂ j ∂ j v * is also a solution to (3.18). Hence, applying the interior Hölder estimate of [START_REF]Elliptic partial dierential equations of second order[END_REF]Theorem 8.24], we have

D 2 v * C 0,β (B(0,1/8)) ≤ C D 2 v * L 2 (B(0,1/4))
, where C and β depend only on a * . Hence,

D 2 v * L ∞ (B(0,1/8)) ≤ C D 2 v * L 2 (B(0,1/4)) . (3.19)
Then applying the Caccioppoli inequality [15, Proposition 2.1, p 76] twice, we infer

B(0,1/4) |D 2 v * | 2 ≤ C B(0,1/2) |v * | 2 . ( 3
v * (x) -v * (0) -x • - B(0,θ) ∇v * ≤ C 0 θ 2 - B(0,1/2) |v * | 2 1/2 .
Hence, choosing θ such that C 0 θ 2 ≤ θ 1+γ 2 1+d/2 , we nd that v * satises (3.17) w ej is replaced by 0, that is,

sup x∈B(0,θ) v * -v * (0) -x • - B(0,θ) ∇v * ≤ θ 1+γ 2 1 2 d/2 - B(0,1/2) |v * | 2 1/2 .
(3.21)

The condition on θ reads θ ≤ 2 1+d/2 C 0 -1/(1-γ) , which depends only on a * , d and γ.

Next, we assume that (3.17) does not hold, that is, there exists sequences ε n → 0, y n ∈ R d and v εn ∈ H 1 (B(0, 1)) such that (3.16) holds (with ε = ε n , y = y n , v ε = v εn ), and

sup x∈B(0,θ) v εn (x) -v εn (0) - d j=1 x j + ε n w ej x ε n + y -w ej (y) - B(0,θ) ∂ j v εn > θ 1+γ - B(0,1)
|v εn | 2 1/2 . (3.22)
Multiplying v εn by some constant if necessary, we may assume that -B(0,1)

|v εn | 2 = 1. Applying the
Caccioppoli inequality, we deduce that v εn is bounded in H 1 (B(0, 1/2)), hence, up to extracting a subsequence, we have v εn -v * in H 1 (B(0, 1/2)) ∩ L 2 (B(0, 1)). Applying Proposition 2.4 (thereby using Assumptions (A1) through (A6)), we prove that v * satises (3.18), hence (3.21). Next, applying Theorem 3.1, we have v εn C 0,β (B(0,1/2)) ≤ C. This allows to pass to the limit in the rst two terms of the left-hand side of (3.22). Weak convergence in H 1 (B(0, 1/2)) allows to pass to the limit in the term -B(0,θ) ∂ j v εn . Moreover, Assumptions (A1) through (A5) allow to apply Lemma 2.1, which implies that, for all j ∈ {1, 2, . . . , d},

sup y∈R d sup x∈B(0,1) ε w ej x ε + y -w ej (y) -→ ε→0 0.
Hence, passing to the limit in (3.22), we nd

sup x∈B(0,θ) v * (x) -v * (0) -x • - B(0,θ) ∇v * ≥ lim inf n→+∞ θ 1+γ - B(0,1) |v εn | 2 1/2 ≥ θ 1+γ - B(0,1) |v * | 2 1/2 = θ 1+γ 2 d/2 - B(0,1/2) |v * | 2 1/2
, and we reach a contradiction with (3.21).

As in [1, Lemma 15] (with f = 0 there), an induction argument allows to prove the following Lemma 3.8. Assume (A1) through (A6), and that γ ∈]0, 1[. Let θ and ε 0 be given by Lemma 3.7.

There exists C > 0 depending only on θ such that, for any

y ∈ R d , if 0 < ε ≤ ε 0 θ n , n ∈ N, and if v ε ∈ H 1 (B(0, 1)) satises (3.16), we have sup x∈B(0,θ n+1 ) v ε (x) -v ε (0) - d j=1 x j + ε w ej x ε + y -w ej (y) κ j (n) ≤ θ (1+n)(1+γ) v ε L ∞ (B(0,1)) ,
where κ j (n) satises

sup 1≤j≤d |κ j (n)| ≤ C v ε L ∞ (B(0,1)) n =0 θ γ . (3.23)
Remark 3.9. In (3.23), the important point is that C depends on θ but not on n. Hence, since θ < 1, it implies sup

1≤j≤d |κ j (n)| ≤ C v ε L ∞ (B(0,1)
) , and will be used as such in the sequel. However, the form (3.23) is more convenient for the induction proof.

Proof of Theorem 3.6. This exactly the proof of [1, Lemma 16], based on Lemma 3.7 and Lemma 3.8.

We therefore omit it.

Estimates in the inhomogeneous case

In this Section, we deal with the non-homogeneous case, that is, the case when the right-hand side of (3.1) is some div(f ), f ∈ L 2 (Ω), with f = 0.

We rst prove estimates on the Green function G ε of the operator -div a x ε ∇• with homogeneous Dirichlet boundary conditions. This uses the results on the homogeneous case, since x → G ε (x, y) and x → ∇ y G ε (x, y) are solution to -div a x ε ∇ x v = 0 in any open set that does not contain y. Then, we use the representation u ε (x) = Ω G ε (x, y)f (y)dy to prove estimates in the case f = 0.

Green function estimates

First, we recall that in [START_REF]wF qrüter nd uFEyF idmn, The Green function for uniformly elliptic equations[END_REF], G ε was proved to exist and be unique in W 1,1 0 (Ω). In addition, the following estimates were established in [START_REF]wF qrüter nd uFEyF idmn, The Green function for uniformly elliptic equations[END_REF][START_REF] Wüller | Estimates for Green's matrices of elliptic systems by L p theory[END_REF]:

∀x = y ∈ Ω, 0 ≤ G ε (x, y) ≤ C |x -y| d-2 , (4.1) ∇ x G ε (•, y) L d d-1 ,∞ (Ω) + ∇ y G ε (x, •) L d d-1 ,∞ (Ω) ≤ C,
where C depends only on a L ∞ and on its ellipticity constant. Here, L p,∞ denotes the Marcinkiewicz space of order p, as dened, e.g., in [START_REF]Interpolation spaces. An introduction[END_REF].

We now show Theorem 4.1. Let d ≥ 3. Assume (A1) through (A6). Let Ω be a C 1,α bounded domain. Denote by G ε the Green function of the operator -div a x ε ∇• on Ω with homogeneous Dirichlet boundary conditions. For any Ω 1 ⊂⊂ Ω, we have the following estimates:

1. ∀x ∈ Ω 1 , ∀y ∈ Ω, x = y, |∇ x G ε (x, y)| ≤ C |x -y| d-1 . (4.2)
2. If in addition a T satises Assumptions (A3), (A4), (A5) and (A6), then we have

∀y ∈ Ω 1 , ∀x ∈ Ω, x = y, |∇ y G ε (x, y)| ≤ C |x -y| d-1 , (4.3) ∀x ∈ Ω 1 , ∀y ∈ Ω 1 , x = y, |∇ y ∇ x G ε (x, y)| ≤ C |x -y| d .
(4.4)

In (4.2)-(4.3)-(4.4), the various constants C depend only on the coecient a, on Ω and on Ω 1 .

The above result is actually contained in [START_REF] Le Bris | Compactness methods in the theory of homogenization[END_REF], if the coecient a is assumed to be periodic. However, it is not stated as such, and its proof, which may be found in the course of the proof of [1, Lemma 17], is dierent from the one we present here.

Proof. We rst prove Assertion 1. We dene δ

= inf {|x -y|, x ∈ Ω 1 , y ∈ ∂Ω} > 0. Let x 0 ∈ Ω 1 , y 0 ∈ Ω \ {x 0 }. We set R = 1 2 min d(x 0 , ∂Ω) , |x 0 -y 0 | We have 2R ≤ |x 0 -y 0 | ≤ CR, (4.5)
where the constant

C is C = 2 if R = 1 2 |x 0 -y 0 |, and C = 2 diam(Ω) δ otherwise.
In particular it depends only on Ω and Ω 1 . Since y 0 / ∈ B(x 0 , R),

-div x a x ε ∇ x G ε (x, y 0 ) = 0 in B(x 0 , R). (4.6)
Applying Theorem 3.6 to x → G ε (x, y 0 ), we have

|∇ x G ε (x 0 , y 0 )| ≤ C R - B(x0,R/2) |G ε (x, y 0 )| 2 dx 1/2
.

Using (4.1), (4.5), and the triangle inequality, |x -

y 0 | ≥ |x 0 -y 0 | -|x -x 0 |, we have - B(x0,R/2) |G ε (x, y 0 )| 2 dx 1/2 ≤ C - B(x0,R/2) 1 R 2(d-2) dx 1/2 = C R d-2 .
Hence,

|∇ x G ε (x 0 , y 0 )| ≤ C R d-1
. Using (4.5) again, we nd (4.2).

Next, we prove Assertion 2. It is well-known (see [START_REF]wF qrüter nd uFEyF idmn, The Green function for uniformly elliptic equations[END_REF]Theorem 1.3]) that the Green function G ε T of the operator -div a T x ε ∇• with homogeneous Dirichlet condition satises G ε T (x, y) = G ε (y, x). Since a T satises Assumptions (A1), (A2), (A3), (A4), (A5), (A6), G ε T satises (4.2). This clearly implies (4.3).

Finally, we note that ∇ y G(x, y 0 ) is also a solution to (4.6). Hence, applying the proof of Assertion 2 to ∇ y G, we nd (4.4).

W 1,p estimates

We now prove W 1,p estimates on the solution v ε of (4.7) below. The following Proposition is the generalization of [START_REF]Periodic homogenization of elliptic problems[END_REF]Theorem 2.4.1] to the present setting. Proposition 4.2. Assume (A1) through (A6).

Let q ∈]2, +∞[, y ∈ R d , R > 0 and H ∈ L q (B(0, 2R), R d ). Assume that v ε ∈ H 1 (B(0, 2R)) is a solution to -div a x ε + y ∇v ε = div(H) in B(0, 2R). (4.7)
Then, there exists C > 0 depending only on the coecient a and on q (in particular it does not depend on y nor on ε) such that

- B(0,R) |∇v ε | q 1/q ≤ C - B(0,2R) |H| q 1/q + C - B(0,2R) |∇v ε | 2 1/2 .
Before we get to the proof of Proposition 4.2, we rst state the following Lemma, which is a simple consequence of [START_REF] Hen | The Calderón-Zygmund lemma revisited[END_REF]Theorem 2.4] (see also [START_REF]Periodic homogenization of elliptic problems[END_REF]Theorem 2

.3.1]): Lemma 4.3. Let B 0 = B(x 0 , R 0 ) be a ball of R d , and F ∈ L 2 (4B 0 ). Let 2 < q 1 < q 2 , f ∈ L q1 (4B 0 ).
Assume that there exists K > 0 such that for any ball B ⊂ 2B 0 with 2|B| ≤ |B 0 |, there exists

F 1 ∈ L 2 (2B) and F 2 ∈ L q2 (2B) such that |F | ≤ |F 1 | + |F 2 | in 2B, - 2B |F 1 | 2 1/2 ≤ K sup B⊂B ⊂4B0 - B |f | 2 1/2 , - 2B |F 2 | q2 1/q2 ≤ K - 4B |F | 2 1/2 + sup B⊂B ⊂4B0 - B |f | 2 1/2
, where the supremum is taken over any ball B such that B ⊂ B ⊂ B(x 0 , 4R 0 ). Then, F ∈ L q1 (B 0 ), and

- B0 |F | q1 1/q1 ≤ C - 4B0 |F | 2 1/2 + - 4B0 |f | q1 1/q1
, where C depends on K, q 1 , q 2 only.

Proof of Proposition 4.2. The proof follows the lines of [31, Theorem 2.4.1]. However, since the setting is slightly dierent, we reproduce it here for the sake of clarity and for the reader's convenience. Let x 0 ∈ B(0, 2R) and R 0 > 0 such that B 0 := B(x 0 , R 0 ) satises 8B 0 ⊂ B(0, 2R). We intend to apply Lemma 4.3 to F = ∇v ε and f = H. For this purpose, we x y 0 ∈ 2B 0 and R 1 > 0 such that

B := B(y 0 , R 1 ) ⊂ 2B 0 . v ε = v ε 1 + v ε 2 , where v ε 1 satises    -div a x ε + y ∇v ε 1 = div(H) in B(y 0 , 4R 1 ), v ε 1 = 0 in ∂ (B(y 0 , 4R 1 )) .
Multiplying this equation by v ε 1 and integrating by parts, we have

- B(y0,4R1) |∇v ε 1 | 2 1/2 ≤ C - B(y0,4R1) |H| 2 1/2 , (4.8)
where C depends only on the ellipticity constant of a. On the other hand, v ε 2 satises

-div a x ε + y ∇v ε 2 = 0 in B(y 0 , 4R 1 ).
Thus, applying Theorem 3.

6 to v ε 2 -- B(y0,4R1) v ε 2 , we have ∇v ε 2 L ∞ (B(y0,2R1)) ≤ C R 1   - B(y0,4R1) v ε 2 -- B(y0,4R1) v ε 2 2   1/2
, where the constant C depends only on the coecient a. Applying the Poincaré-Wirtinger inequality, this implies

∇v ε 2 L ∞ (B(y0,2R1)) ≤ C - B(y0,4R1) |∇v ε 2 | 2 1/2 .
The constant C is equal to C = C R1 C P W (B(y 0 , 4R 1 )) = CC P W (B(y 0 , 4)), due to the scaling of the constant C P W in the Poincaré-Wirtinger inequality. Hence, C depends only on a. On the other hand, using (4.8) and the triangle inequality, we have

- B(y0,4R1) |∇v ε 2 | 2 1/2 ≤ - B(y0,4R1) |∇v ε 1 | 2 1/2 + - B(y0,4R1) |∇v ε | 2 1/2 ≤ C - B(y0,4R1) |H| 2 1/2 + - B(y0,4R1) |∇v ε | 2 1/2 .
Thus,

∇v ε 2 L ∞ (B(y0,2R1)) ≤ C - B(y0,4R1) |H| 2 1/2 + - B(y0,4R1) |∇v ε | 2 1/2 . ( 4.9)
Collecting (4.8) and (4.9), we may apply Lemma 4.3 (with

B 0 = B(x 0 , R 0 ), q 1 = q, q 2 = 2q 1 , f = H, F 1 = ∇v ε 1 , F 2 = ∇v ε 2 , and B = B(y 0 , R 1 )) nding - B0 |∇v ε | q 1/q ≤ C - 4B0 |H| q 1/q + - 4B0 |∇v ε | 2 1/2
. This is valid for any x 0 and R 0 > 0 such that B(x 0 , 8R 0 ) ⊂ B(0, 2R). Hence, covering B(0, R) by a nite number of such balls, we conclude the proof.

Lipschitz estimates

Note that Proposition 4.2 does not include the case q = +∞. However, using the estimates we have proved on the gradient of G ε in Theorem 4.1, we are able to now derive Lipschitz estimates:

Proposition 4.4. Assume that the coecients a and a T satisfy Assumptions (A1) through (A6). Let β > 0 and R > ε > 0, and assume that H ∈ C 0,β (B(0, 2R)). Then, there exists a constant C > 0 depending only on a and β such that, if v ε satises (4.7), then

∇v ε L ∞ (B(0,R)) ≤ C - B(0,2R) |∇v ε | 2 1/2 + Cε β [H] C 0,β (B(0,2R)) + C ln 1 + R ε H L ∞ (B(0,2R)) . (4.10)
We recall here that [•] C 0,β (B(0,2R)) denotes the Hölder semi-norm on B(0, 2R) (see (2.12)).

Proposition 4.4 is a generalization of [25, Lemma 3.5], in two ways. First, we replace, here, the periodicity assumption by (A1) through (A6). Second, in [START_REF] Hen | Periodic homogenization of Green and Neumann functions[END_REF], Lemma 3.5 is stated only for the specic case where v ε = R ε dened by (1.11), hence H = H ε dened by (2.7). Due to these dierences, we provide below a complete proof, although the ideas are contained in [START_REF] Hen | Periodic homogenization of Green and Neumann functions[END_REF].

Proof. We split the proof in several steps: rst, introducing a cut-o function, we write v ε as an integral of G ε , which is the Green function of the operator -div a x ε + y ∇• with homogeneous Dirichlet boundary conditions on B(0, 2R). Then, we use this representation and Theorem 4.1 to prove (4.10).

Step 1: introduction of a cut-o function and use of the Green function. We dene φ ∈ C ∞ c (B(0, 3R/2)) such that

0 ≤ φ ≤ 1, φ = 1 in B(0, 5R/4), ∇φ L ∞ (B(0,2R)) ≤ C R , D 2 φ L ∞ (B(0,2R)) ≤ C R 2 .
We clearly have

∇(φv ε ) L ∞ (B(0,R)) = ∇v ε L ∞ (B(0,R)) . Moreover, -div a z ε + y ∇ (φv ε ) = -div v ε a z ε + y ∇φ -a z ε + y ∇v ε • ∇φ + φ div(H).
Hence, multiplying by G ε (x, z) and integrating with respect to z over B(0, 2R),

φ(x)v ε (x) = - B(0,2R) G ε (x, z) a z ε + y ∇v ε (z) • ∇φ(z)dz + B(0,2R) ∇ z G ε (x, z) • v ε (z)a z ε + y ∇φ(z) dz - B(0,2R) ∇ z (G ε (x, z)φ(z)) • H(z)dz =: v ε 1 (x) + v ε 2 (x) + v ε 3 (x).
Step 2: bound on v ε 1 . Let x ∈ B(0, R). Since ∇φ vanishes in B(0, 5R/4) and outside B(0, 3R/2), we have

|∇v ε 1 (x)| ≤ B(0,3R/2)\B(0,5R/4) |∇ x G ε (x, z)| a z ε + y |∇v ε (z)| |∇φ(z)|dz.
Successively using |∇φ| ≤ C R , estimate (4.2), and B(0, 3R/2) \ B(0, 5R/4) ⊂ B(0, 2R) \ B(0, R), we deduce

|∇v ε 1 (x)| ≤ C R d B(0,2R)\B(0,R) |∇v ε | ≤ C - B(0,2R) |∇v ε | 2 1/2 . ( 4.11)
Step 3: bound on v ε 2 . Similar arguments allow to prove that

|∇v ε 2 (x)| ≤ B(0,3R/2)\B(0,5R/4) |∇ x ∇ z G ε (x, z)| |v ε (z)| a z ε + y |∇φ(z)|dz ≤ C R B(0,3R/2)\B(0,5R/4) |∇ x ∇ z G(x, z)| 2 dz 1/2
v ε L 2 (B(0,2R)) , (4.12)

the last inequality coming from the Cauchy-Schwarz inequality. We then apply (4.4), which implies B(0,3R/2)\B(0,5R/4)

|∇ x ∇ z G(x, z)| 2 dz 1/2 ≤ C R d/2 . (4.13)
We point out that adding a constant to v ε does not change (4.7), hence we may assume that B(0,2R) v ε = 0. So, using the Poincaré-Wirtinger inequality, we have

v ε L 2 (B(0,2R)) ≤ C R 2 B(0,2R) |∇v ε | 2 1/2 = CR 1+d/2 - B(0,2R) |∇v ε | 2 1/2
, where C does not depend on R. Inserting this inequality and (4.13) into (4.12), we infer

|∇v ε 2 (x)| ≤ C - B(0,2R) |∇v ε 2 | 2 1/2 . (4.14)
Step 4: bound on v ε 3 . We x here again x ∈ B(0, R). Integrating by parts, we have

B(0,2R) ∇ z (G ε (x, z)φ(z)) dz = 0, (4.15) hence v ε 3 (x) = B(0,2R) ∇ z (G ε (x, z)φ(z)) • (H(z) -H(x))dz.
We dierentiate this equalilty with respect to x, and use (4.15) again, nding

∇v ε 3 (x) = B(0,2R) ∇ z (∇ x G ε (x, z)φ(z)) • (H(z) -H(x))dz - B(0,2R) ∇ z (G ε (x, z)φ(z)) • ∇ x H(x)dz =0 .
Thus,

|∇v ε 3 (x)| ≤ B(0,2R) |φ(z)| |∇ z ∇ x G ε (x, z)| |H(z) -H(x)|dz + B(0,2R) |∇φ(z)| |∇ x G ε (x, z)| |H(z) -H(x)|dz
Using that ∇φ vanishes in B(0, 5R/4) and outside B(0, 3R/2), that |∇φ| ≤ C/R, and (4.2), we have

B(0,2R) |∇φ(z)| |∇ x G ε (x, z)| |H(z) -H(x)|dz ≤ C R H L ∞ (B(0,2R)) B(0,3R/2)\B(0,5R/4) 1 |x -z| d-1 dz ≤ C H L ∞ (B(0,2R)) .
Moreover, using (4.4) and the fact that H is β-Hölder continuous, we also have,

B(0,2R) |φ(z)| |∇ z ∇ x G ε (x, z)| |H(z) -H(x)|dz ≤ C[H] C 0,β (B(0,2R)) B(x,ε) |x -z| β |x -z| d dz + 2 H L ∞ (B(0,2R)) B(0,2R)\B(x,ε) dz |x -z| d .
The integral in the right-most term of the right-hand side is bounded as follows (we use here |x| ≤ R):

B(0,2R)\B(x,ε) dz |x -z| d ≤ C max(3R,ε) ε r d-1 dr r d = C ln max(3R, ε) ε ≤ C ln 1 + R ε .
Hence, Remark 4.5. In Propoisition 4.4, we have assumed that both coecients a and a T satisfy Assumptions (A1) through (A6). The result however still holds if only a satises those assumptions. Indeed, the assumption on a T is only used for the proof of (4.13) and (4.16): in both cases, we have used the pointwise bound (4.4) on ∇ x ∇ y G, but the only relevant bound for proceeding with the proof of Proposition 4.4 is an L 2 bound, which can alternately be obtained using (4.2) and the Cacciopoli inequality (see [START_REF] Wf Tosien | Etude mathématique et numérique de quelques modèles multi-échelles issus de la mécanique des matériaux [Mathematical and numerical study of some multiscale models for the materials sciences[END_REF]Section 2.5.3] for the details).

|∇v ε 3 (x)| ≤ Cε β [H] C 0,β (B(0,2R)) + C ln 1 + Rε -1 H L ∞ (B(0,2R)) . ( 4

Convergence rates for Green functions

We now prove the following convergence result of G ε to the Green function G * of the operator -div(a * ∇•) with homogeneous Dirichlet conditions on Ω. It is the extension, in our setting, of [START_REF] Hen | Periodic homogenization of Green and Neumann functions[END_REF]Theorem 3.3] Theorem 4.6. Assume that the matrix-valued coecients a and a T satisfy Assumptions (A1) through (A6), and (A7)-(A8) for some ν > 0.

Let Ω be a domain of class C 2,1 , and denote by G ε and G * the Green functions of the operators -div a x ε ∇• and -div(a * ∇•), respectively, with homogeneous Dirichlet boundary conditions on Ω. Then there exists a constant C > 0 depending only on a, Ω and ν such that

∀x = y ∈ Ω, |G ε (x, y) -G * (x, y)| ≤ C ε ν |x -y| d+ν-2 .
(4.17)

The proof of Theorem 4.6 replicates that of [START_REF] Hen | Periodic homogenization of Green and Neumann functions[END_REF]Theorem 3.3], but we need to everywhere keep track of the use of Assumptions (A7)-(A8) and check that these properties are sucient to proceed at each step of the arguments.

We prove the following lemma, which is a generalization of [25, Lemma 3.2]: Lemma 4.7. Assume that the matrix-valued coecient a satises (A1) through (A6), and (A7)-(A8) for some ν > 0. Let Ω be a C 2,1 bounded domain, x 0 ∈ Ω, R > 0, q 1 > d and q

2 ∈]1, +∞[. Assume that u ε ∈ H 1 (Ω ∩ B(x 0 , 4R)) and u * ∈ W 2,q1 (Ω ∩ B(x 0 , 4R)) satisfy    -div a x ε ∇u ε = -div(a * ∇u * )
in Ω ∩ B(x 0 , 4R),

u ε = u * on (∂Ω) ∩ B(x 0 , 4R).
Then, there exists a constant C depending only on a, Ω, q 1 and q 2 such that

u ε -u * L ∞ (Ω∩B(x0,R)) ≤ CR -d q 2 u ε -u * L q 2 (Ω∩B(x0,4R)) + Cε ν R 1-ν ∇u * L ∞ (Ω∩B(x0,4R)) + Cε ν R 2-d q 1 -ν D 2 u * L q 1 (Ω∩B(x0,4R)) (4.18)
Proof. We follow the proof of [25, Lemma 3.2], adapting it when necessary. First, since the problem is translation invariant, we may assume that x 0 = 0. Then, we dene a smooth open set Ω such that

Ω ∩ B(0, 2R) ⊂ Ω ⊂ Ω ∩ B(0, 4R).
We dene the remainder R ε by (1.11). We know that it satises (2.6), with H ε dened by (2.7). Next, we split

R ε into R ε = R ε 1 + R ε 2 , where R ε 1 is dened as the unique solution of    -div a x ε ∇R ε 1 = -div(H ε ) in Ω, R ε 1 = 0 on ∂ Ω. (4.19) Hence, R ε 2 satises          -div a x ε ∇R ε 2 = 0 in Ω, R ε 2 (x) = ε d j=1 w ej x ε ∂ j u * (x) on ∂ Ω ∩ ∂Ω. (4.20)
We use a scaling argument, dening R

ε 2 (x) = R ε 2 (x/R), a(x) = a(x/R), w ej (x) = w ej (x/R
), and u * (x) = u * (x/R). Writing down the equation satised by R ε 2 , we are thus in the case R = 1 and we may apply De Giorgi-Nash estimate. Scaling back to the original unknown R ε 2 , this implies

R ε 2 L ∞ (Ω∩B(0,R)) ≤ C ε d j=1 w ej x ε ∂ j u * (x) L ∞ ( Ω) + C R d/q2 R ε 2 L q 2 ( Ω) .
Using Assumption (A7) and the triangle inequality, this implies

R ε 2 L ∞ (Ω∩B(0,R)) ≤ Cε ν R 1-ν ∇u * L ∞ ( Ω) + C R d/q2 R ε L q 2 ( Ω) + C R ε 1 L ∞ ( Ω) . (4.21)
Next, according to the denition (1.11) of R ε , and using Assumption (A7) again, we have

R ε L q 2 ( Ω) ≤ u ε -u * L q 2 ( Ω) + CR d/q2 ε d j=1 w ej x ε ∂ j u * (x) L ∞ ( Ω) ≤ u ε -u * L q 2 ( Ω) + Cε ν R d/q2+1-ν ∇u * L ∞ ( Ω) . (4.22)
Inserting (4.22) into (4.21), we thus have

R ε 2 L ∞ (Ω∩B(0,R)) ≤ Cε ν R 1-ν ∇u * L ∞ ( Ω) + C R d/q2 u ε -u * L q 2 ( Ω) + C R ε 1 L ∞ ( Ω) . (4.23)
Next, we bound R ε 1 . Denoting by G ε the Green function of the operator -div a x ε ∇• on Ω with homogeneous Dirichlet boundary conditions on Ω, we have, for any x ∈ Ω,

R ε 1 (x) = - Ω ∇ y G ε (x, y) • H ε (y)dy.
Using the Hölder inequality and the estimate (2.10) of Lemma 2.7 (this is where we use Assumption (A8)), we have

|R ε 1 (x)| ≤ Cε ν R 1-ν ∇ y G(x, •) L q 1 ( Ω) D 2 u * L q 1 ( Ω) .
Since q 1 > d, we have q 1 < d d-1 , hence, using [20, Equation (1.12)] and Theorem 4.1,

|R ε 1 (x)| ≤ Cε ν R 1-ν Ω 1 q 1 -d-1 d ∇ y G(x, •) L d d-1 ,∞ ( Ω) D 2 u * L q 1 ( Ω) ≤ Cε ν R 2-d q 1 -ν D 2 u * L q 1 ( Ω) . (4.24)
Collecting (4.23) and (4.24), we have proved

R ε L ∞ (Ω∩B(x0,R)) ≤ CR -d/q2 u ε -u * L q 2 (Ω∩B(x0,4R)) + Cε ν R 1-ν ∇u * L ∞ (Ω∩B(x0,4R)) + Cε ν R 2-d q 1 -ν D 2 u * L q 1 (Ω∩B(x0,4R)) . (4.25)
Next, we write

u ε (x) -u * (x) = R ε (x) + ε d j=1 w ej x ε ∂ j u * (x),
which implies, using the triangle inequality and Assumption (A7),

v ε -v * L ∞ (Ω∩B(x0,R)) ≤ R ε L ∞ (Ω∩B(x0,R)) + Cε R ε 1-ν ∇u * L ∞ (Ω∩B(x0,R)) .
Inserting (4.25) into this estimate, we nd (4.18).

The following result is the generalization of [25, Theorem 3.4] (with q = ∞ there) to the present setting. Here, the proof is substantially dierent from [START_REF] Hen | Periodic homogenization of Green and Neumann functions[END_REF]Lemma 3.2].

Lemma 4.8. Under the assumptions of Theorem 4.6, let q > d, x 0 = y 0 ∈ Ω, R = |x 0 -y 0 |/16. Assume that f ∈ C ∞ c (Ω ∩ B(y 0 , 4R)), and that u ε and u * are solutions to

   -div a x ε ∇u ε = -div(a * ∇u * ) = f in Ω, u ε = u * = 0 on ∂Ω.
Then,

u ε -u * L ∞ (Ω∩B(x0,R)) ≤ CR 2-d q -ν ε ν f L q (Ω) , (4.26)
where C depends only on the coecient a, q and Ω.

Proof. Due to translation invariance, we may assume that y 0 = 0. We apply Lemma 4.7 with q 1 = q.

Hence, u ε -u * satises (4.18), that is,

u ε -u * L ∞ (Ω∩B(x0,R)) ≤ CR -d q 2 u ε -u * L q 2 (Ω∩B(x0,4R)) + Cε ν R 1-ν ∇u * L ∞ (Ω∩B(x0,4R)) + Cε ν R 2-d q -ν D 2 u * L q (Ω∩B(x0,4R)) , (4.27)
for any q 2 > 1. We x q 2 < 2, and we are going to estimate separately each term of the right-hand side of (4.27).

Step 1: bound on ∇u * L ∞ . Denoting by G * the Green function of the operator -div (a * ∇•) with homogeneous Dirichlet boundary conditions on Ω, we have

∀x ∈ Ω, ∇u * (x) = Ω ∇ x G * (x, y)f (y)dy = Ω∩B(0,4R) ∇ x G * (x, y)f (y)dy. Hence, |∇u * (x)| ≤ ∇ x G * (x, •) L q (Ω∩B(0,4R)) f L q (Ω) . Applying [20, Theorem 3.3 (iv)], we have ∇ x G * (x, •) L q (Ω∩B(0,4R)) ≤ C B(0,4R) 1 |x -y| q (d-1) dy 1/q ≤ CR d q -d+1 if |x| < 8R, CR d q |x| -d+1 if |x| ≥ 8R.
Hence,

|∇u * (x)| ≤ C f L q (Ω) R d-d q max (R d-1 , |x| d-1 ) . (4.28)
In particular, we have

∇u * L ∞ (Ω) ≤ CR 1-d q f L q (Ω) , (4.29)
where C depends only on a * and Ω.

Step 2: bound on D 2 u * L q . According to standard elliptic regularity results (see for instance [16, Lemma 9.17]), we have

D 2 u * L q (Ω) ≤ C f L q (Ω) , (4.30)
where C depends only on a * and Ω. In addition, using the Green function representation again, [20, Theorem 3.3 (vi)], and an argument similar to the proof of (4.28), we have, if |x| > 8R,

|D 2 u * (x)| ≤ Ω ∇ 2 x G * (x, y)f (y)dy ≤ C Ω∩B(0,4R) 1 |x -y| d |f (y)|dy ≤ C R d-d q |x| d f L q (Ω) .
(4.31)

Pointing out that |x| ≥ R for all x ∈ B(x 0 , 4R), this implies, using the Hölder inequality,

D 2 u * L q (Ω∩B(x0,4R)) ≤ CR d-d/q 1 R d R d/q |B(x 0 , 4R)| d/q f L q (Ω) = CR d-2d/q f L q (Ω) ≤ C (diam(Ω)) d-2d/q f L q (Ω) . (4.32)
Step 3: bound on u ε -u * L q (Ω∩B(x0,4R)) . As in the proof of Lemma 4.7, we dene R ε by (1.11), and write R ε = R ε 1 + R ε 2 , where R ε 1 and R ε 2 are solutions to (4. [START_REF]Error estimate and unfolding for periodic homogenization[END_REF]) and (4.20), respectively (with Ω = Ω), and H ε is dened by (2.7). Mutliplying the rst line of (4.19) by R ε 1 and integrating, we have

∇R ε 1 L 2 (Ω) ≤ C H ε L 2 (Ω) , (4.33)
where C depends only on the ellipticity constant of the coecient a. We claim that

H ε L 2 (Ω) ≤ Cε ν R d 2 -d q +1-ν f L q (Ω) . (4.34)
We rst deal with H ε L 2 (Ω\B(0,8R)) , then with H ε L 2 (Ω∩B(0,8R)) . Using Assumptions (A7) and (A8), we have, for all x ∈ Ω,

|H ε (x)| ≤ ε a L ∞ sup 1≤j≤d w ej x ε |D 2 u * (x)| + ε B x ε D 2 u * (x) ≤ Cε |x| ε 1-ν |D 2 u * (x)| ≤ Cε ν |x| 1-ν |D 2 u * (x)|.
We then compute the L 2 norm of H ε on Ω \ B(0, 8R), and use (4.31), together with |x| ≥ R:

H ε L 2 (Ω\B(0,8R)) ≤ Cε ν Ω\B(0,8R) |x| 2(1-ν) |D 2 u * (x)| 2 dx 1/2 ≤ Cε ν Ω\B(0,8R) 1 |x| 2d+2ν-2 R 2d-2d q dx 1/2 f L q (Ω) ≤ Cε ν R 1-ν+ d 2 -d q f L q (Ω) . (4.35)
In addition, successively using Lemma 2.7 (with q = 2 there), the Hölder inequality, and [16, Lemma 9.17],

H ε L 2 (Ω∩B(0,8R)) ≤ Cε ν R 1-ν D 2 u * L 2 (Ω∩B(0,8R)) ≤ Cε ν R d 2 -d q +1-ν f L q (Ω) .

(4.36)

Collecting (4.35) and (4.36), we infer (4.34). Inserting (4.34) into (4.33), we thus have ∇R ε 1 L 2 (Ω) ≤ Cε ν R d 2 -d q +1-ν f L q (Ω) . Hence, using the Hölder inequality again and Sobolev embeddings,

R ε 1 L q 2 (Ω∩B(0,4R)) ≤ CR d q 2 -( d 2 -1) R ε 1 L 2d d-2 (Ω∩B(0,4R)) ≤ CR d q 2 +1-d 2 ∇R ε 1 L 2 (Ω) ≤ Cε ν R 2+ d q 2
d q -ν f L q (Ω) . (4.37)

We estimate R ε 2 . Using the maximum principle,we have

R ε 2 L ∞ (Ω) ≤ ε d j=1 w ej • ε ∂ j u * L ∞ (∂Ω)
.

This estimate, together with (4.28) and Assumption (A7), imply

R ε 2 L ∞ (Ω) ≤ Cε ν R d-d q f L q (Ω) sup x∈∂Ω |x| 1-ν max (R d-1 , |x| d-1 ) ≤ Cε ν R d-d q f L q (Ω) R 2-d-ν .
Thus, R ε 2 L ∞ (Ω) ≤ Cε ν R 2-ν-d q f L q (Ω) . + R ε 1 L q 2 (Ω∩B(x0,4R)) + R ε 2 L q 2 (Ω∩B(x0,4R)) . (4.39)

The rst term is bounded using Assumption (A7) and (4.29):

ε d j=1 w ej • ε ∂ j u * L q 2 (Ω∩B(x0,4R)) ≤ Cε ν R 2+ d q 2
d q -ν f L q (Ω) .

(4.40)

Hence, inserting (4.37), (4.38), (4.40) into (4.39), we infer

u ε -u * L q 2 (Ω∩B(x0,4R)) ≤ Cε ν R 2+ d q 2
d q -ν f L q (Ω) .

(4.41)

Finally, we collect (4.27), (4.29), (4.32) and (4.41), which proves (4.26).

We are now in position to prove Theorem 4.6.

Proof of Theorem 4.6. Let q > d, x 0 , y 0 ∈ Ω, R = |x 0 -y 0 |/16 and f ∈ C ∞ c (Ω ∩ B(y 0 , 4R)). We apply Lemma 4.8. We have Since q > d, we may apply inequality (4.26). This gives Ω (G ε (x, y) -G * (x, y)) f (y)dy ≤ Cε ν R 2-d q -ν f L q (Ω∩B(y0,R)) .

Thus, a duality argument allows to prove G ε (x, •) -G * (x, •) L q (Ω∩B(y0,4R)) ≤ Cε ν R 2-d q -ν . Hence, we may apply Lemma 4.7 with q 2 = q . This implies

|G ε (x 0 , y 0 ) -G * (x 0 , y 0 )| ≤ CR -d q G ε (x 0 , •) -G * (x 0 , •) L q (Ω∩B(y0,4R))
+ Cε ν R 1-ν ∇ y G * (x 0 , •) L ∞ (Ω∩B(y0,4R)) + Cε ν R 2-d q -ν D 2 y G * (x 0 , •) L q (Ω∩B(y0,4R)) .

Applying once again [START_REF]wF qrüter nd uFEyF idmn, The Green function for uniformly elliptic equations[END_REF]Theorem 3.3] to G * , we have ∇ y G * (x 0 , •) L ∞ (Ω∩B(y0,4R)) ≤ CR 1-d and D 2 y G * (x 0 , •) L q (Ω∩B(y0,4R)) ≤ CR -d q . Thus, using (4.42), we get |G ε (x 0 , y 0 ) -G * (x 0 , y 0 )| ≤ Cε ν R 2-d-ν , which concludes the proof, since 16R = |x 0 -y 0 |.

We now prove that a potential B dened by (1.21) exists and has suitable properties in the present setting.

Lemma 5.3. Assume that 1 < q < +∞, and that M = M i k 1≤i,k≤d ∈ L q (R d ) satises ∀k ∈ {1, . . . , d}, div M k = 0.

Then, the potential B ij k dened by

B ij k (x) = R d 1 dω d x i -y i |x -y| d M j k (y) - 1 dω d
x j -y j |x -y| d M i k (y) dy, (5.6) where the constant ω d is the surface of the unit sphere in R d , satises ∇ B ∈ L q (R d ), and (1.21), hence (1.18)-(1. [START_REF]Error estimate and unfolding for periodic homogenization[END_REF])-(1.20). In addition, there exists a constant depending on d and q only such that

∇ B L q (R d ) ≤ C M L q (R d )
.

(5.7)

Finally, if q < d and if M ∈ L ∞ (R d ), then B ∈ L ∞ (R d ), and there exists a constant depending only on d and q such that

B L ∞ (R d ) ≤ C M L q (R d ) + M L ∞ (R d )
.

(

R d |∇B| 2 = R d -M i k ∂ j B ij k + M j k ∂ i B ij k ≤ M L 2 (R d ) ∇B L 2 (R d ) .
Hence, a density argument allows to dene it as a continuous operator from L Hence, applying the Hölder inequality,

B L ∞ (R d ) ≤ C M L ∞ (R d ) 1 |x| d-1
L 1 (B(0,1)

+ C M L q (R d ) 1 |x| d-1
L q (B(0,1) C . We point out that, on the one hand, |x| 1-d ∈ L 1 (B(0, 1)), and on the other hand, since q < d, q > d/(d -1), whence |x| 1-d ∈ L q (B(0, 1) C ). We have thus proved (5.8). Proposition 5.4. Assume that the matrix-valued coecient a satises (1.2) and (1.3) for some r ≥ 1. Let

M i k (x) = a * ik - d j=1
a ij (x) (δ jk + ∂ j w e k (x))

be dened by (1.18). Then there exists B ij k , 1 ≤ i, j, k ≤ d, solution to (5.9)

L q 2 (

 2 (4.38) We next bound u ε -u * . Applying the triangle inequality,u ε -u * L q 2 (Ω∩B(x0,4R)) ≤ ε Ω∩B(x0,4R))

  u ε (x) = Ω G ε (x, y)f (y)dy and u * (x) = Ω G * (x, y)f (y)dy.

-

  div y a T y ε ∇ y G ε (x 0 , y) = 0 in Ω ∩ B(y 0 , 4R), -div y (a * ) T ∇ y G * (x 0 , y) = 0 in Ω ∩ B(y 0 , 4R), G ε (x 0 , •) = G * (x 0 , •) = 0 on (∂Ω) ∩ B(y 0 , 4R).

  2 (R d ) to itself. Furthermore, T is a Calderon-Zygmund operator (see[START_REF] Weyer | Ondelettes et opérateurs. II, Actualités Mathématiques[END_REF] Def. 1, p 224]). Hence, (5.7) holds.It remains to prove(5.8). We split the integral in (5.6) into the integral over |x -y| < 1 and the integral over |x -y| > 1, and ndB(x) ≤ C M * 1 |x| d-1 1 |x|<1 (x) + C M * 1 |x| d-1 1 |x|>1 (x).

  (1.19)-(1.20), that is,∀i, j, k ∈ {1, . . . , d}, B ij k = -B ji k , d i=1 ∂ i B ij k = M j k .Moreover, if r = d, then there exists C > 0 such that∀x ∈ R d , ∀y ∈ R d , |B(x) -B(y)| ≤ C|x -y| 1-νr .

  2,1 . However, as far as estimates (1.13)-(1.14)-(1.15) are concerned, a C 1,1 regularity is sucient. The regularity C 2,1 is only necessary to

  .[START_REF]wF qrüter nd uFEyF idmn, The Green function for uniformly elliptic equations[END_REF] In(3.19) and(3.20), the constant C depends only on a * . Using a Taylor expansion, and applying(3.19) and (3.20) to bound the remainder, we nd that there exists a constant C 0 depending only on a * such that

	sup
	x∈B(0,θ)

  .8) Proof. First, it is clear that (5.6) is a well-dened function if M has compact support. Next, we consider the operator T , which to M associates ∇ B. Moreover, (1.19)-(1.20) are clearly satised by B, hence, we have (1.21). Multiplying it by B and integrating by parts, we have

Acknowledgements

The work of the third author is partially supported by ONR under Grant N00014-15-1-2777 and by EOARD, under Grant FA-9550-17-1-0294.

Next, we prove the following result, which is a consequence of Theorem 4.6, and is the generalization of [START_REF] Hen | Periodic homogenization of Green and Neumann functions[END_REF]Theorem 3.4] to the present setting.

Corollary 4.9. Assume that the matrix-valued coecients a and a T satisfy Assumptions (A1) through (A6), and (A7)-(A8) for some ν > 0. Let Ω be a bounded C 2,1 domain and q ∈ [1, +∞[. Then there exists a constant C > 0 depending only on a, ν, Ω and q, such that for any f ∈ L q (Ω), if u ε and u * are solution to (1.1) and (1.4), respectively, then

where

and

Proof. First, assume that

we use Theorem 4.6 and a simple application of Young-O'Neil inequality [START_REF]Convolution operators and L(p, q) spaces[END_REF][START_REF]Some remarks on convolution operators and L(p, q) spaces[END_REF], which gives

which proves the result. The case s = +∞ is treated by a similar argument.

5 Proof of the main results

Proof of Theorem 1.4

We give in this section the Proof of Theorem 1.4. We rst prove (1.22). Applying Corollary 4.9, we clearly have

.

Hence, using Assumption (A7) and the fact that ∇u * L 2 (Ω) ≤ C f L 2 (Ω) , we deduce (1.22). Next, we prove (1.23). For this purpose, we write again

, where R ε 1 and R ε 2 are dened by (4. [START_REF]Error estimate and unfolding for periodic homogenization[END_REF]) and (4.20), respectively (with Ω = Ω.) Multiplying the rst line of (4.19) by R ε 1 and integrating by parts, we have

. Hence, applying Lemma 2.7, we have (2.10), which implies

The right-most estimate is a consequence of standard elliptic regularity estimates [START_REF]Elliptic partial dierential equations of second order[END_REF]Lemma 9.17].

Next, we apply the Caccioppoli inequality (actually, we need to cover Ω 1 ⊂⊂ Ω by balls B r (x i ) such that B 2r (x i ) ⊂ Ω for each i, and apply the Caccioppoli inequality on each of theses balls), getting

where we applied the Poincaré inequality to R ε 1 . The constant C in the above inequality only depends on Ω 1 , Ω, and the coecient a. Using (1.22) and (5.1), we prove (1.23).

31

We now turn to the proof of (1.24). We x Ω 2 such that Ω 1 ⊂⊂ Ω 2 ⊂⊂ Ω. We cover Ω 1 by balls B r (x j ) such that B 2r (x j ) ⊂ Ω 2 for all j. Applying Proposition 4.2 to R ε , we have

Hence, using (1.23) and (2.10) again, this implies

Here again, elliptic regularity [16, Lemma 9.17] implies D 2 u * L q (Ω) ≤ f L q (Ω) , and we conclude using the Hölder inequality.

Finally, we prove (1.25). We assume f ∈ C 0,β (Ω). We rst assume β ≤ α. Here again, we dene Ω 2 such that Ω 1 ⊂⊂ Ω 2 ⊂⊂ Ω. We cover Ω 1 by balls B r (x j ) such that B 2r (x j ) ⊂ Ω 2 for all j. Applying Proposition 4.4 to v ε = R ε , we nd

. We apply (1.23), (2.10) and (2.11), whence

Here again, we apply standard elliptic estimates [START_REF]Elliptic partial dierential equations of second order[END_REF]Corollary 8.36], thereby proving (1.25).

We assume now that β > α. In particular, we have f ∈ C 0,α (Ω). Thus, we may apply the above result with β = α, and we have

, which completes the proof.

Application to local perturbations of periodic problems: proof of Theorem 1.1

We prove here that the setting dened by (1.2), (1.3) is covered by Theorem 1.4 with ν = ν r dened by (1.12), thereby proving Theorem 1.1. First, we recall that [START_REF]On correctors for linear elliptic homogenization in the presence of local defects[END_REF] (see also [START_REF]Local proles for elliptic problems at dierent scales: defects in, and interfaces between periodic structures[END_REF]) shows that in such a setting, the corrector equation (1.8) has a solution w p which reads as (1.9), where

and with the property if r < d, w p ∈ L ∞ (R d ).

(5.4) Proposition 5.1. Assume that the matrix-valued coecient a satises (1.2) and (1.3), with r = d.

Then there exists a constant C > 0 depending only on a such that

where ν r is dened by (1.12). Remark 5.2. In Proposition 5.1, the case r = d is not covered. However, since in fact a ∈ L r ∩ L ∞ , this case can be addressed using the fact that a ∈ L r for any r > d.

Proof. Since p → w p is a linear map, it is sucient to prove (5.5) in the case |p| = 1. First, elliptic regularity [START_REF]Elliptic partial dierential equations of second order[END_REF]Theorem 8.32] implies that w p,per ∈ C 1,α unif (R d ), hence it clearly satises (5.5). Therefore, we only prove that w p satises (5.5).

If r < d, ν r = 1, and (5.5) is a direct consequence of (5.4). If r > d, we apply Morrey's Theorem [14, Theorem 4.10] to w p :

Applying the triangle inequality, (5.5) is proved.

Proof. We dene B = B per + B, where B per is the periodic solution to

This solution is proved to exist in [22, pages 6-7]. In addition, B per is solution to

Our Assumption (A2) and classical elliptic regularity (applied to w p,per ) show that (M per ) j k is in C 0,α unif (R d ). Hence, still using elliptic regularity [START_REF]Elliptic partial dierential equations of second order[END_REF]Corollary 8.32], we have ∇B per ∈ C 0,α unif (R d ). Arguing as in the proof of Proposition 5.1, we obtain that B per satises (5.9).

We now turn to B. In order to dene it, we rst set, for all j, k,

(5.10)

In view of (5.2) and (5.3), we have M ∈ L q (R d ), for any q ∈]r, +∞[, with q = r allowed if r > 1.

Hence, M satises the assumptions of Lemma 5.3, hence there exists B, dened by (5.6). We have ∇ B ∈ L q (R d ), and one easily proves that B is a solution to

In the case r < d, we simply apply (5.8), nding that B ∈ L ∞ (R d ), which implies (5.9), since ν r = 1.

In the case r > d, we have ∇ B ∈ L q (R d ), and we may apply Morrey's Theorem as we did above for w p . This proves (5.9).

Collecting the results of Proposition 5.1 and Proposition 5.4, we have thus proved the following Proposition, which in turn implies Theorem 1.1. Proposition 5.5. Assume that r ∈ [1, +∞[, r = d, and that the coecient a satises (1.2) and (1.3). Then a satises Assumptions (A1) through (A6), and (A7)-(A8), with ν = ν r dened by (1.12).

Proof. It is clear that (1.3) implies (A1) and (A2). As mentioned above, the results of [START_REF]Local proles for elliptic problems at dierent scales: defects in, and interfaces between periodic structures[END_REF][START_REF]On correctors for linear elliptic homogenization in the presence of local defects[END_REF] imply that (A3) and (A4) are satised. Proposition 5.1 implies (A7), and Proposition 5.4 implies (A8). Finally, Lemma 2.2 implies (A5) and (A6).