On correctors for linear elliptic homogenization in the presence of local defects

We consider the corrector equation associated, in homogenization theory , to a linear second-order elliptic equation in divergence form --$\partial$i(aij$\partial$ju) = f , when the diffusion coefficient is a locally perturbed periodic coefficient. The question under study is the existence (and uniqueness) of the corrector, strictly sublinear at infinity, with gradient in L r if the local perturbation is itself L r , r<+$\infty$. The present work follows up on our works [7, 8, 9], providing an alternative, more general and versatile approach , based on an a priori estimate, for this well-posedness result. Equations in non-divergence form such as --aij$\partial$iju = f are also considered, along with various extensions. The case of general advection-diffusion equations --aij$\partial$iju + bj$\partial$ju = f is postponed until our future work [10]. An appendix contains a corrigendum to our earlier publication [9].


Introduction
Motivation.In a series of works [7,8,9] (see also the related works [6,25]), we have shown that the solution to a highly oscillatory equation of the type −div (a(x/ε) ∇u ε ) = f (1) may be efficiently approximated using the same ingredients as classical periodic homogenization theory when the coefficient a in ( 1) is a perturbation of a periodic coefficient, say to fix the ideas a = a per + ã where a per is periodic and ã ∈ L r , 1 ≤ r < +∞, is a local perturbation that formally vanishes at infinity.The quality of the approximation (that is, the rate of convergence in H 1 norm of u ε minus its approximation based on homogenization theory) is entirely based upon the existence of a corrector function w p , strictly sublinear at infinity (that is, , solution, for each p ∈ R d , to the corrector equation associated to (1), namely Such a situation comes in sharp contrast to the general case of homogenization theory where only a sequence of "approximate" correctors w p,ε , satisfying − div (a(x/ε) (p + ∇w p,ε )) ε→0 ⇀ 0, is needed to conclude, but where the rate of convergence of the approximation is then unknown.See more details in our previous works and in [6,25].A quick inspection on (2) shows that the corrector w p is expected to read as w p = w p,per + wp with w p,per the periodic corrector (solution to − div (a per (p + ∇w p,per )) = 0) and wp solution with ∇ wp ∈ L r to − div (a ∇ wp ) = div (ã (p + ∇w p,per )) in R d . ( In turn, the setting being linear, the existence and uniqueness of wp solution to (3) in the correct functional space is formally equivalent to the existence of an a priori estimate for the exponent q = r, and u solution to The purpose of this article is to show how the estimate (4) (and similar estimates) can be established with a good degree of generality (in particular q needs not be equal to r, the Lebesgue exponent such that ã ∈ L r , but can be any exponent 1 < q < +∞), using a quite versatile approach based on a simple version of the concentration-compactness principle [28].Intuitively, estimate (4) holds true because the perturbation ã within the coefficient a in (5) vanishes in a loose sense at infinity, while, by the celebrated results of Avellaneda and Lin (see [2,3] and more specifically [4]), the estimate holds true when a = a per .Thus, the integrability in R d of the solution remains unchanged.The approach introduced here not only provides an alternate proof of the results of our earlier works for local perturbations of periodic coefficients, but also allows for considering, instead of (1), equations not in divergence form − a ij ∂ ij u = f , which were not approached in our works so far.This also provides an approach for the case of advection-diffusion equations − a ij (•/ε) which will be discussed in a forthcoming publication [10].
Mathematical setting.More precisely, (1) is supplied with homogeneous Dirichlet boundary conditions and posed on a bounded regular domain Ω ⊂ R d , with a right-hand-side term f ∈ L 2 (Ω).For our exposition, we will assume d ≥ 2. Of course, as always, dimension 1 is specific and can be addressed by (mostly explicit) analytic arguments that we omit here.In our earlier publications [7,8,9], the coefficient a considered is of the form a = a 0 + ã where a 0 denotes the unperturbed background, and ã the perturbation.For some of our results there, the unperturbed background can be quite general provided it enjoys the "natural" properties that make homogenization explicit.Similarly, we consider different cases of perturbations ã, and can prove some of the results in the absence of some regularity of the coefficients.We refer the reader to [7,8,9] for all the precise settings and statements regarding the above informal claims.In the present contribution, however, we only address the case a = a per + ã (6) where a per denotes a periodic unperturbed background, and ã the perturbation, with a per (x) + ã(x) and a per (x) are both uniformly elliptic, in for some α > 0, (7) Note that, actually, in the above assumptions, the fact that ã is bounded is implied by the assumption ã ∈ C 0,α unif (R d ) ∩ L r .We nevertheless state it as above to highlight the fact that ã ∈ L q for any q ≥ r.
The reason why we make the above set of assumptions (7) is that (a) we need our results to hold true in the absence of the perturbation ã and the periodic case a 0 = a per with a per sufficiently regular is the only one where we are actually aware of (thanks to the works of Avellaneda and Lin) that this is the case (see however Remark 7 below), and (b) the perturbation ã has to formally vanish at infinity for our specific arguments to hold.
We note, on the other hand, that we readily consider the case of matrixvalued coefficients, instead of scalar-valued coefficients as in our previous works.The modifications are only a matter of technicalities.
All the results of the present article are stated and proved for equation, not for systems.However, as we point out in Remark 6 below, the result of Proposition 2.1 (i.e the divergence form case) carries over to systems.This is not the case of our proof for the non-divergence form (see Remark 8 below), since our proof makes essential use of the maximum principle or of its consequences.
Given the above assumptions, it is well known [5] that there exists a periodic corrector w p,per unique up to the addition of a constant, that solves posed for each fixed vector p ∈ R d .In these particular conditions, ∇w p,per ∈ C 0,α unif (R d ) ∩ L ∞ .This corrector allows to consider the following first-order approximation to u ε issued from the so-called two-scale expansion truncated at the first order where e i are the canonical vectors of R d and where u * denotes the homogenized limit of u ε , that is the solution to with homogeneous Dirichlet boundary conditions on ∂Ω, where a * is the homogenized matrix-valued coefficient (actually also computed from local averages of the solution w p,per to (8)).We have that u ε − u ε,1 per converges to zero (at least) in H 1 norm and, precisely because of the existence of the corrector, the rate of the convergence u ε − u ε,1 per H 1 as ε vanishes can be made precise in terms of ε.We refer the reader to our previous works and the classical textbooks [5,24] for more details.
It has been pointed out in our works that this quality of approximation carries over to the case of a local perturbation of the coefficient in (1).A proof of this fact is contained in [6,25].Problem (8), now reading as (2), is therefore a key step in the understanding, and approximation, of the solution u ε both locally and globally.This fact is intuitively clear when one has realized that this problem is obtained by zooming-in from (1) to the small scale.Using linearity, ( 2) is equivalent to (3) and the key question is thus to prove existence for the latter equation.
Plan.Our contribution is organized as follows.To start with, we consider in Section 2 the case of the equation in divergence form (1) under the conditions made precise in (7).We establish the estimate announced in (4) for the solutions to (5).Our result is stated in Proposition 2.1.The subsequent Section 3 is devoted to the analogous estimate, stated in Proposition 3.1, for the equation in non-divergence form.The fact that each of the estimate implies the well-posedness of the corresponding corrector problem (and thus, subsequently and using the arguments of our other works, the agreement of the first order approximation (9) with the oscillatory solution u ε in various norms and at a certain well defined rate in ε) is made precise in Section 4. Finally, we take the opportunity of the present article to provide, in Appendix A, a corrigendum of our previous work [8,9].Although this did not at all affect the main results of our works, we made there, for some intermediate technical result (namely Lemma 4.2 of [9] and Lemma 1 of [8]), some erroneous claims.We correct this here.

Estimate for operators in divergence form
As mentioned in our introduction, we wish to prove existence and uniqueness of the (strictly sublinear at infinity) corrector w p , solution for p ∈ R d fixed, to the corrector equation (2).Assuming that the coefficient a is of the form (6) and satisfies the assumptions (7), we readily introduce wp = w p − w p,per where the latter denotes the periodic corrector associated to a per , the existence and uniqueness (up to the addition of a constant) of which is a classical fact [5].For further reference, we note that under the regularity conditions (7) for the coefficient a per , elliptic regularity implies that the periodic corrector w p,per is a W 1,∞ function.Indeed, the classical Hilbert theory gives that w p,per ∈ H 1 loc and is periodic.Harnack inequality then implies that w p,per ∈ L ∞ .Finally, [21,Theorem 8.32] implies that ∇w p,per is Hölder continuous, hence in particular is in W 1,∞ .Equation (2) reads as (3), that is, − div (a ∇ wp ) = div (ã (p + ∇w p,per )) in R d , which we now have to solve.Formally simplifying both sides of the equation leads to considering the equation −∆ wp = div (ã p) and we thus expect, for r > 1, to find ∇ wp in the same space as ã, namely L r .This of course will in particular ensure that w p is strictly sublinear at infinity.This expectation is confirmed by the results of our earlier contributions, which we now prove in a different manner here.Our main result is the following: Such a solution is unique up to the addition of a constant.In addition, there exists a constant C q , independent on f and u, and only depending on q, d and the coefficient a, such that u satisfies (4), namely The existence and uniqueness (up to the addition of a constant) of wp (and thus of the corrector w p ) is an immediate consequence of Proposition 2.1.For r > 1, the proposition is applied to q = r, f = ã (p + ∇w p,per ), given that ã ∈ (L r (R d )) d×d and ∇w p,per ∈ (L ∞ (R d )) d .The case r = 1 is considered in Remark 1 below.
Remark 1 As stated in assumption (7), the case ã ∈ L r for r = 1 is allowed in Proposition 2.1.However, the proposition then only gives existence of ∇ wp ∈ L q (R d ) for any q > 1, and not ∇ wp ∈ L 1 as ã.Writing − div(a per ∇ wp ) = div(ã(∇w p + p)) and using ∇w p ∈ L ∞ , a fact that is established there, the results of [4] , contrary to what is mistakenly stated in [9,8].A counterexample for a per = 1, ã with compact support, is provided in Remark 4 below.
The proof of Proposition 2.1 to which we now proceed is based upon the following intuitive property.When the defect ã identically vanishes, the coefficient a is the periodic coefficient a per .In this particular case, estimate (4) has been established in [4].The estimate is shown there, using the representation of the solution u in terms of the Green function G per (x, y) associated to the operator − div (a per ∇.), and the properties of approximation of this Green function obtained from the results of [2].Next, when ã = 0, one notices that, , we have that ã(x) |x|→∞ −→ 0. Intuitively, the operator − div (a ∇.) is therefore close to the operator − div (a per ∇.) at infinity, and estimate ( 4) is likely to hold true there.On the other hand, locally, estimate (4) is a consequence of elliptic regularity and the fact that it holds true in the Hilbertian case q = 2.The actual rigorous proof of Proposition 2.1 implements this strategy of proof, using a continuation argument, the celebrated results of [4] and our results [7] on the case q = 2.
Proof of Proposition 2.1 We argue by continuation.We henceforth fix some 2 ≤ q < +∞.The case 1 < q ≤ 2 will be obtained by duality at the end of the proof.We define a t = a per + t ã and intend to prove the statements of Proposition 2.1 for t = 1.For this purpose, we introduce the property P defined by: we say that the coefficient a, satisfying the assumptions (6)-(7) (for some 1 ≤ r < +∞) satisfies P if the statements of Proposition 2.1 hold true for equation (5) with coefficient a.We next define the interval We intend to successively prove that I is not empty, open and closed (both notions being understood relatively to the closed interval [0, 1]), which will show that I = [0, 1], and thus the result claimed.
To start with, we remark that the results of Avellaneda and Lin in [4, Theorem A] show that I = ∅ since 0 ∈ I. Notice that the property u ∈ L 1 loc is a straightforward consequence of elliptic regularity using f ∈ L 1 loc (R d ) d and the Hölder regularity of the coefficient a per because of (7).This property immediately carries over to all t ∈ [0, 1] as soon we know there is a solution in the following argument.
Next, we show that I is open (relatively to the interval [0, 1]).For this purpose, we suppose that t ∈ I and wish to prove property P on [t, t + ε[ for some ε > 0. In order to solve, for we write it as follows: where Φ t is the linear map which to f ∈ L q (R d ) associates ∇u ∈ L q (R d ), where u is the solution to (5).Since Φ t is continuous from L q to L q , with norm C q , it is clear that, for C q ε ã L ∞ (R d ) < 1, the above map is a contraction.Hence, applying the Banach fixed-point theorem, (12) has a unique solution in L q (R d ), which satisfies the estimate (4), in which C q has been replaced by .
We now show, and this is the key point of the proof, that I is closed.We assume that t n ∈ I, t n ≤ t, t n −→ t as n −→ +∞.For all n ∈ N we know that, for any f ∈ L q (R d ) d , we have a solution (unique to the addition of a constant) u with ∇u ∈ L q (R d ) d of the equation and that this solution satisfies ∇u C n depending on n but not on f nor on u.We want to show the same properties for t.
We first temporarily admit that the sequence of constants C n is uniformly bounded from above in n and conclude.For f ∈ L q (R d ) d fixed, we consider the sequence of solutions u n to which we may write as The sequence of gradients ∇u n is bounded in L q (R d ) d , and therefore weakly converges (up to an extraction) to some ∇u.We may pass to the weak limit in the above equation (recall that a tn − a t converges strongly in L ∞ ) and we find a solution to − div (a t ∇u) = div f .The solution also satisfies the estimate (because the sequence C n is bounded and because the norm is weakly lower semi continuous).There remains to prove uniqueness, that is, with ∇u ∈ L q (R d ), implies ∇u ≡ 0 in the present setting.To this end, we notice that (13) also reads as Using that, in the right-hand side, ã ∇u ∈ L q1 (R d ) (by the Hölder inequality), and the results of [4] on the operator with periodic coefficient, this implies that, in turn, ∇u ∈ L q1 (R d ) d .One may then iterate this argument, and inductively obtain ∇u ∈ L qn (R d ) d for 1 . We recall that we have assumed q ≥ 2, thus 1 q ≤ 1 2 .If in addition r ≥ 2, it is then always possible to find n ≥ 0 such that 1 ≤ q n ≤ 2. In the case r < 2, we note that ã ∈ L r ∩ L ∞ ⊂ L 2 , hence we apply the argument with r = 2.In both cases, we have 1 ≤ q n ≤ 2 for some adequate n ≥ 0, and we obtain, by interpolation, ∇u ∈ But, for such an L 2 function, (13) immediately implies ∇u ≡ 0 by ellipticity (a precise argument may be found in [7]).This concludes the argument of uniqueness.
In order to prove that the sequence of constants C n is indeed bounded, we argue by contradiction and assume that the sequence C n is unbounded, which amounts to assuming there exist We readily notice that (15) also reads as where, as n −→ 0, the rightmost term inside the divergence strongly converges to zero in Therefore, without loss of generality, we may change f n into f n + (a t − a tn ) ∇u n and replace (15) by We now concentrate our attention on the sequence ∇u n .In the spirit of the method of concentration-compactness, we now claim that where B R of course denotes the ball of radius R centered at the origin.We again argue by contradiction (we recall the main argument we are conducting here is also an argument by contradiction) and assume that, contrary to (19), Since ã satisfies the properties in (7), it vanishes at infinity and thus, for any δ > 0, we may find some sufficiently large radius R such that where B c R denotes the complement set of the ball B R .We then write ã ∇u n q using ( 17) and ( 21) for the latter majoration.On the other hand, (20) implies that the first term in the right hand side vanishes, and since δ is arbitrary, this shows that ã ∇u n converges strongly to zero in L q (R d ) d .Inserting this and ( 16) into (18), which, for this specific purpose, we rewrite as and using the continuity result in L q (R d ) d for the periodic setting established by Avellaneda and Lin in [4], we deduce that ∇u n (strongly) converges to zero in L q (R d ) d .This evidently contradicts (17) .We therefore have established (19).
Because of the bound (17), we may claim that, up to an extraction, ∇u n weakly converges in L q (R d ) d , to some ∇u.Passing to the limit in the equation in the sense of distributions, we have − div(a t ∇u) = 0. Next, we show that this convergence in L q loc (R d ) d is indeed strong.By Sobolev compact embeddings, we know this convergence implies the strong convergence of the sequence u n to u (up to a sequence of irrelevant constants c n which, with a slight abuse of notation, we may include in u) in L q loc (R d ).Since we have we multiply this equation by ( We may pass to the limit in each term of the right hand side, since Using the fact that a is elliptic, we therefore obtain that ∇u n −→ ∇u in L 2 (B R ).Finally, using the elliptic estimate (see e.g.[19,Theorem 7.2]) for all 0 < R < +∞, and all solutions v to − div (a t ∇v) = divf , we obtain, applying (23) to u n − u, that ∇u n strongly converges in (L q (B R )) d to ∇u.We infer from (19) and the local strong convergence that ∇u cannot identically vanish, while it solves − div(a t ∇u) = 0.The argument following (13) implies that ∇u = 0, and we reach a final contradiction.This concludes the proof in the case 2 ≤ q < +∞.
For the case 1 < q < 2, we argue as announced by duality.At this stage, we have established the claims within Proposition 2.1 for 2 ≤ q < +∞.We may apply them to the case of the transposed coefficient a T of a, and the operator − div a T ∇. , since obviously, a T satisfies the assumptions (7) if a does.Let us now fix f ∈ L q (R d ) d , for some 1 < q ≤ 2, and denote by 2 ≤ q ′ < +∞, the conjugate exponent of q, that is, , we may associate the unique (up to an , to − div a T ∇v = div g.Its gradient ∇v depends linearly, continuously, on g.We may therefore define by g −→ L f (g) := R d f .∇v a linear form on this linear form is therefore a continuous map on and we read on estimate (24) that We now identify more precisely U .Assuming that g additionally satisfies and thus, by the estimate, ∇v ≡ 0. It follows that, for such g, L f (g) = R d f .∇v = 0, thus R d g .U = 0.This property shows that there exists some u such that U = ∇u, and thus ∇u ∈ L q (R d ) d with We finally show that u satisfies − div (a ∇u) = div f .To this end, we consider the specific case where v ∈ D(R d ) (that is, v is smooth and has compact support) and set g = a T ∇v, so that, in effect, − div a T ∇v = div g holds true.
Applying the above, we have , while, by definition of g, the right-hand side reads as Since this holds true for all v ∈ D(R d ), this shows − div (a ∇u) = div f and concludes our proof.♦ Remark 2 Although, in the above proof, the case 1 < q < 2 is proved by duality, it is also possible to use a direct approach similar to the above argument.The heart of the above proof is the following result: if u is a solution to − div(a∇u) = 0 and if ∇u ∈ L q (R d ), then u is constant.We use it for q = 2, but it is still valid for any q > 1, as it is stated in Lemma 2.2 below.Note however that we do not know if this lemma, which is proved only for equations, carries over to systems.
Lemma 2.2 Assume d ≥ 2, and that the matrix a is C 0,α unif (R d ), uniformly elliptic and bounded.If u satisfies − div(a∇u) = 0 and ∇u ∈ L q (R d ), for some 1 ≤ q < d, then ∇u = 0.
Proof: We first note that, according to the Galiardo-Nirenberg-Sobolev inequality [18, Section 5.6.1,Theorem 1], u ∈ L q * (R d ), up to the addition of a constant, with 1 q * = 1 q − 1 d .Hence, we have u ∈ W 1,q * loc , with sup Here, B 2 (x 0 ) is the ball of radius 2 centered at x 0 .Next, we apply [14, Theorem 1], which states that, if u satisfies the above properties, then for any ball B 1 (x 0 ) of radius 1, we have u ∈ W 1,s (B 1 (x 0 )) for all 1 < s < +∞, with the following estimate where C depends only on the ellipticity constant of a, on a C 0,α (B2) , on d and on s, q * .In particular it does not depend on u nor on x 0 .Applying the De Giorgi-Nash estimate, we have , where C does not depend on x 0 , for the same reasons as above.Hence, applying (25) for s = 2, u is bounded, which, by Liouville theorem (see for instance [30]), implies that u is constant.♦ Remark 3 In the proof of Proposition 2.1, we have used some of our results established in the case q = 2 in [7].We actually only made use of the uniqueness result (in order to prove that ∇u ≡ 0 in (13) above), while we did establish [7] existence and uniqueness of the solution (although not stated as such, the proofs of [7] and [9] imply continuous dependency on the datum).If we allow ourselves to also use the existence result, then the above proof may be slightly simplified.One may then prove by continuation "only" that estimate (4) holds true (in particular, it in turn implies uniqueness), while the existence part is a consequence of a density argument: we approximate for each f n , we have a solution u n the gradient of which is in L 2 (R d ) d ; using the estimate, the sequence ∇u n is a Cauchy sequence in L q (R d ) d ; we may finally pass to the limit and obtain a solution for We chose to present the proof of Proposition 2.1 because its pattern is more general and applies (see Section 3) to operators that are not in divergence form (to which our arguments of [7] do not carry over).
Remark 4 It is well known that, even when a = a per ≡ 1, estimate (4) is wrong for q = 1 (in dimensions d > 1).For instance, for d ≥ 3, let us define where B 1 is the unit ball of R d , |B 1 | its volume, and e = 0 is a fixed vector in R d .It is clear that −∆u = div (1 B1 e) in the sense of distributions.However, a simple computation shows that ∇u(x) ≈ 1 , as |x| → +∞.
Remark 5 We shall see in Section 4 below that a consequence of Proposition 2.1 is the existence of a corrector in the adequate functional space, and, in turn, a quantitative theory of homogenization where the rates of convergence may be made precise, both for the Green functions associated to the divergence operators and for the solutions to the homogenized problems.Actually, the existence of a corrector in the adequate functional space conversely implies Proposition 2.1.By the arguments introduced for the periodic case in [2], further made precise in [27] for various boundary conditions, and adapted in [6] to the case of divergence operators with perturbed periodic coefficients satisfying (7), it is indeed possible to establish, from the existence of a suitable corrector, the approximation properties for the Green function G(x, y).Then, the arguments of [4], using the representation formula for the solution of (5), can be replicated to obtain the results of Proposition 2.1.

Remark 6
The statement and proof of Proposition 2.1 concern equations, but an analogous result holds for systems in divergence form.Indeed, all our arguments carry over to systems, including the central result by Avellaneda and Lin of [4] (based on the results of [2]) on the continuity of operators with periodic coefficients, and the uniqueness result that is a consequence of our arguments of [7] when ∇u ∈ L 2 (R d ) d .The only point above at which we have used the fact that we deal with an equation (actually, applying the Harnack inequality) is when we prove that ∇w p,per ∈ L ∞ .But this is also implied by the results of Avellaneda and Lin [2].However, as stated in Remark 2, for systems, we do not have a direct proof of the case q < 2, and only can prove it by a duality argument.

Remark 7
All what we need in the above proof of Proposition 2.1 is that (i) the gradient ∇w per of the periodic corrector is L ∞ (and the latter fact is, in particular, true when a per itsef is Hölder continuous), (ii) the coefficient a is uniformly continuous (in order to be able to use the local elliptic regularity result (23)), (iii) the result of Avellaneda and Lin concerning the continuity (4) in the case of a periodic coefficient.The recent results of [22] allow to provide a functional analysis setting for non Hölder coefficients.

Estimate for operators in non-divergence form
The purpose of this section is to prove the result analogous to that of Proposition 2.1 in the case of the equation, not in divergence form, Note that because of the specific form of (26), we may assume, without loss of generality, that the matrix-valued coefficient a in ( 26) is symmetric.
Our result is: Such a solution is unique up to the addition of an affine function.In addition, there exists a constant C q , independent on f and u, and only depending on q, d and the coefficient a, such that u satisfies Proposition 3.1 will be used in the next section (and also in [10]) to proceed with the homogenization of the equation The correctors associated to (28) may be put to zero, since they are solutions to −a ij ∂ ij (p.x + w p (x)) = 0.Because it is not in divergence form, the homogenization of (28) (or the precise understanding of the behavior of its solution u ε for ε small) however requires to understand the adjoint problem defining the invariant measure associated to (28).The latter reads as or equivalently, decomposing, in the same spirit as we decomposed the corrector earlier, the measure as m = m per + m , where −∂ ij (a per ij m per ) = 0.The existence and uniqueness of m per , under the constraints m per ≥ 0 and m per = 1, is proved in [5].The existence of m in the suitable functional space is readily related to Proposition 3.1.We will see the details in Section 4.

From the non-divergence form to the divergence form
To start with and as a preparatory work (both for the proof of Proposition 3.1 and the homogenization of equation ( 28) in Section 4), we recall here, for convenience of the reader, a classical algebraic manipulation (see e.g.[5]) that transforms an equation in non divergence form to an equation in divergence form provided an invariant measure (that is, a solution to the adjoint equation) exists and enjoys suitable properties.We perform the transformation here in full generality and abstractly, in the case of the general equation where a ij , 1 ≤ i, j ≤ d, b i , 1 ≤ i ≤ d, are general coefficients.We will actually use the transformation at several distinct stages of our work in the present article, and also in our forthcoming article [10].The coefficients a ij , b i will either be periodic, or include the local perturbation.They will either be at scale one (meaning a ij (x)), or be rescaled by ε as in a ij (x/ε), etc.The firstorder coefficients b i , 1 ≤ i ≤ d, will identically vanish (as in the case here), or not (in [10]).
Consider (31), posed on a (not necessarily) bounded domain Ω and supplied with some suitable boundary conditions (or conditions at infinity) we do not make precise in this formal generic argument.Assume that there exists a positive solution m, actually bounded away from zero, inf m > 0, to the adjoint equation on the same domain, with boundary conditions that we do not make precise either, and suitably normalized.Multiplying (31) by m, we obtain, without even using the specific properties of m, that where Precisely because of (32 Equation ( 35) (again formally) implies the existence of a skew-symmetric matrix In the particular case of dimension d = 3, this is equivalent to the existence of a vector field where B and B are related by Using B and relation (36), it is then immediate to observe that and thus (33) reads as the equation in divergence form with Since inf m > 0, a is elliptic.Moreover, B is skew-symmetric, hence the matrix A is elliptic.
As mentioned earlier, we will make the above transformation explicit, and justify it, in each specific instance we need.The first of these instances, and actually a very classical and well known one, is a simple periodic setting.Consider (31) posed on the entire space R d for periodic second-order coefficients a ij = a per ij , 1 ≤ i, j ≤ d, with period the unit cell of the periodic lattice Z d , b i ≡ 0, 1 ≤ i ≤ d, and c ≡ 0. The adjoint equation ( 32) to be considered is posed also on the entire space, for periodic solutions, and reads as It is established, e.g. in [5], that there exists a unique nonnegative periodic solution m per that is normalized, regular and is indeed bounded away from zero.Performing the above manipulations, we note that b ≡ 0 and div a per is of zero mean in (36).Thus, the matrix B = B per may be assumed periodic, and we write the equation originally considered in the divergence form We notice that, because b i ≡ 0, 1 ≤ i ≤ d, here, div B per = div(m per a per ), and therefore div A per = div(m per a per ) − div a property we shall use in the next section.

Proof of Proposition 3.1
The proof of Proposition 3.1 essentially follows the same pattern as that of Proposition 2.1.We again argue by continuation, (this time for all 1 < q < +∞ since, in this case, the exponent q = 2 does not play any specific role), and show that the interval defined by ( 11) is again the entire interval [0, 1].Of course, this time Property P is based on the statements of Proposition 3.1 and not those of Proposition 2.1 any longer.
The fact that 0 ∈ I is a consequence of the results of [4, Theorem B], precisely because of the algebraic manipulations we recalled above, which allow to rewrite the equation under the conservative form (43), with the specific property (44).The local integrability u ∈ L 1 loc (R d ) is like in Section 2 obtained by elliptic regularity.
Next, we show that I is open (relatively to the interval [0, 1]).In order to do so, we proceed exactly as in the proof of Proposition 2.1, writing the equation where φ t is the application f → D 2 u, where u is the solution to Here again, the map appearing in (45) is proved to be a contraction for ε > 0 sufficiently small, thereby showing existence and uniqueness of the solution, together with the continuity estimate.
Regarding the closeness of I, the heart of the matter is, similarly to the case of an operator in divergence form, to show that if we have then we reach a contradiction.To this end, we first prove, using the same argument as in the proof of Proposition 2.1 and the result by Avellaneda and Lin [4] on the operator with periodic coefficient (this time in non divergence form), that Because of the bound (48), we may claim that, up to an extraction, D 2 u n weakly converges in L q (R d ) d×d , to some D 2 u.Passing to the limit in the sense of distribution implies that u is a solution to − (a t ) ij ∂ ij u = 0. Hence, The Poincaré-Wirtinger inequality and (48) imply that, up to the addition of an affine function, u n is bounded in W 2,q (B R ).Applying Rellich Theorem, we know that, up to extracting a subsequence, u n converges strongly in L q (B R ), for any R > 0. Elliptic regularity results [21, Theorem 9.11] then imply Here, the constant C(R) depends on R, on the ellipticity constant of a and of its C 0,α unif (R d ) norm, but not on f n , u n , u.The right-hand side of this inequality tends to 0 as n → +∞, hence we have strong convergence of u n to u in W 2,q (B R ).In particular, (49) implies that u ≡ 0. Concluding the proof amounts to reaching a contradiction with − (a t ) ij ∂ ij u = 0.This requires a significantly different proof from the case of operators in divergence form, because here we cannot bootstrap some L 2 integrability and use coerciveness to conclude.The proof, in the present case, relies on the maximum principle.
We first give the end of the proof assuming d ≥ 3. We will explain below how to adapt it to the case d = 2.
First, we assume q < d/2.In such a case, we claim that This is proved by induction on n.The case n = 0 is true by assumption.If we assume that (51) is true for n − 1, with n < d/(2q), the Gagliardo-Nirenberg-Sobolev inequality [18, Section 5.6.1] and the fact that imply that, up to the addition of an affine function, In other words, s * * n−1 = s n .We then apply [21,Theorem 9.11] again (that is, inequality (50) with u n = 0 and f n = 0), finding where C does not depend on u nor on the center x 0 of the balls.Summing up all these estimates for x 0 ∈ δZ d , with δ > 0 sufficiently small, we obtain (51).Next, we choose n such that which is always possible because d/q > 2.Then, we have s n > d/2.Hence, Morrey's Theorem [18, Section 5.6.2]implies that u ∈ C 0,α unif (R d ).Since we also have u ∈ L s * * n−1 (R d ), we infer that u vanishes at infinity: for any δ > 0, we have, for R sufficiently large, |u(x)| < δ if |x| > R. Applying the maximum principle, we infer that −δ ≤ u ≤ δ in R d .This being valid for any δ > 0, we find u ≡ 0, reaching a final contradiction.
Second, we assume that q ≥ d/2.We claim that (52) Here again, we prove this by induction: for n = 0, we have σ 0 = q and assumption (48) implies D 2 u ∈ L q (R d ) d×d .Assuming that (52) holds for n − 1, with n < r − r/q, we write the equation satisfied by u as Applying the results of [4], we thus have This concludes the proof of (52).
Next, we choose n such that Since ã ∈ L r ∩L ∞ , we may in fact increase r so that this condition is fulfilled.For such a value of n, we have σ n < d/2, and we may therefore apply our argument of the case q < d/2.Here again, we reach a contradiction.
Let us now assume that d = 2.We cannot, as we did above, assume that q < d/2.However, the proof of ( 52) is still valid.We apply this inequality for the largest possible value of n, that is, n = r − r q , where ⌊•⌋ denotes the integer part.We thus have As we already pointed out above, since our assumption is that ã we may increase r if we wish.Since q r r − r q → q − 1 as r → +∞, we infer that Next, we note that, since the ambient dimension is d = 2, the fact that the matrix a is elliptic and symmetric implies that there exists two positive constants C 0 and C 1 such that, for any symmetric matrix e = e ij , C 0 (a ij e ij ) 2 ≥ e 2 ij + C 1 det(e), with summation over repeated indices.This inequality is easily proved by elementary considerations, and was used for instance in [15], and stated in [32,Equation (4)].We apply it to e = D 2 u, multiply by R+1 , and |∇χ R | ≤ 2. We integrate over R 2 and find We note that the integrand in the last term is equal to Integrating by parts, we thus have Hence, If q ≥ 4/3, (53) implies D 2 u ∈ L 4/3 , and, by the Gagliardo-Nirenberg-Sobolev inequality, ∇u ∈ L 4 .As a consequence, |∇u||D 2 u| ∈ L 1 , and, letting R → +∞ in (54), we infer that D 2 u = 0.If q < 4/3, then, by the Gagliardo-Nirenberg-Sobolev inequality, ∇u ∈ L q * , where 1 q * = 1 q − 1 2 .In particular, the conjugate exponent of q * , denoted by (q * ) ′ , satisfies 4/3 < (q * ) ′ < 2. Thus, q < (q * ) ′ < 2. Hence, successively applying Hölder inequality and the interpolation inequality to (54), we infer Once again, we have reached a contradiction.This shows that I is closed.As it is also open and non empty, it is equal to [0, 1] and this concludes the proof of Proposition 3.1.♦

Remark 8
In sharp contrast to the case of operators in divergence form, Proposition 3.1 and its proof as presented above cannot be extended to the case of systems.In particular, we have made use of the result by Avellaneda and Lin for non-divergence form operators, which is, to the best of our knowledge, specific to equations.In addition, even though some systems satisfy the maximum principle, we do not see how to adapt our proof of uniqueness to the generic case of systems.Note also that, besides the usefulness of Proposition 3.1 on its own, the specific use we will make of that proposition in homogenization theory is exposed in Section 4. The treatment of non-divergence form operators there will require the use of the invariant measure associated to their adjoint, a concept we do not even know how to define for systems.
4 Application to homogenization

Divergence form
We return to the corrector equation ( 2), namely which we write under the form (3): Since w p,per is the periodic corrector, that is the solution to (8) − div (a per (x) (p + ∇w p,per (x))) = 0, with the coefficient a per satisfying assumptions (7), we have, as pointed out at the beginning of Section 2, ∇w p,per ∈ L ∞ (R d ) d .We insert this information in the right-hand side of (3), and may therefore conclude, using Proposition 2.1 for the specific exponent q = r that there exists a function wp , uniquely defined up to the addition of a constant, that solves (3), with ∇ wp ∈ L r (R d ) d and with, considering (4), Setting w p = w p,per + wp , returning to equation ( 2) and using the regularity ( 7), we also have that We have therefore provided an alternative proof of our main results in Theorem 4.1 of [9].The arguments of [6] then allow to prove quantitative homogenization results.Recall however Remark 1: the above argument does not cover the case r = 1, since in Proposition 2.1, q = 1 is excluded.
To end this section, let us mention that, using the above computation, if G is the Green function associated to (31), and if G is the Green function associated to (40), we have G(x, y) = m(y)G(x, y).
Corollary 4.2 (of Proposition 3.1) Assume ( 6)- (7).For all 1 < q < +∞ and f ∈ L q (R d ) d×d , there exists a unique u ∈ L q (R d ) solution, at least in the sense of distributions, to This function u satisfies for a constant C q independent of f .
Proof of Proposition 4.1.Applying Corollary 4.2, we know that there exists a solution m to (57), with m ∈ L r (R d ).In the case r = 1, we have m ∈ L q (R d ), for any q > 1.The measure m per solution to ∂ ij a per ij m = 0 is already known to exist (see [17]), to be Hölder continuous thanks to standard elliptic regularity results, and to be bounded away from 0. The measure m = m per + m is a solution to (56).We prove now that m is positive.For this purpose, we first point out that m is uniformly Hölder continuous, thanks to the results of [12,13] Hence, for R sufficiently large, we have Applying the maximum principle on B R , we infer that m ≥ 0 in the whole space R d .Next, we apply the Harnack inequality [12], which implies that m is bounded away from 0. This concludes the proof of Proposition 4.1.♦ Next, we rescale m, considering m ε (x) = m(x/ε) and multiply ( 28) by m ε .The standard manipulations ((33) through (41)) recalled in Section 3.1 yield with the elliptic matrix valued coefficient and the skew-symmetric matrix-valued coefficient B defined by (38).In the specific case considered, where m = m per + m, B is defined as the sum B = B per + B, where the periodic part B per is obtained solving the periodic equation div B per = div(m per a per ) (the right-hand side being divergence-free because of (42), we recall) and where div B = div ( m a per + m per ã + m ã) .
The latter equation (which also has a divergence-free right-hand side because of (30), that is, (57)) admits a skew-symmetric solution B ∈ L r (R d ) d×d which is unique up to the addition of a constant.This is an application of the Calderón-Zygmund operator theory.Indeed, we introduce the solution to which is known to exist thanks to [29,31], with the additional property that the corresponding operator is continuous from L r to L r : where C is a universal constant.Now, using that div(div(ma)) = 0, a simple computation gives Since, according to (65) and the fact that ã, m ∈ L r (R d ), T ∈ W −1,r ′ (R d ), we necessarily have T = 0, hence B satisfies (63).Finally, we point out that the regularity assumed on a per and ã implies that m per and m are both Hölder continuous, and consequently that A = ma − B satisfy the assumptions (7).On the other hand, the right-hand side m ε f of (61) strongly converges (to f ) in H −1 (R d ) as ε vanishes.We may therefore apply the results of [9,6] to (61) and obtain the homogenized limit, with actual rates of convergence.
We conclude this section with the proof of Corollary 4.2.
Proof of Corollary 4.2 We fix f ∈ L q (R d ) d×d , for some 1 < q < +∞, and denote by q ′ the conjugate exponent of q, that is, 1 q + 1 q ′ = 1.To any arbitrary function g ∈ L q ′ (R d ), we may associate the unique (up to the addition of an is linear continuous.We may therefore define by this linear form is therefore a continuous map on L q ′ (R d ).Hence there exists some u ∈ L q (R d ) such that and we read on estimate (66) that There remains to show that u satisfies To this end, we consider the specific case where v ∈ D(R d ) (that is, v is smooth and has compact support) and set g = −a ij ∂ ij v. Applying the above, we have by definition of g, the right-hand side reads as Since this holds true for all v ∈ D(R d ), this shows −∂ ij ( a ij u) = ∂ ij f ij and concludes our proof.♦ and b the perturbation.Assume that 0 < µ ≤ a 0 (x) + b(x), 0 < µ ≤ a 0 (x), a.e., for some fixed constant µ, a (i) Then, for all 1 ≤ q ≤ 2, there exists a constant C such that, for all R > 0 and all x ∈ R d , G satisfies where B 2R (x)\B R (x) = {y, R ≤ |x − y| ≤ 2R} denotes the annular region enclosed between the balls of radius R and 2R.
(ii) Assume in addition that a 0 = a per is periodic and Hölder continuous, and Three comments are in order: [a] the estimate (68) is correctly stated in [9] (as estimate (26) therein), but the proof there has a flaw.For clarity, we provide the entire, corrected proof here.In the course of the proof of (68) in [9], it is indeed claimed that the estimate holds true.Note the gradient in x and not in y in the integrand.It is actually unclear that the latter estimate is correct, and we suspect it is not.
[b] it is claimed in [9] that the estimate holds; we are only able to establish this estimation as a consequence of a more precise, namely pointwise, estimation of ∇ x ∇ y G(x, y) which is a consequence of arguments in both [9] and [6] and provided the additional assumption b ∈ L r (R d ), for some 1 ≤ r < +∞ holds.
[c] we therefore replace this estimate on annular regions by the estimate (69), and show it is sufficient to conclude the proof of Lemma 4.2 in [9], thereby checking there is no circular argument.
Proof: (i) The proof of (68) is exactly that of [9]: we first prove that, for 1 ≤ q ≤ 2, Note that in (68) and ( 72), the role played by x and y are reversed.We will see below that (72) indeed implies (68).
We first proceed for dimensions d ≥ 3.In the case q = 2, we use the Caccioppoli inequality (see e.g.[9,Lemma 4.3]), which implies that for any x 0 ∈ R d such that y / ∈ B R (x 0 ).Next, we fix y and we cover B 2R \ B R = {x, R < |x − y| < 2R} by balls B R/2 (x i ), for some points x i such that 5R/4 < |x i | < 7R/4, in such a way that (i) a finite number of such x i is sufficient to cover the ring B 2R \ B R and that (ii) any point in B 2R \ B R belongs to at most K balls B R/2 (x i ), for some K that is independent of the radius R.This is easily seen to be possible.
The above estimate holds for any couple of balls (B R/2 (x i ), B R (x i )).We sum all such estimates over the finite number of indices i and obtain Since d ≥ 3, using the classical pointwise estimate (established in [23,26] and recalled in [9, estimate (28)]), we get, This proves the case q = 2.For q < 2, we simply apply the Hölder inequality and use (75): We thus have proved (72) for d ≥ 3.
We next prove (72) for d = 2.For this purpose, we use the following inequality, valid for any β ∈ (0, 2], and which expresses and quantifies the continuous embedding of L 2,∞ into L r for r < 2 on bounded domains: where C β = 4 1+2 −β (2 β −1) (2−β)/2 is suitable.This estimate is proved for instance in the Appendix of [11].We are going to apply it to f = ∇ x G and Ω = B 2R \ B R .Since, in sharp contrast to the situation in dimensions d ≥ 3, G(x, y) does not vanish when |x − y| −→ +∞, we use the estimate to bound from above the right hand side.We find: This implies (72) for q = 2 − β ∈ [0, 2).Finally, in order to prove (72) for q = 2, we fix y and first point out that, integrating the equation − div x (a(x)∇ x G(x, y)) = δ y (x) on the set {x, G(x, y) ≥ s} (which contains y) for some s ∈ R, that where n s denotes the outward normal to the set {x, G(x, y) ≥ s}.Note that, here, we have implicitly assumed that the set {x, G(x, y) ≥ s} is Lipschitzcontinuous, so that its outer normal is well defined and we can integrate by parts.This may not be the case, given the regularity of G.However, using the co-aera formula (see [1,Theorem 3.40]), it is simple to prove that, since G ∈ C 0,α away from x = y, this set is Lipschitz-continuous for almost all s ∈ R.This is sufficient for our purpose here.Next, we multiply the equation by G and integrate on {m ≤ G ≤ M } for some m ≤ M .This gives (79) Hence, using (78), we have We apply the estimate (30) of [16], namely here for all r such that B 2r ⊂ B 2R \ B R .In view of (77), that estimate implies that M R − m R is bounded independently of R.This proves (72) in the case q = 2.
At this stage, we have proved (72).We next point out that, in all generality and for non necessarily symmetric matrix-valued coefficients a, H(x, y) = G(y, x) is the Green function of the operator − div(a T ∇•), where a T is the transpose matrix of a, which satisfies the same assumptions as a.Hence, we may apply (72) to H, finding (68).
(ii) We now turn to the proof of (69).
Here again, we first proceed with the case d ≥ 3, and deal with d = 2 separately.
We note that G satisfies − div x (a∇ x G(•, y)) = 0 in the set |x − y| > 1/2.Hence, applying [21,Theorem 8.32], we have where the constant C does not depend on x 0 , and α > 0 is defined by (7).Using the classical estimate we recalled in (74), we deduce that there exists some constant C > 0 such that This estimate, applied to H(x, y) = G(y, x), the Green function of the operator − div a T ∇• , yields Next, we apply the proof of (i) to ∂ y k G.More precisely, since − div x (a∇ x ∂ y k G(x, y)) = 0 in the set {|x − y| > 1/2}, we may apply Caccioppoli inequality and the whole sequence of arguments that successively lead to (73) through (76) to ∂ y k G instead of G and we obtain, for 1 ≤ q ≤ 2, where C depends on q but not on y nor on R. In particular, summing up all these inequalities for R = 2 k , k ≥ 0, we have The right-hand side is a converging series if and only if q > d/(d − 1).Hence, We are now going to prove that (84) is valid for any q ≤ 2, that is, for a constant C that, like in (83) and (84), depends on q, but not on y.In order to prove (85), we write where We successively prove that G per , G 1 and G 2 satisfy (85), for |x − y| > 4. From this we will infer that G satisfies (85).
Step 2: bound on G 1 .We prove that G 1 satisfies an estimate similar to (90) for |x − y| > 3 (see (100) below).For this purpose, we first prove a bound on ∇ y G 1 , from which we deduce a bound on ∇ x ∇ y G 1 .We write, from (87), Hence, differentiating this equality with respect to y k , we have The term H 2 (x, y) is easily estimated using (82) and (89), and the fact that ∇χ(z − y) vanishes outside 1 < |z − y| < 2: for any x such that |x − y| > 3. Next, we write H 1 (x, y) as follows In order to estimate the first term of the right-hand side of (93), we point out that b ∈ C 0,α unif (R d ), and that G per satisfies, according to [23, Theorem 3.5] and since a Actually, (94) is proved in [23,Theorem 3.5] for a problem in a bounded domain with homogeneous boundary conditions.But a careful examination of the proof shows that the constant does not depend on the size of the domain, implying (94).In addition, [23,Theorem 3.3] The first term of the right-hand side of (95) is dealt with using the fact that, if |x − y| > 3 and |y − z| < 2, then |x − y| ≤ 3|x − z|.Hence, where C does not depend on x nor on y.The second term of the right-hand side of (95) is estimated using that χ has compact support, and integrating by parts: where C is independent of x and y.Here, we have used (89) and (81).Inserting (97) and ( 96) into (95), we have Collecting ( 92) and (98), and inserting them into (91), we find that Lemma 16], which implies that Step 3: G 2 satisfies (85).Differentiating (88) with respect to y k , we have that is, for some constant C independent of x and y.Since this is a point-wise estimate, we may apply it to G(y, x), which is the Green function of the adjoint operator − div a T ∇ .Hence, (111) also holds for ∇ y G. Since ∇ y G satisfies − div(a∇ x ∇ y G) = 0 in the set |x−y| > 1, we apply here again elliptic regularity results, as for instance [21,Theorem 8.32].We thus infer that we have, instead of (84), Then, we adapt the proof of the case d ≥ 3: we define here again G per , G 1 and G 2 by ( 86)-( 87)-(88).The first and second steps, which deal with G per and G 1 , are identical, and we do not reproduce them.The third step is different, since it is in this step that we use (84).We write (101), and point out, here again, that (102) holds.We replace (104) by the fact that where C does not depend on y.Applying [2, Theorem A] to (101), we infer that ∇ x ∇ y G 2 (•, y) L q 1 ({|x−y|>1}) ≤ C.Here again, this, together with (90) and (100), imply that where C does not depend on y.We repeat the argument following (112), where we use (113) instead of (112).This gives ∇ x ∇ y G(•, y) L q 2 ({|x−y|>1}) ≤ C, ∀ q 2 ∈ r 2 , +∞ provided r/2 ≥ 1. Otherwise we have ∇ x ∇ y G(•, y) L q 2 ({|x−y|>1}) ≤ C, for all q 2 > 1. Repeating the argument n times, we thus have ∇ x ∇ y G(•, y) L qn ({|x−y|>1}) ≤ C, ∀ q n ∈ max 1, r n , +∞ .
We have proved (69), thereby concluding the proof of Lemma A.1.♦ Given Lemma A.1, we now explain how one needs to modify the proof of Theorem 4.1 of [9], which we recall here: Theorem A.2 (Theorem 4.1 of [9]) Assume that a = a 0 + b satisfies 0 < µ ≤ a 0 (x)+b(x), 0 < µ ≤ a 0 (x), a.e., for some fixed constant µ, a 0 ∈ C 0,α unif (R d ), b ∈ C 0,α unif (R d )∩L r (R d ), for some r ∈ [1, +∞[.Assume that a 0 = a per is periodic.Then, problem (2) has a solution w p such that w p = w p,0 + wp , where w p,0 is the periodic corrector, that is, the solution to (3), and • if 1 ≤ r < d, then ∇ wp ∈ L r , lim |x|→+∞ wp (x) = 0, and the solution w p is unique among those satisfying w p = v per + v, where v per is periodic and ∇v ∈ L r ; • if 2 ≤ r, then ∇ wp ∈ L r .In addition, the solution w p is unique in the class of solutions w p = v per + v, where v per is periodic and ∇v ∈ L r .
We recall that in the proof of [9, Theorem 4.1], the corrector w p = w p,per + wp is proved to exist, writing wp as wp (y) = ∇ x G(y, x) [b (p + ∇w p,per (x))] dx.
A crucial ingredient of the proof is to establish that ∇ wp ∈ L ∞ .The function wp is splitted as wp = w 1 + w 2 , where where we have chosen q = r ′ , that is, 1 q + 1 r = 1, and used (69) in the right-hand side.This shows that ∇ wp is bounded and the proof of Theorem 4.1 of [9] then proceeds unchanged.
Remark 10 We think that (70) is not true in full generality.Indeed, applying the De Giorgi-Nash estimate (see for example [30,  ( This estimate is true for a periodic coefficient (see [11,4]).It is unclear for a general coefficient.The results of [11] give an example in which (114) and the estimate |∇ x ∇ y G(x, y)| ≤ C|x − y| −d cannot hold together.This is why we do not expect (70) to hold for a general coefficient a.Note however that, as stated in Remark 11 below, it is true if the coefficient a is a local perturbation of a periodic coefficient.
Remark 11 For the case of a coefficient that reads a = a per + ã, with a per periodic and ã ∈ L r (R d ), it is possible to adapt the proofs of [2,3,27], thereby proving directly that inequality (114), thus (70), hold.The central estimate for this is Lemma 16 of [2].We will prove in [6,25] that it is valid in this special case.