Some quantitative homogenization results in a simple case of interface

Abstract Following a framework initiated by Blanc, Le Bris and Lions, this article aims at obtaining quantitative homogenization results in a simple case of interface between two periodic media. By using Avellaneda and Lin’s techniques, we provide pointwise estimates for the gradient of the solution to the multiscale problem and for the associated Green function. Also we generalize the classical two-scale expansion in order to build a pointwise approximation of the gradient of the solution to the multiscale problem (up to the interface), and, adapting Kenig, Lin and Shen’s approach, we obtain convergence rates.


Introduction
In this article, we are concerned with the quantitative homogenization of the following elliptic equations in divergence form: in a simple case of interface between two periodic media. Equation (1) is a prototypical equation for various physical phenomena (like electrostatics or when generalized to systems, elastostatics) set on a material with a microstructure of characteristic scale e ( 1: Homogenization of (1), which aims at studying the behavior of the solution u e when e ! 0; has attracted much attention for half a century. Two particular structures are especially studied: the periodic structure and the stationary ergodic structure (see, e.g, the reference books [1,Chap. 1] for the periodic case, and [2,Chap. 7] for a the stationary stochastic case). Both of these frameworks can be used for actual numerical computations: the homogenization theory is an efficient tool for approximating numerically the solution u e of (1) and its gradient, for a fixed e > 0: Recently, Blanc, Le Bris and Lions proposed in [3] two other cases that can be amenable to numerical computations (see [4]). In the first case, the matrix A is periodic but perturbed by a defect at the microscopic scale (see also [5] for an extension to the advection-diffusion case, and [6] for quantitative homogenization results). In the second case, which might be a fair model for bicrystals, the matrix A is obtained by gluing two periodic structures with H€ older continuous coefficients along a planar interface. This particular framework has the specificity that the associated homogenized equation involves a matrix A ? that is piecewise constant with a discontinuity across the interface (in the generic case). From this perspective, this second case is very different from the aforementioned settings, where the homogenized matrix is constant. The authors of [3] proposed a definition of the correctors and showed that they exist and enjoy some desirable properties of regularity and boundedness. This is a first step in order to obtain quantitative homogenization results. The present article is an attempt to go further, by taking advantage of the literature in periodic homogenization (in particular, the celebrated work of Avellaneda and Lin [7] and the recent article of Kenig, Lin and Shen [8]).
The type of results we show here are familiar to the experts of periodic or stochastic homogenization. But the main idea of this article is the following: in a simple case of bicrystals, the generalized two-scale expansion yields an approximation that possesses the same qualitative and quantitative properties as the two-scale expansion in the periodic setting when considering the gradient of the multiscale solution. From a theoretical point of view, this might be useful for understanding the homogenization of elliptic equations in the case where the homogenized matrix is discontinuous. We also hope this may be of interest for the numerical practitioner.
Our aim is twofold: estimate and approximate the gradient ru e in L 1 norm up to the interface. Obviously, far from the interface, the classical theory of periodic homogenization provides a way to fulfill these goals, first by Avellaneda and Lin's results [7], and then by using the two-scale expansion. Hence, the very difficulty of our study is located close to the interface. This is the reason why we strive for pointwise estimates and approximations (for u e but also on the level of the multiscale Green function).
Our first purpose is to obtain pointwise estimates on the gradient ru e of the multiscale problem (1). In the periodic setting, such results are provided by Avellaneda and Lin's theory [7]. But, as shown in [9] (see also [6,10]), the periodicity assumption is not necessary to these local estimates: they can be obtained in various frameworks, as long as the correctors and the potential (defined by (14) and (22)) associated with the matrix A are strictly sublinear and as long as the homogenized matrix is constant.
The fact that the homogenized matrix is constant is a useful but mere contingent assumption due to the framework used by the authors (the matrix A is supposed to be periodic, possibly perturbed by a defect, or stationary ergodic). Actually, the crucial ingredient is that the multiscale problem inherits regularity properties from the homogenized problem, which are very favorable when the homogenized matrix is constant. But the solution of an elliptic equation the coefficient of which is piecewise H€ older continuous with discontinuities only on smooth interfaces also enjoys some regularity properties (see e.g. [11]), which are sufficient for Avellaneda and Lin's approach. Yet, there is another impediment: in the case of a discontinuous homogenized coefficient A ? ; the A ? -harmonic functions (i.e. satisfying ÀdivðA ? Á ru ? Þ ¼ 0) might have a discontinuous gradient (as a consequence, its second gradient may involve a singular measure supported on the interface). As discussed below, this fact prevents the classical two-scale expansion to work properly. This motivates us to introduce a generalized two-scale expansion. Equipped with this expansion and with the regularity result of [11], we can proceed with Avellaneda and Lin's proof.
Our second purpose is to show to what extent the generalized two-scale expansion yields an accurate pointwise approximation of the gradient ru e ; as does the classical two-scale expansion in the periodic setting, where the convergence rate can be quantified in e (see e.g. [8,Lem. 3.5]). We aim at deriving the same type of convergence rate in the case of bicrystals, up to the interface.
This article is organized as follows. In Section 2, we describe precisely our mathematical setting. Then, in Section 3, we introduce and motivate the generalized two-scale expansion. It is formulated by appealing to the A-harmonic functions (which involve the so-called correctors) and to the A ? -harmonic functions (which are piecewise linear). This expansion is meant to approximate the solution u e of (2) by means of the solution u ? of the homogenized problem. As in the classical cases, the residuum solves an elliptic equation with a R.H.S. in divergence form. We state our main results in Section 4. They concern first pointwise estimates on u e and on ru e and then pointwise approximations of these quantities by the generalized twoscale expansion. These results are also interpreted on the level of the Green functions. We conclude this section by discussing some aspects, limitations and possible extensions of those results. The following sections are devoted to the proofs. More precisely, we collect some elementary results in Section 5 concerning the correctors and the H-convergence of the matrix AðÁ=eÞ; and concerning the regularity properties of the solutions of elliptic equations involving discontinuous coefficients. Then, in Section 6, we use Avellaneda and Lin's techniques to prove pointwise estimates on u e and ru e : Finally, in Section 7, we follow Kenig, Lin and Shen's approach [8] to estimate the residuum between u e and the generalized two-scale expansion. There, the Green function plays a central role.

Mathematical setting
From now on, R d is endowed with a canonical basis ðe 1 ; :::; e d Þ: Since we want to focus on the interface and avoid the problem of boundaries, we set following the equation on the whole ambient space R d ; with d ! 3: (the more difficult case d ¼ 2 will be mentioned in some results). In the above expression, f 2 C 1 c ðR d Þ is a smooth function with compact support, 0 < e < 1; and A is an elliptic and bounded matrix modeling an interface between two infinite crystals that share a common periodic cell on the interface I :¼ f0g Â R dÀ1 : As is also classical in Avellaneda and Lin's theory, we assume that the matrix A is H€ older continuous on the left and on the right of the interface. These assumptions, formalized below, correspond to the simplest case of interface in [3, Sec. 5]: Assumption 1 (ellipticity and boundedness). There exists a constant l>0 such that, for all x; n 2 R d ; the matrix A(x) is invertible and n Á A x ð Þ Á n ! ljnj 2 and n Á A À1 x ð Þ Á n ! ljnj 2 : Assumption 2 (periodicity with commensurable periods). The matrix A(x) satisfies Assumption 3 (regularity). For a fixed a>0; there holds Remark 1. The above regularity assumption can be weakened as in [11,Th. 1.9]: A À and A þ can be assumed to be uniformly a-H€ older continuous everywhere but on the (regular) boundaries of disjoint inclusions.
By using the Lax-Milgram theorem, it can be shown that there exists a solution u e 2 H 1 loc ðR d Þ to (2) such that ru e 2 L 2 ðR d ; R d Þ: This solution is unique up to the addition of a constant that we set by imposing that the mean of u e on R d vanishes.
Under Assumptions 1 and 2, the homogenized problem associated with (2) when e ! 0 is the following: where the homogenized matrix A ? is defined by and A ? 6 are the homogenized matrices associated with the periodic matrices A 6 : In general, the matrix A ? is discontinuous across the interface.
By standard arguments, it can be shown (see Lemma 5.6) that the gradient ru e weakly converges to ru ? in L 2 ðR d ; R d Þ: In the periodic case (namely if A þ ¼ A À ), obtaining strong convergence is more difficult and requires the so-called two-scale expansion: where here, and in the sequel, the Einstein summation convention is used. The functions w i are the so-called correctors, which are the strictly sublinear solutions (unique up to the addition of a constant) to the following equation: We explain in the next section how to generalize the definition of correctors and the two-scale expansion.

Definition of the correctors and the two-scale expansion
A fundamental ingredient of Avellaneda and Lin's proof is that the so-called correctors "correct" sublinear A ? -harmonic functions to A-harmonic sublinear functions. Hence, the first step is to build the sublinear A ? -harmonic functions, i.e. the functions P j satisfying: They induce a natural definition of correctors, which slightly differs from [3]. Unfortunately, with these correctors, the classical formula (6) for the two-scale expansion is algebraically inadequate. As a consequence, we propose a generalization of this formula which takes into account the fact that the homogenized matrix is not constant and that allows for a divergence-form representation of the residuum u e;1 Àu e :

A ? -harmonic functions
When A ? is constant, the sublinear A ? -harmonic functions are the affine functions. (We say that a function f is sublinear if lim sup jxj!þ1 jxj À1 jf ðxÞj ¼ l < þ 1 and strictly sublinear if l ¼ 0 in the previous limit.) In our case, the space of sublinear A ? -harmonic functions is spanned by the constant functions and the following piecewise linear functions: for j 2 ½½1; d; where a is related to the transmission matrix through the interface I and reads: If a ¼ 0 (which strictly encompasses the case where A ? is constant), the functions P j are linear.
It is straightforward that the functions P j are solution to (8). Indeed, by definition, the functions P j are continuous and their gradients read Hence, the functions P j are A ? -harmonic in R Ã À Â R dÀ1 and in R Ã þ Â R dÀ1 ; and they satisfy the transmission conditions across the interface: for all x 2 I and k 2 ½½2; d:

Definition of the correctors
Since the correctors are meant to turn the A ? -harmonic functions P j into A-harmonic sublinear functions, they should solve the following equation: Using the techniques of [3], we show in Section 5.3 the following proposition: Proposition 3.1. Suppose that the matrix A satisfies Assumptions 1-3. Then, there exists a solution w j 2 H 1 loc ðR d Þ to (14), which satisfies the following estimates: and for any 0 < b < minða; 1=4Þ: If a ¼ 0; definition (14) coincides with the classical one (7) and with [3, (48)], that we recall here: However, in the case where a 6 ¼ 0; these three definitions lead to different objects. We motivate our choice in the next section.

A possible generalization of the two-scale expansion
Now, we introduce a generalization of the two-scale expansion. From above, it appears clearly that the corrected version of the sublinear A ? -harmonic functions u ? ðxÞ ¼ a j P j ðxÞ (for ða j Þ 2 R d ) is the following where we use the convention ðrPÞ ij :¼ @ i P j : This suggests to set, for the solution u ? to (4), the following generalized two-scale expansion In (18), the quantity is actually a gradient in harmonic coordinates. Indeed, if we set then, it obviously holds that @ z jũ ðzÞ ¼ U ? j ðP À1 ðzÞÞ: Moreover, by the transmission conditions through the interface (see (12) and (13)), the function U ? j is continuous across the interface I (for f sufficiently regular).
Notice that we recover the classical two-scale expansion when a ¼ 0: The classical argument for assessing the quality of the two-scale expansion is that it allows for a divergence-form representation of the residuum u e;1 Àu e (see e.g. [2, pp. 26-27]). We justify that this algebraical structure is preserved by the generalized expansion (18), with a right-hand term involving the gradient rU ? : In this perspective, it shall be underlined that the formal computation of [2] with the classical two-scale expansion (6) and with definition (17) of [3] involves the quantity r 2 u ? (which, in our case, might involve a singular measure supported on the interface I ) multiplied by quantities that might be discontinuous across the interface I : As a consequence, the mathematical significance of this formal computation is not clear for bicrystals, even when resorting to the theory of distributions.
We now proceed with the computation of ÀdivðAð x e Þ Á rðu e;1 ðxÞ À u e ðxÞÞÞ: For simplicity, we set e ¼ 1 and drop the argument x of the functions below. By (2) and (4), we have We now use definitions (19) and (18) to expand the above right-hand term: Next, using once more (19), we obtain: k : Yet, by definition of P j and w j ; there holds Hence, as will be justified by Proposition 5.5, there exists a tensor B ijk that is antisymmetric in its first two indices and that satisfies Therefore, using the antisymmetry of B, one can express: As a conclusion, while restoring the scale e, we obtain: In the above expression, it can be seen that every term is well-defined in the weak sense. Moreover, the right-hand term is multiplied by e so that, formally, one can expect that the error jru e;1 À ru e j scales like e in various L p norms. This justifies the introduction of the generalized two-scale expansion (18).

Main results
We are now in a position to state our main results. The first ones concern Lipschitz estimates. They can be used in a second step to quantify the error residuum between the generalized two-scale expansion and the actual solution of the multiscale problem.

Estimation
Our first result is a generalization of the local Lipschitz estimates [7, Lem. 16]: Theorem 4.1. Suppose that d ! 2 and that the matrix A satisfies Assumptions 1, 2 and 3. Let e>0; x 0 2 R d and R > 0. Assume that the function u e 2 H 1 ðBðx 0 ; 2ÞÞ is a solution to Then, there exists a constant C that only depends on A and d such that If the ball Bðx 0 ; RÞ does not intersect the interface I ; the above result concerns nothing but the classical periodic setting. But, in Theorem 4.1 the ball Bðx 0 ; RÞ may intersect the interface I ; where the gradient ru e ðxÞ might be discontinuous: in this case, a Lipschitz estimate holds up to the interface. On the first hand, this result might seem surprising: one could have expected that the discontinuity of A through the interface would interact with the oscillations of the small scale so that ru e would not remain bounded when e goes to 0. But, on the other hand, in the periodic setting, it is known that some Lipschitz estimates can also be obtained up to the boundary of a smooth domain (see e.g. [7, Th. 2]), which, from a geometric point of view, might be seen as a kind of interface. Moreover, the way of building the correctors themselves (see [3,Th. 5.1] and Section 5.3) is reminiscent of boundary layers. However, we have not been able to take this apparent similarity further.

Remark 2.
Since the function u e is continuous in Bðx 0 ; RÞ; Theorem 4.1 actually induces a local L 1 estimate in the following sense: Similarly, Corollary 4.2 and Theorem 4.5 can be understood in a local L 1 sense. We prove Theorem 4.1 by using the compactness method of [7]. Two scales should be separated: the small scales, where R=e ( 1; where the Schauder estimates provided by [11] comes into play, the large scales, for R=e ) 1; where we use the compactness method of Avellaneda and Lin.
The large-scale control on ru e is due to a structural property of the matrix A, which uniformly H-converges to its associated homogenized matrix A ? (this statement is made precise in Lemma 5.6). The idea of the proof is to compare u e to a locally A ? -harmonic function u ? (since A ? is piecewise constant, this function enjoys sufficient regularity properties for our purpose). By the uniform H-convergence, u e can be made sufficiently close to u ? ; and thus inherit a medium-scale regularity estimate from it. Then, by "linearizing" u e in the spirit of the two-scale expansion (18) (here we need the correctors w j to be strictly sublinear), one can iterate the medium-scale regularity estimate on balls of exponentially increasing radii to obtain a large-scale regularity estimate. There, it is of the uttermost importance to use a AðÁ=eÞ-harmonic approximation of u e in order to iterate the reasoning (this is another motivation for using the correctors defined by (14)). Finally, a blow-up argument turns the large-scale regularity estimate into an estimate on the gradient ru e by resorting to the Schauder estimates of [11].
As is well-known in the periodic setting (see e.g. [8]), pointwise estimates on the Green function can be derived from the Lipschitz estimates. The Green function Gðx; yÞ (also called fundamental solution) associated with the operator ÀdivðA Á rÞ is a solution of the following equation weak formulation (see [12] for a precise definition): Remark that the Green function x7 !Gðx; yÞ is locally A-harmonic for x 6 ¼ y: Therefore, by applying Theorem 4.1, we deduce the following estimates on the gradient and the mixed gradient of the Green function: Suppose that the matrix A satisfies Assumptions 1, 2 and 3. Let G be the Green function of the operator ÀdivðA Á rÞ on R d . Then, there exists a constant C > 0 depending only on d and A such that, for any x 6 ¼ y 2 R d nI, there holds It should be noted that, by a dilatation argument, the Green function G e of the operator ÀdivðAðÁ=eÞ Á rÞ can be written as Whence the Green function G e also satisfies (27), (28), and (29), with a constant C that does not depend on e.
Remark 3. Remark 3 (Case d ¼ 2). The conclusions of Corollary 4.2 also hold in the case d ¼ 2. It can be retrieved from the case d ¼ 3 by expressing the two-dimensional Green function by means of a 3-dimensional Green function with well-chosen coefficients. This is not shown here but can be found in [7,Th. 13] (see also [12,Prop. 5]).
The proofs of the Theorem 4.1 and Corollary 4.2 are respectively postponed until Sections 6.1, and 6.2.

Approximation
We now estimate the residuum u e;1 Àu e (or equivalently u e Àu ? ) in the L 1 norm by combining the algebraical expression (23) and the estimates on the Green function provided by Corollary 4.2: Proposition 4.3. Let d ! 3; x 0 2 R d and e>0: Suppose that the matrix A satisfies Assumptions 1, 2 and 3. Let f 2 L p ðR d Þ with support inside Bðx 0 ; 1Þ; for p > d. Assume that the functions u e and u ? are respectively the zero-mean solutions to (2) and (4). Then, there exists a constant C that only depends on A, d and p such that For the sake of concise notations, we define the matrices W(x) and W † ðxÞ by where d ij stands for the Kronecker symbol, and the functions P † and w † are the analogous of P and w, but with respect to the transposed matrix A T . Then, the gradient ru e;1 can be expressed by means of W and U ? respectively defined by (32) and (19) as Since the last right-hand term of the above identity scales like e, we expect ru e ðxÞ to be well approximated by Wðx=eÞ Á ru ? ðxÞ: We justify it first on the level of the Green function, in the same vein as the recent results of [8] (see also [13] in the stationary ergodic case). Indeed, as a consequence of Going backwards to the solutions u ? and u e ; this implies an L 1 estimate on the gradient of the residuum: Corollary 4.6. Let d ! 3; x 0 2 R d and e>0. Suppose that the matrix A satisfies Assumptions 1, 2 and 3. Let f 2 L 1 ðR d Þ with support inside Bðx 0 ; 1Þ. Assume that the function u e is the zero-mean solution to (2) and that u ? is the zero-mean solution to (4). Then, there exists a constant C that only depends on A and d such that The proofs of Propositions 4.3 and 4.4, respectively Theorem 4.5 and Corollary 4.6 are postponed until Sections 7.1, respectively 7.2.

Remarks and possible extensions
We conclude this Section by discussing some aspects of this study.
First, we shall underline that the above results concern the problem on R d ; so that there is no boundary. In this regard, if we denote the cell Q :¼ ½À1=2; 1=2 Â ½0; T 2 Â Á Á Á Â ½0; T d and set e :¼ 1=n for n 2 N; then the above results can be generalized to the problem (2) set on Q with periodic boundary conditions (see [14] for a related work in the case of a periodic coefficient). But it seems more difficult to treat the case where (2) is set on a regular bounded domain X along with Dirichlet boundary conditions. Indeed, in this case, we need to show boundary estimates, which might not be true in the neighborhood of the intersection point between the boundary @X and the interface I : At the moment, it is not clear for the author which results may still hold in this case.
Second, in all the results above, the constant C of the estimates is said to "depend on A". This rather vague dependence is a consequence of the fact that the compactness method of Avellaneda and Lin relies on a proof by contradiction. However, one can likely be more precise by proceeding with the proof on the class Eðl; a; s; ðT 6 i ÞÞ of matrices A 2 L 1 ðR d ; R dÂd Þ satisfying Assumptions 1, 2 and 3 with kA 6 k C 0;a ðR d Þ s (rather than by working on a fixed matrix). Thus, the dependence on A would be replaced by a dependence on ðl; a; s; ðT 6 i ÞÞ: Such assumptions have been developed in [8], for example.
Once these limitations are left aside, we remark that, as in [6,9], the main ingredients used here are the long-range behavior of the correctors and the regularity of the homogenized problem. Actually, our proofs only require the fact that A is uniformly elliptic and bounded and uniformly H€ older continuous up to the interface I (Assumption 1, 3) and that there exist correctors w j and a potential B that are bounded. Therefore, the structural Assumption 2 can certainly be weakened. In particular (see [3,Th. 5.7]), one can reasonably think that assuming that the ratios T þ i =T À i are not Liouville-Roth numbers would be sufficient to build bounded correctors w j and a bounded potential B.
The regularity of the matrix A is a key ingredient in the proof of Avellaneda and Lin to show Lipschitz estimates, which encompass the small scales and the large scales. However, as shown in [9], no regularity assumption is necessary to obtain large-scale regularity down to the scale e. Therefore, this assumption could be removed to obtain a weaker version of the above results. In this regard, the approach of [9] could be adapted to obtain regularity estimates (instead of Avellaneda and Lin's approach). One can optimistically think that this would pave the way to quantitative homogenization results in the case of "stochastic" bicrystals.
Finally, one could also think of systems of elliptic equations in divergence form, for which Avellaneda and Lin's approach as well as the regularity results of [11] are adapted. One can extend Theorem 4.1 to the case of systems by a slight adaptation -namely, by showing that the result of C 0;a regularity [7, Th. 1] still holds in our case and then by invoking this regularity estimate instead of the De Giorgi-Nash Moser theorem in the proofs below. Generalizing the other above results would require first to generalize the W 2;p estimates for piecewise constant coefficients in [15,16] (see Lemma 5.2) to the case of systems. To the best of our knowledge, this has not been done yet.

Preliminary considerations
In this section, we collect some results that will be used throughout this article. First, we introduce a few notations. Then, we state some regularity results concerning elliptic equation with piecewise regular (or constant) coefficients. In particular, we show some estimates on U ? defined by (19) and we build a procedure for "linearizing" locally A ? -harmonic functions by appealing to the A ? -harmonic sublinear functions P j : Next, we build the correctors defined by (14) and a solution B to (22) (that we call the potential) and we show that they enjoy some regularity properties. Finally, we justify that the matrices AðÁ=eÞ uniformly H-converge to A ? when e ! 0:

Notations
We introduce here some useful notations for building the correctors and the potential. From now on, the matrix A satisfies Assumptions 1, 2 and 3.
For i 2 ½½2; d; we denote by T i the least common multiple of T À i and T þ i : We define the domains We say that u is D-periodic if u is T i -periodic in x i , for i ! 2: We denote w 6 j ; respectively B 6 ; the correctors, respectively the potential associated with the periodic matrices A 6 : By definition, B 6 ijk is a tensor antisymmetric in its first two indices that solves We recall that both the correctors w 6 j and the potential B 6 are ½0; T 6 1 Â Á Á Á Â ½0; T 6 d -periodic and of regularity C 1;a : Last, if X is a bounded domain, we define the rescaled integral À Ð X u ¼ jXj À1 Ð u; where jXj is the Lebesgue measure of X.

Regularity results
We borrow a regularity result from [11] (see also [17]): Theorem 5.1 (Local version of Theorem 1.1 of [11]). Let A 2 L 1 ðR d ; R dÂd Þ be a matrix defined by (3), where the matrices A 6 satisfy Assumption 3 (but are not necessarily periodic), and that satisfies Assumption 1. Let 0 < b < minða; 1=4Þ: Suppose that f 2 L 1 ðBðx; 2ÞÞ; and that g 2 C 0;b ðBðx; 2ÞnI Þ: If u solves then there exists a constant C only depending on d; a; b, l and kAk C 0;a ðBðx;2ÞÞ such that Then, we provide some W 1;p estimates on the quantity U ? defined by (19): x 0 2 R d ; p 2 ðd; þ1Þ, and A ? be a matrix defined by (5) and satisfying Assumption 1. Suppose that f 2 L p ðR d Þ is supported into Bðx 0 ; 1Þ. Let u ? 2 H 1 loc ðR d Þ be the zero-mean solution to (4) and define U ? by (19). Then there exists a constant C > 0 depending only on d and A ? 6 such that Moreover, there holds The proof (37) rests on a regularity result [16] on non-divergence elliptic equations with coefficients that are constant on the half-spaces R À Â R dÀ1 and R þ Â R dÀ1 : One turns (4) into such an equation by means of the A ? -harmonic coordinates P j : We need to treat separately Estimation (38) since, if d ¼ 3 or d ¼ 4, it is not guaranteed that u ? defined above lies in L 2 ðR d Þ: Proof of Lemma 5.2. We first show an L p estimate on u ? : By definition, there holds: Since the Green function G ? associated with the operator ÀdivðA ? Á rÞ is such that jG ? ðx; yÞj CjxÀyj Àdþ2 ; and since the function f is in L q ðR d Þ for all q 2 ½1; p (by the H€ older inequality, recalling that the support of f is inside Bðx 0 ; 1Þ), the Young inequality yields Next, we define the functionũ by (20). It satisfies the following elliptic equation: whereÃðzÞ is defined bỹ and J(z) is the Jacobian of P evaluated on P À1 ðzÞ: By construction,ÃðzÞ is elliptic and constant on the half-spaces R Ã 6 Â R dÀ1 ; and the product jJðzÞj À1Ã ðzÞ is divergence-free in R d : Whence, (41) can be rewritten as As a consequence, we can apply [16,Th.] (see also [15,Lem. 2.4]): there exists a constant C so that Thus, by (40), we deduce A simple change of variable yields the desired estimate (37). We now show (38). Since f is compactly supported in Bðx 0 ; 1Þ; thenũ isÃ-harmonic on R d nBðz 0 ; qÞ; where q :¼ kðrPÞ À1 k L 1 ðR d Þ and z 0 :¼ P À1 ðx 0 Þ: Therefore, for z 1 2 R d such that jz 0 Àz 1 j>2q; one can apply [11, Prop. 1.7] on Bðz 1 ; jz 0 À z 1 j=2Þ so that Now, recalling that u ? satisfies (40), then, by using (27) and the Cauchy-Schwarz inequality, we obtain that, if jxÀx 0 j>2; there holds Transposing it on the level ofũ yields that, for any z 2 Bðz 1 ; jz 0 Àz 1 j=2Þ; we have jũ z ð Þj CjzÀz 0 j Àdþ2 : Therefore, we deduce from (43) that kr 2ũ k L 1 ðB z 1 ;jz 0 Àz 1 j=4 ð Þ Þ Cjz 0 Àz 1 j Àd : As a consequence, since we already know that r 2ũ 2 L p ðR d Þ for p > 2, we finally obtain that r 2ũ 2 L 2 ðR d Þ: This proves (38).
w We now explain how locally A ? -harmonic functions can be "linearized" by using the sublinear A ? -harmonic functions P j : Lemma 5.3. Let A ? be a matrix defined by (5) and satisfying Assumption 1. Let x 0 2 R d ; and assume that the function u ? 2 H 1 ðBðx 0 ; 1ÞÞ satisfies in Bðx 0 ; 1Þ. Then, there exists a constant C depending only on d and l such that, for all h 2 ð0; 1=2Þ, there holds We underline that the above formula (46) gives a first-order approximation of u ? that is also A ? -harmonic. In this regard, it is a generalization of [7, (3.5)]. This estimates will play a central role in the proof of Theorem 4.1 by encapsulating some regularity properties of the homogeneous problem (4).
The (simple) proof below interprets the A ? -harmonic functions P j as new coordinates, in which (46) appears as a first-order Taylor expansion.
Proof of Lemma 5.3. The key ingredient of the proof is that the functionũ defined by (20) satisfies Indeed, by the same argument as for establishing (41) above, we obtain thatũ satisfies Moreover, since the matrix jJj À1Ã is divergence-free, the gradient rũ is continuous across the interface (inside Bðx 0 ; 1=2Þ). Hence, (49) can be improved as (47).
Therefore, a first-order Taylor expansion onũ yields Finally, since rũðPðxÞÞ ¼ ðrPðxÞÞ À1 Á ru ? ðxÞ; we obtain (46). i. There exists a solution w j to Eq. (14). This solution satisfies w j is D-periodic: The function w j satisfying both (14) and (50) is unique up to the addition of a constant.
ii. There exist constants C > 0 and j>0 such that We now build a potential B:

Uniform H-convergence
Equipped with the correctors, we are in a position to state a first qualitative homogenization result: Lemma 5.6. Suppose that the matrix A satisfies Assumptions 1, 2 and 3. Let sequences x n 2 R d and e n 2 R Ã þ satisfy x n Á e 1 ! l 2 R and e n ! 0. Then, the sequence A n :¼ AððÁ À x n Þ=e n Þ H-converges to A ? ðÁÀle 1 Þ on every regular bounded domain of R d : The proof is classical and relies on the div-curl lemma [18,Lem. 1.1 p. 4]. Therefore, we only emphasize on its main ingredient: the matrix A admits correctors w j such that and that satisfy the following weak convergences in L 2 ðX; R d Þ: for any bounded domain X, for any j 2 ½½1; d and for all sequences x n 2 R d and e n ! 0: The above facts (53), (54), and (55) are consequences of Proposition 5.4, using the properties of the periodic correctors w 6 j :

Estimation
This section is devoted to proving the Lipschitz estimates of Theorem 4.1, from which we derive the estimates on the multiscale Green function of Corollary 4.2.

Lipschitz estimates
Our proof of Lipschitz estimates closely follows the proof of Avellaneda and Lin [7]. It is based on the method of compactness and it is done in the following three steps: 1. The initialization step (see Section 6.1.1), in which we take advantage of the uniform H-convergence (Lemma 5.6) of the multiscale problem to the homogeneous problem (4). Thus, the multiscale solution u e inherits the medium-scale regularity property of the solution u ? of (4) encapsulated in (46). This property is reinterpreted in terms of a "linearization" of u e by AðÁ=eÞ-harmonic functions (here, it is crucial that the correctors w j are strictly sublinear). 2. The iteration step (see Section 6.1.2), in which the previous estimates are iterated to obtain Lipschitz regularity of u e down to scale e (this is also called "excess decay" in [9,Lem. 2]). In this step, it is crucial to resort to an AðÁ=eÞ-harmonic approximation of u e (otherwise, we could not iterate). 3. A blow-up step (see Section 6.1.3), in which we use the regularity result Theorem 5.1 to obtain Lipschitz regularity on scales smaller than e.
6.1.1. Initialization: "linearization" of locally AðÁ=eÞ-harmonic functions For the sake of conciseness, we define the A-harmonic coordinates v by We prove first that the multiscale problem inherits regularity from the homogenized problem: Lemma 14 in [7] Now, by absurd, we assume that there exist e n ! 0; x n 2 R d and u e n satisfying (56) in Bðx n ; 1Þ and such that, for any n 2 N; (We recall that ePð x e Þ ¼ PðxÞ for all x 2 R d and e>0:) We renormalize u e n by ð Up to a subsequence, there holds x n Á e 1 ! l 2 R: Since the cases l ¼ 61 are the classical periodic cases, we assume that l 2 R: We denote x 1 :¼ le 1 : The sequence u e n ðÁ þ x n Þ is bounded in the space L 2 ðBð0; 1ÞÞ and, by the Cacciopoli estimate, in the space H 1 ðBð0; 1=2ÞÞ: Therefore, up to a subsequence (that we do not relabel), it weakly converges to u ? ðÁ þ x 1 Þ 2 H 1 ðBð0; 1=2ÞÞ and in L 2 ðBð0; 1ÞÞ: On the one hand, by the De Giorgi-Nash Moser theorem [19, Th. 8.24 p. 202], there exists b 2 ð0; 1Þ such that the sequence u e n ðÁ þ x n Þ is bounded in C 0;b ðBð0; 1=4ÞÞ: By weak convergence, we also have ð À B x n ;1 ð Þ ju e n j 2 Moreover, the quantity Pðx n þ zÞÀPðx n Þ only depends on z and x n Á e 1 and rPðzÞ only depends on signðz Á e 1 Þ: As a consequence, one can take the limit n ! þ1 in (59). This yields On the other hand, by Lemma 5.6, u ? satisfies (45) in Bðx 1 ; 1=2Þ: Therefore, it also satisfies (58). This is in contradiction with (61) (since u ? cannot be uniformly equal to 0 on Bðx 1 ; 1=2Þ by (59) and (60)). As a consequence, our supposition (59) was absurd. This establishes the existence of e 0 such that (46) is valid for any e < e 0 and x 0 2 R d : w

Iteration
We iterate Lemma 6.1 to obtain the following: Lemma 6.2 (see Lemma 15 in [7]). Suppose that the matrix A satisfies Assumptions 1, 2 and 3. Let c 2 ð0; 1Þ. Let h and e 0 as in Lemma 6.1. Assume that u e satisfies (56) in Bðx 0 ; 1Þ, for x 0 2 R d , and e h n e 0 . Then, there exist a constant C that only depends on d, h and l, and a sequence jðnÞ 2 R d such that A central argument of the proof is that the functions v j are A-harmonic, so that Lemma 6.1 can be iterated.
Proof. We proceed by induction. If By Lemma 6.1, (62) is satisfied. Moreover, since rP only takes two values, we have: and, by the Stokes' theorem ð A similar formula is obtained for the other part of the ball Bðx 0 ; hÞ \ ðR þ Â R dÀ1 Þ: As a consequence, (63) is satisfied for n ¼ 0.
We assume now that Lemma 6.2 is true for n ! 0: Let 0 < e h nþ1 e 0 and u e 2 H 1 loc ðBðx 0 ; 1ÞÞ satisfying (56) in Bðx 0 ; 1Þ: Applying Lemma 6.2, there exists j j ðnÞ associated to u e such that (62) and (63) are satisfied. We setẽ : Since the functions v j are A-harmonic and by (56), we deduce that the function v is AðÁ=ẽÞ-harmonic in Bðx 0 ; 1Þ: Hence, thanks to Lemma 6.1, Yet, by the induction hypothesis (62) and by definition (64), We set so that inserting (64) and (67) in (65) and using (66) yields (62) for the n þ 1-th step. Moreover, thanks to Stokes' theorem (see above) and to (66), where the constant C only depends on d and h (but not on n). This proves (63) for the n þ 1-th step and concludes the proof of Lemma 6.2. Proof of Theorem 4.1. The proof is done by a blow-up argument, in two steps: the first aims at controlling the oscillation of u e down to the scale e. It relies on Lemma 6.2 and on the fact that the correctors are strictly sublinear; the second step uses the first step along with the regularity of the operator ÀdivðAðÁ=eÞ Á rÞ at a scale finer than e -the latter being provided by Theorem 5.1.
Without loss of generality, we assume that R ¼ 4 and that e < e 0 : Step 1: We set c ¼ 1=2; and obtain e 0 and h from Lemma 6.1. Let x 1 2 Bðx 0 ; 2ÞnI : We first show that, if 1 ! r ! e=e 0 ; there holds We set n 2 N such that h nþ1 r h n ; and x 2 Bðx 1 ; rÞ: Thanks to Lemma 6.2, we obtain By Proposition 3.1, the correctors w j are bounded. Therefore, we deduce from the above estimate (69) that ju e x ð Þ À u e x 1 ð Þ j C jx À x 1 j þ e þ r 1þc À Á ku e k L 1 B where G † is the Green function associated with the transposed operator ÀdivðA T Á rÞ: Therefore, without loss of generality, it is sufficient to estimate r x Gðx; yÞ in order to establish (28). By definition, GðÁ; yÞ is A-harmonic in Bðx; jxÀyj=2Þ: Hence, applying Theorem 4.1 and using (27) yields (28) as follows: Finally, differentiating (70) with respect to y implies that r y GðÁ; yÞ is also A-harmonic in Bðx; jxÀyj=2Þ: Therefore, as a consequence of Theorem 4.1, we obtain ; which implies (29), by resorting to (28). w

Approximation
In this section, we prove Proposition 4.3, Proposition 4.4, Theorem 4.5 and Corollary 4.6. The proofs of this section follow the strategy of [8]. For simplicity, we denote henceforth the residuum: where U ? is defined by (45 jrG e x; y ð Þ j 2 dy Since ðdÀ1Þp=ðpÀ1Þ < d and 2ðdÀ1Þ>d; then the above integrals converge. Moreover, by Lemma 5.2, and since f is supported in Bðx 0 ; 1Þ there holds Therefore, (73) yields Furthermore, by a Sobolev injection (recall that p > d), we estimate As a consequence of (74) and (75), and since the correctors w j are bounded, definition (18) of u e;1 implies that We now show a localized version of (30), which is a key step to prove pointwise error estimate on the Green function (27) in Bðx 0 ; 1Þ: Then, there exists a constant C independent of e so that for R e and U ? respectively defined by (71) and by (19).
Proof. We decompose R e :¼ R e 1 þ R e 2 where R e 1 is the zero-mean solution on R d to the following equation: and where the vector-valued function H e is defined by As a consequence of (27) and (28), and since the quantity B ijk ÀA ij w k is bounded, there holds We now estimate the function R e 2 : By (23) and (78), it satisfies Àdiv A x=e Therefore, by applying the triangular inequality and then (80) and (82), we get The triangular inequality and then (80) yield As a consequence, we obtain (77) by combining (83) where u and u ? are respectively the zero-mean solutions to (2) where W is defined by (32).
The proof is divided in four steps. The first step concerns the case where x 2 Bðx 0 ; 1=2Þ is far from the interface: we suppose distðx; IÞ ! d (where d 2 ð0; e=2Þ will be fixed at the end of the proof). We define R e by (71). In this case, thanks to the estimates on the Green function provided by Corollary 4.2 combined with the identity (23), we show that This step closely follows the proof of [8,Lem. 3.5]. However, two points should be underlined: First, the function r 2 U ? might involve a singular measure supported on I ; so that it is necessary to assume that distðx; IÞ ! d: Second, we shall play with the extra parameter d (not present in [8,Lem. 3.5]) to get sufficiently close to the interface I (the salient point is that the R.H.S. of (88) blows up very slowly when d ! 0). The second step is concerned with x 2 Bðx 0 ; 1=2Þ close to the interface (i.e. at a distance smaller than d). Then we use a regularity result at the scale e (namely Theorem 5.1) to compare rR e ðxÞ with rR e ðx 0 Þ; for x 0 farther from the interface. Appealing to the previous step for x 0 and using a triangular inequality provides the desired bound. In the third step, we estimate the derivatives of U ? in (88) by invoking the regularity results of [11]. Finally, in the fourth step, we choose an optimal parameter d and establish (87) by means of the two previous steps.
Step 1: Estimates far from the interface In this step, we assume that the distance distðx; IÞ between x and the interface I ; is larger than d and we show (88). As in the proof of Lemma  This establishes (36). w and using (11), Before going to next step, we need to rewrite divðgÞ in a more suitable form. For the sake of simplicity, we only perform the computations on g þ (the computations concerning g À can be obtained by replacing the index þ by À). Recall that there exist ½0; T 6 1 Â ½0; T 2 Â Á Á Á Â ½0; T d -periodic potentials ðB 6 Þ ijk associated with A 6 : These potentials are antisymmetric in i and j, and they satisfy @ i B 6 ð Þ ijk ¼ A ?

6
À Á jk À A 6 ð Þ jl d lk þ @ l w 6 k À Á : Therefore, divðg þ Þ reads: Recall that rP j is constant everywhere but on the interface. Thus, by using the antisymmetry of B þ and the Schwarz theorem, we rewrite the above divergence term as: div g þ ð Þ ¼ À@ i / þ @ k B þ ð Þ kil @ l P j À/ þ @ i @ k B þ ð Þ kil @ l P j As a consequence, going back to divðgÞ; there holds: