S Wolf

Homogenization of the p-Laplace equation in a periodic setting with a local defect

In this paper, we consider the homogenization of the p-Laplace equation with a periodic coefficient that is perturbed by a local defect. This setting has been introduced in [6, 7] in the linear setting p = 2. We construct the correctors and we derive convergence results to the homogenized solution in the case p > 2 under the assumption that the periodic correctors are non degenerate.

Introduction

This paper is concerned with the homogenization of non-linear degenerate elliptic equations in a periodic setting with defects. More precisely, we are interested in p-Laplacian type equations that are defined, for some p ≥ 2, as

   -div a • ε ∇u ε |∇u ε | p-2 = f in Ω u ε ∈ W 1,p 0 (Ω) (1.1)
for a fixed bounded domain Ω ⊂ R d , d ≥ 1 and f ∈ L p (Ω). For p = 2, we recover the standard linear conductivity equation. In (1.1), the scalar-valued coefficient a is assumed to be of the form

a = a per + a, (1.2)
where a per is a periodic coefficient with standard coercivity and boundedness condition and a is a perturbation of a per such that a ∈ L q (R d ) for some 1 ≤ q ≤ p p-1 . We assume that the coefficient a itself is coercive and bounded and we choose λ > 0 such that ∀y ∈ R d , λ -1 < a per (y) < λ and λ -1 < a(y) < λ.

(1.3)

For fixed ε > 0, Problem (1.1) is well-posed and corresponds to the Euler-Lagrange equation of the minimization Problem min v∈W 1,p 0 (Ω)

1 p Ω a • ε |∇v| p - Ω f v . (1.4)
The behaviour of (1.1) when ε → 0 has been studied in the absence of perturbation, i.e. when a = a per . It corresponds to a particular case of the homogenization of the equation

-divA • ε , ∇u ε = f (1.5)
under general growth and continuity conditions for the operator A(y, ξ) (in our case, we have that A(y) = a per (y)ξ|ξ| p-2 ). The homogenized limit of (1.5) is derived in [START_REF] Fusco | Further results on the homogenization of quasilinear operators[END_REF][START_REF] Fusco | On the homogenization of quasilinear divergence structure operators[END_REF]. It is proved that u ε converges in the W 1,p -weak topology, when ε → 0, to u * which is defined by the homogenized equation

-divA * (∇u * ) = f u * ∈ W 1,p 0 (Ω), (1.6)
where, for ξ ∈ R d , the homogenized operator is

A * (ξ) := Q A y, ξ + ∇w ξ (y) dy,
and the function w ξ ∈ W 1,p per (Q) is the corrector in the direction ξ given as the periodic solution (up to an additive constant) to the equation -divA(•, ξ + ∇w ξ ) = 0.

(1.7)

The strong convergence of the gradient ∇u ε -∇u * -∇w ∇u * (./ε) -→ ε→0 0 in L p (Ω) (1.8) has been obtained in [START_REF] Dal | Correctors for the homogenization of monotone operators[END_REF] with ∇u * replaced by its discretization at small scale ε, for measurability reasons, see Section 2 below for the details. The periodic homogenization of the integral functionals corresponding to (1.1) is exposed in e.g. [START_REF] Braides | Gamma-convergence for Beginners[END_REF]. The stochastic case has been studied qualitatively in [START_REF] Dal | Nonlinear stochastic homogenization[END_REF]. Recently, quantitative results for non-linear stochastic problems have been obtained in [START_REF] Fischer | Optimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems[END_REF] with optimal convergence rates for non-degenerate non-linear operators with quadratic growth, see also [START_REF] Wang | Quantitative estimates on periodic homogenization of nonlinear elliptic operators[END_REF] for the deterministic case. The case of stochastic non-degenerate operators with p-growth, p > 2, is addressed in [START_REF] Clozeau | Quantitative nonlinear homogenization: control of oscillations[END_REF].

In this paper, we study Equation (1.7) when the perturbation a belongs to the space L q (R d ) for 1 ≤ q ≤ p p-1 and to some Hölder space (see Theorem 2.3 below). We then derive the homogenized limit of the sequence (u ε ) ε>0 and we study the convergence of the two-scale expansion (1.8) when we use, on the one hand, the periodic corrector and, on the other hand, the non-periodic corrector (corresponding respectively to the solutions of (1.7) when A(y, ξ) = a per (y)ξ|ξ| p-2 and A(y, ξ) = a(y)ξ|ξ| p-2 ). We also illustrate the quantitative convergence of the two-scale expansion (1.8) in the one-dimensional setting and prove that, in this case, using the non-periodic corrector instead of the periodic corrector in fact improves the quality of convergence of (1.8). The main difficulty of this work is that Equation (1.7) is posed on the whole space R d . One major tool to obtain the strong convergence (1.8) in the nonperiodic case is the continuity of the application ξ -→ ∇w ξ (see Theorem 2.4 below). This will be proved under one of the two Assumptions (A4) or (A4)' below.

Before stating our main results, we would like to comment on the special case p = 2 for the homogenization of Problem (1.1). This problem is very standard since the 70's for a periodic coefficient a, see e.g. [START_REF] Bensoussan | Asymptotic Analysis for Periodic Structures[END_REF] for qualitative results and [START_REF] Avellaneda | Compactness methods in the theory of homogenization[END_REF] for quantitative results. It is worth mentioning that, in this case, the homogenization objects such as correctors and homogenized limits are explicit and very easy to compute. The setting (1.1)-(1.2) has first been introduced in [START_REF] Blanc | A possible homogenization approach for the numerical simulation of periodic microstructures with defects[END_REF] for q = 2. It models local defects that could appear, at the microscale, in a periodic background. The results obtained have been generalized to the case 1 ≤ q < +∞ in [START_REF] Blanc | Local profiles for elliptic problems at different scales: defects in, and interfaces between periodic structures[END_REF][START_REF] Blanc | On correctors for linear elliptic homogenization in the presence of local defects[END_REF] and convergence rates have been proved in [START_REF] Blanc | Precised approximations in elliptic homogenization beyond the periodic setting[END_REF]. In [START_REF] Goudey | A periodic homogenization problem with defects rare at infinity[END_REF], a new non-periodic setting has been introduced to model defects that are not local but rare at infinity. We stress that, in [START_REF] Blanc | A possible homogenization approach for the numerical simulation of periodic microstructures with defects[END_REF][START_REF] Blanc | Local profiles for elliptic problems at different scales: defects in, and interfaces between periodic structures[END_REF][START_REF] Blanc | On correctors for linear elliptic homogenization in the presence of local defects[END_REF][START_REF] Blanc | Precised approximations in elliptic homogenization beyond the periodic setting[END_REF][START_REF] Goudey | A periodic homogenization problem with defects rare at infinity[END_REF], as in the present work, the macroscopic behaviour of the oscillating solution remains the same as in the case of a periodic coefficient. This will be expressed, for the non-linear case, in Theorem 2.7 below.

The paper is organized as follows. The main results of the paper are presented in Section 2. We develop in Section 3 explicit calculations in the one dimensional setting and obtain convergence results. We then turn in Section 4 to the existence of the non-periodic correctors in any dimension. The properties of the non-periodic corrector are proved in Section 5. We then derive qualitiative homogenization results in Section 6. We finally prove in Section 7 a weaker continuity result for the mapping ξ -→ ∇w ξ that is enough to derive qualitative homogenization. We recall in Appendix A the proof of classical results in the periodic case. Technical inequalities are gathered in Appendix B.

Main results

Notations

In the whole paper, d ≥ 1 will be the dimension of the ambient space. The standard unit cube - 1 2 , 1 2 d will be denoted by Q. The euclidian norm will be written | • | as well as the Lebesgue measure of a measurable subset of R d . Let Ω be a bounded domain of R d . If 1 < q < +∞ is an exponent, we define its conjugate by q := q/(q -1). The euclidian open ball of R d centered in x and of radius r > 0 will be written B(x, r). If x = 0, we write B r := B(0, r). We use similar notations for cubes, namely Q(x, r) and Q r . We define the mean-value operation for a measurable and integrable function u by

- B(x,r) u := 1 |B(x, r)| B(x,r)
u.

The indicator function of a measurable set A is denoted 1 A .

The standard Lebesgue and Sobolev spaces are denoted by L q (Ω) and W 1,q (Ω). The associated norms are

u L q (Ω) := Ω |u| q 1/q and u W 1,q (Ω) := Ω |u| q 1/q + Ω |∇u| q 1/q .
The space L q per (resp. W 1,q per ) denotes the set of functions that are periodic and locally belong to L q (resp. W 1,q ). Theses two spaces are endowed with the norms

u L q per := Q |u| q 1/q and u W 1,q per := Q |u| q 1/q + Q |∇u| q 1/q .
The space of uniformly L q (resp. W 1,q ) functions is denoted by L q unif (resp. W 1,q unif ). These spaces are endowed with the norms

u L q unif (R d ) := sup x∈R d u L q (x+Q) and u W 1,q unif (R d ) := sup x∈R d u W 1,q (x+Q) .
For 0 < α < 1, the space C 0,α refers to the standard Hölder space endowed with the norm

u C 0,α := u L ∞ + sup x =y |u(x) -u(y)| |x -y| α .
We define, for δ > 0, the discretization operator M δ : L q (Ω) -→ L q (Ω) introduced in [START_REF] Dal | Correctors for the homogenization of monotone operators[END_REF][START_REF] Fusco | On the homogenization of quasilinear divergence structure operators[END_REF]. If φ ∈ L q (Ω), we set

M δ φ := k∈Z d s.t. δ(Q+k)⊂Ω - δ(Q+k) φ 1 δ(k+Q) . (2.1)
It is clear that M δ is linear and bounded over L q (Ω) and that M δ φ -→ δ→0 φ in L q (Ω).

The periodic case

We assume in this paragraph that a = 0 in (1.2). In this case, the corrector equation is, according to (1.7):

-div a per (y)(ξ + ∇w per ξ )|ξ + ∇w per ξ | p-2 = 0.

(2.

2)

The equation (2.2) admits a unique solution w per ξ in the space W 1,p per (Q)/R. Indeed, the weak formulation of (2.2) is

∀φ ∈ W 1,p per (Q)/R, Q a per (y)(ξ + ∇w per ξ )|ξ + ∇w per ξ | p-2 • ∇φ = 0, (2.3)
which is exactly the Euler-Lagrange equation of the minimization Problem min

v∈W 1,p per (Q)/R 1 p Q a per (y) ξ + ∇v p dy . (2.4)
It is easy to see that the functional appearing in Problem (2.4) is strictly convex, coercive and continuous with respect to ∇v. Thus, (2.4) admits a minimizer w per ξ , the gradient of which is unique. We impose that -Q w per ξ = 0 so that w per ξ is itself unique. Besides, we have the following Proposition (see [START_REF] Fusco | On the homogenization of quasilinear divergence structure operators[END_REF][START_REF] Fusco | Further results on the homogenization of quasilinear operators[END_REF][START_REF] Dal | Correctors for the homogenization of monotone operators[END_REF] or Appendix A below for a proof) gathering the main properties of the application ξ -→ ∇w per ξ :

Proposition 2.1. Let a per : R d -→ R be a periodic and Lipschitz continuous coefficient satisfying (1.3).

(i) The map ξ -→ ∇w per ξ is homogeneous in the sense that for all ξ ∈ R d and t ∈ R,

∇w per tξ = t∇w per ξ . (2.

5)

(ii) There exists an exponent α = α(d, p, a per ) > 0 such that for all ξ ∈ R d , ∇w per ξ ∈ C 0,α (R d ). Moreover, there exists a constant C = C(d, p, a per ) > 0 such that

∇w per ξ L p unif (R d ) ≤ C|ξ| and ∇w per ξ C 0,α (R d ) ≤ C|ξ|. (2.

6)

(iii) There exists a constant C = C(d, p, a per ) > 0 such that for all ξ, η ∈ R d ,

∇w per ξ -∇w per η L p unif (R d ) ≤ C |ξ| 1-β + |η| 1-β |ξ -η| β , β := 1 p -1 . (2.

7)

(iv) There exists a constant C = C(d, p, a per ) > 0 such that for all ξ, η ∈ R d ,

∇w per ξ -∇w per η L ∞ (R d ) ≤ C |ξ| 1-γ + |η| 1-γ |ξ -η| γ , γ := βp p + d/α , (2.8)
where β is defined in (2.7) and α is given by (ii).

It is proved in [START_REF] Fusco | On the homogenization of quasilinear divergence structure operators[END_REF] that u ε converges weakly in W 1,p (Ω) to u * which is defined by (1.6). Note that (1.6) is well posed due to the monoticity of A * (see [START_REF] Dal | Correctors for the homogenization of monotone operators[END_REF] and [START_REF] Le | Nonlinear Elliptic Partial Differential Equations[END_REF]Corollary 8.1]). Convergence in the L ∞ -norm may be obtained in the one-dimensional setting, see Section 3 below.

Results in the non-periodic case

The first result of this contribution concerns the corrector equation (1.8) in the setting (1.1)-(1.2). For a fixed direction ξ ∈ R d , this equation, posed on the whole space R d , is

-div a(y)(ξ + ∇w ξ )|ξ + ∇w ξ | p-2 = 0, (2.9)
where the coefficient a is of the form a := a per + a and a per is a periodic coefficient. We assume that a and a per satisfy the following assumptions:

(A1) there exists λ > 0 such that (1.3) is satisfied;

(A2) the coefficients a and a per are Lipschitz-continuous;

(A3) the perturbation a vanishes at infinity in the sense that a ∈ L p (R d ).

A few comments are in order. First, if a satisfies a ∈ C 0,1 (R d ) and a ∈ L q (R d ) for some q ≤ p then a satisfies (A3) by interpolation. Second, Assumption (A2) allows to ensure local regularity (see Proposition 2.1 above) of the periodic and non-periodic correctors. Finally, the assumptions of [START_REF] Blanc | A possible homogenization approach for the numerical simulation of periodic microstructures with defects[END_REF] in the linear setting correspond to the case p = 2 in the assumptions (A1)-(A2)-(A3) above.

We now consider the equation (2.9) when the coefficient a has the non-periodic structure (1.2). For u ∈ L ∞ (R d ), we define the spaces

W u := v ∈ W 1,1 loc (R d ), R d ∇v p + R d u p-2 ∇v 2 < +∞ and W u := W u /R. (2.10)
The space W u is endowed with the norm

v Wu := ∇v L p (R d ) + |u| p-2 2 ∇v L 2 (R d ) .
(2.11)

In the sequel, we denote undifferently functions and equivalence classes for the relation: f ∼ g if and only if f -g is almost everywhere constant. Lemma 4.1 below gathers some properties satisfied by spaces of the form (2.10). In order to solve (2.9), we seek for w ξ of the form w ξ = w per ξ + w ξ where w per ξ is the solution to (2.3) such that -Q w per ξ = 0. We transform the equation (2.9) into

-diva ξ + ∇w per ξ + ∇ w ξ p-2 (ξ + ∇w per ξ + ∇ w ξ ) -ξ + ∇w per ξ p-2 (ξ + ∇w per ξ ) = div(h), (2.
, R d a ξ + ∇w per ξ + ∇ w ξ p-2 (ξ + ∇w per ξ + ∇ w ξ ) -ξ + ∇w per ξ p-2 (ξ + ∇w per ξ ) • ∇w = - R d h • ∇w.
We easily check using Appendix B that each integral appearing in Definition 2.2 is convergent. Note that if w ξ is a solution to (2.12) in the sense of Definition 2.2, then it is a solution to (2.12) in the distribution sense but it is not clear that the converse holds true. This is true if the weight ξ + ∇w per ξ satisfies Assumption (A4)' below (see also Remark 2.13).

Theorem 2.3 (Existence of the non-periodic correctors). Assume that the coefficient a = a per + a satisfies Assumptions (A1)-(A2)-(A3). Then, for all ξ ∈ R d , there exists a unique solution w ξ to equation (2.9) such that w ξ ∈ W In view of Theorem 2.3, we denote in the sequel w ξ ∈ W ξ+∇w per ξ the unique function such that Q w ξ = 0. The function w per ξ + w ξ is a solution to (2.9) and w ξ solves (2.12)-(2.13) in the sense of Definition 2.2. We also define

w ξ := w per ξ + w ξ ∈ W 1,p per (Q) + W ξ+∇w per ξ . (2.14)
The analogous properties of those given in Proposition 2.1 are given in Theorem 2.4 below for the non-linear correctors w ξ , ξ ∈ R d . In order to obtain continuity results for the application ξ -→ ∇w ξ , we need the following assumption:

(A4) There exists c > 0 independent of ξ ∈ R d such that |ξ + ∇w per ξ | ≥ c|ξ| on Q. We comment in Subsection 2.4 on this assumption. We are able to prove the following Theorem.

Theorem 2.4. Let a := a per + a be a non-periodic coefficient satisfying Assumptions (A1)-(A2)-(A3). For ξ ∈ R d , let w ξ be defined by (2.14).

(i) The map ξ -→ ∇w ξ is homogeneous in the sense that for all ξ ∈ R d and t ∈ R,

∇w tξ = t∇w ξ . (2.

15)

(ii) There exists a constant C = C(d, p, a) > 0 and an exponent α = α(d, p, a) > 0 such that for all

ξ ∈ R d , ∇w ξ ∈ L p unif (R d ), ∇w ξ ∈ C 0,α (R d
) and, moreover, we have the estimates

∇w ξ L p unif (R d ) ≤ C|ξ| and ∇w ξ C 0,α (R d ) ≤ C|ξ|.
(2.16)

(iii) Assume that Assumption (A4) is satisfied. Then there exists a constant C = C(d, p, a, c) > 0 independent of ξ and η such that, for all ξ, η ∈ R d ,

∇ w ξ -∇ w η L p (R d ) ≤ C |ξ| 1-β + |η| 1-β |ξ -η| β , β = γ p -1 min(1, p -2).
(2.17

)
where γ is given by (2.8).

(iv) Assume that Assumption (A4) is satisfied. Then there exists a constant C = C(d, p, a, c) > 0 and an exponent γ > 0 both independent of ξ and η such that, for all ξ, η ∈ R d ,

∇w ξ -∇w η L ∞ (R d ) ≤ C |ξ| 1-γ + |η| 1-γ |ξ -η| γ . (2.18)
An important tool to obtain Theorem 2.4 (iii) is the following Theorem:

Theorem 2.5. Let a := a per + a be a non-periodic coefficient such that (A1)-(A2)-(A3)-(A4) are satisfied. For all ξ ∈ R d , we have that ∇ w ξ ∈ L p (R d ) and the estimate

∇ w ξ L p (R d ) ≤ C|ξ| (2.19)
holds true where C = C(d, p, a, c) > 0 is a constant independent of ξ.

Remark 2.6. Note that, under Assumption (A4), the non-periodic part ∇ w ξ of the corrector has the same integrability as the defect a at infinity. This is reminiscent of the linear case p = 2, see [START_REF] Blanc | Local profiles for elliptic problems at different scales: defects in, and interfaces between periodic structures[END_REF].

Using Theorem 2.4, we can prove qualitative results concerning the homogenization of (1.1) in the non-periodic setting.

Theorem 2.7. Let Ω be a bounded smooth domain, f ∈ L p (Ω), a := a per + a be a scalar-valued coefficient satisfying Assumptions (A1)-(A2)-(A3). For ε > 0, let u ε ∈ W 1,p 0 (Ω) be the solution to (1.1).

(i) We have that u ε - ε→0 u * weakly in W 1,p (Ω) and u ε -→ ε→0 u * strongly in L p (Ω), where u * solves Problem (1.6) and ∀ξ ∈ R d , a * (ξ) = Q a per (y)(ξ + ∇w per ξ )|ξ + ∇w per ξ | p-2 dy. (2.20)
Besides, we have the L p (Ω)-weak convergence a(./ε)∇u

ε |∇u ε | p-2 - ε→0 a * (∇u * ).
(ii) Assume that (A4) is satisfied. Then, we have the strong convergence

∇u ε -∇u * -∇w Mε∇u * • ε -→ ε→0 0 in L p (Ω), (2.21)
where M ε is defined by (2.1).

(iii) We have the strong convergence

∇u ε -∇u * -∇w per Mε∇u * • ε -→ ε→0 0 in L p (Ω), (2.22)
where M ε is defined by (2.1).

We stress that, instead of assuming (A4), Theorem 2.7 can be proved under the assumption that the mapping

Φ p : R d -→ L p unif (R d ) ξ -→ ∇w ξ (2.23)
is continuous. This continuity can be obtained under the following Assumption (A4)' which is clearly weaker than Assumption (A4):

(A4)' For ξ ∈ R d , there exists a constant C > 0 that may depend on ξ and ∇w per ξ such that the following weighted Poincaré-Wirtinger inequality holds true: there exists r min > 0 such that for all R > r min and w ∈ H

1 Q \ Q 1/2 , |ξ + ∇w per ξ (R•)| p-2 2
w --

Q\Q 1/2 w L 2 (Q\Q 1/2 ) ≤ C |ξ + ∇w per ξ (R•)| p-2 2 ∇w L 2 (Q\Q 1/2 ) . (2.24)
We comment in Subsection 2.4 on Assumption (A4)' and we will provide a sufficient condition on ξ + ∇w per ξ so that (2.24) is satisfied. We are able to prove the following Theorem: Theorem 2.8. Assume that (A1)-(A2)-(A3)-(A4)' are satisfied. Then the mapping Φ p defined by (2.23) is continuous. Hence the conclusion of Theorem 2.7 holds true.

We close this section by mentioning that the results of Theorem 2.7 can be improved in the onedimensional setting. We devote Section 3 to convergence results in this particular case.

Remark 2.9. To prove Theorems 2.7 and 2.8, Assumption (A4)' can further be weakened into the following one: the set of smooth functions with compact support over R d , denoted by C ∞ 0 (R d ), is dense in W ξ+∇w per ξ . We show in Lemma B.3 (see Appendix B) that, as pointed out in [START_REF] Vasilevich | Weighted sobolev spaces[END_REF], the density result is implied by Assumption (A4)'. Note that, under Assumption (A4)', we can easily prove (by density) that (2.12)-(2.13) admits a unique solution in the distribution sense in W ξ+∇w per ξ .

Remark 2.10. The method of proof of this paper allows to build the non-periodic correctors for a defect a that belongs to the dual space of W ξ+∇w per ξ , see Lemma 4.1 (iii). This is in particular the case if a ∈ L 2 (R d ). We are however not able to show that the non-periodic corrector satisfies ∇ w ξ ∈ L 2 (R d ) but only that ∇ w ξ ∈ L 2 (|ξ + ∇w per ξ | p-2 dλ), see Remark 2.11 below. More generally, building the nonperiodic correctors for a defect a ∈ L 2+δ ∩ C 0,α (R d ) is a challenging problem that we are unable to address for now. In the linear setting p = 2, this was achieved in [START_REF] Blanc | On correctors for linear elliptic homogenization in the presence of local defects[END_REF] by studying the continuity from L q (R d ) to L q (R d ) for q > 2 of the Riesz operator associated to the coefficient a.

Remark 2.11. The space W ξ+∇w per ξ is in general different from the space

• W 1,p ∩ • H 1 (R d ), where • W 1,p and • H 1 (R d
) are the standard homogeneous Sobolev spaces, unless ξ + ∇w per ξ does not vanish. Assume that there exists x 0 ∈ Q such that ξ + ∇w per ξ (x 0 ) = 0. We can assume by invariance translation that x 0 = 0. Owing to Proposition 2.1 (ii), we have that |ξ + ∇w per ξ (x)| ≤ C|x| α in Q. Let φ ∈ D(Q) be such that φ = 1 on B(0, 1/4). We define Ψ := k∈Z d \{0}

1 |k| δ+ν φ(|k| ν (• -k))
, where δ, ν > 0 will be chosen later. We have

∇Ψ p L p (R d ) = k∈Z d \{0} 1 |k| pδ |k| -ν Q ∇φ(|k| ν x) p dx = k∈Z d \{0} ∇φ p L p (Q) |k| pδ+dν . (2.25)
Besides, we have that

∇Ψ 2 L 2 (|ξ+∇w per ξ | p-2 dλ) = k∈Z d \{0} 1 |k| 2δ |k| -ν Q ∇φ(|k| ν x) 2 |ξ + ∇w per ξ (x)| p-2 dx ≤ k∈Z d \{0} C ∇φ 2 L 2 (Q) |k| αν(p-2)+dν+2δ .
(2.26)

Finally, ∇Ψ 2 L 2 (R d ) = k∈Z d \{0} 1 |k| 2δ |k| -ν Q ∇φ(|k| ν x) 2 = k∈Z d \{0} ∇φ 2 L 2 (Q) |k| dν+2δ . (2.27) We fix ν ∈ ( d d+2 , 1) and δ ∈ max{ d(1-ν) p , d(1-ν)-αν(p-2) 2 }, d(1-ν) 2 so that ∇Ψ ∈ L 2 (|ξ+∇w per ξ | p-2 dλ)∩ L p (R d ) and ∇Ψ / ∈ L 2 (R d ). Note that ∇Ψ ∈ C 0,δ/ν (R d
) so that this counter-example is consistent with the result of Theorem 2.4 (ii) since δ < ν.

Comments on the Assumptions

On Assumption (A4). Assumption (A4) is quite restrictive but is known to be true in dimension 1. Besides, it is proved in [START_REF] Kirill | On full two-scale expansion of the solutions of nonlinear periodic rapidly oscillating problems and higher-order homogenised variational problems[END_REF]Lemma 2,p. 404] that it is also satisfied in dimension d = 2.

We show here that Assumption (A4) is satisfied for laminate materials (in any dimension). Suppose that a per (x) = a 0 (x 1 ) where a 0 : R -→ R is a periodic function. Let ξ = 0. In this case, the periodic corrector w per ξ is a function of the first variable i.e. w per ξ (x) = w 0 ξ (x 1 ) and (2.2) becomes

- d dx 1   a0(x1) ξ 1 + dw 0 ξ dx 1 ξ 1 + dw 0 ξ dx 1 2 + ξ 2 2 + • • • + ξ 2 d p-2 2    = 0. (2.28) If there exists i ≥ 2 such that ξ i = 0, then |ξ + ∇w per ξ | ≥ |ξ i | > 0.
In the other cases, ξ i = 0 for i ≥ 2, thus ξ 1 = 0 and (2.28) reduces to:

- d dx 1   a 0 (x 1 ) ξ 1 + dw 0 ξ dx 1 ξ 1 + dw 0 ξ dx 1 p-2   = 0. (2.29)
There exists a constant C(ξ) such that ξ 1 +

dw 0 ξ dx1 p-1 = C(ξ)/a 0 (x 1 ), where z p-1 := sgn(z)|z| p-1 . If C(ξ) = 0, then w 0 ξ (x 1 ) = -ξ 1 x 1
which contradicts the periodicity of w 0 ξ . In any cases, we have shown that |ξ + ∇w per ξ | > 0. We then prove easily that this implies (A4).

On Assumption (A4)'. This Assumption is satisfied in dimension d = 1, 2 because (A4) is satisfied.

For higher dimensions, we provide here a sufficient condition implying (A4)': Lemma 2.12 (see [START_REF] Cardone | Estimates in homogenization of degenerate elliptic equations by spectral method[END_REF] and [START_REF] Vasil | Homogenization of degenerate elliptic equations[END_REF]). Assume that d ≥ 2 and that |ξ

+ ∇w per ξ | 2-p ∈ L d/2 (Q), then (A4)' is satisfied.
Proof. We refer to [10, Lemma 8].

If we assume that {ξ + ∇w per ξ = 0} is a finite number of points (in the case d > 2) and that all critical points have finite order, denoting by m the maximum order of the corresponding zero points, we have that |ξ

+ ∇w per ξ | 2-p ∈ L d/2 (Q) if and only if m d 2 (p -2) < d i.
e. p < 2 + 2/m. Thus, in this case, Assumption (A4)' can be replaced by assuming that p < 2 + 2/m. Note also that if ξ + ∇w per ξ vanishes at order m along a line (or a curve) in dimension d,

then |ξ + ∇w per ξ | 2-p ∼ |x| m(2-p) which is L d/2 (Q) if and only if d 2 m(p -2) < d -1 i.e. p < 2 + 2(d-1)
dm . Remark 2.13. The Assumption (A4)' is used in the proof of Lemma 7.1 which allows to pass from solutions in the distribution sense to solutions in the sense of Definition 2.2 for PDEs of the form (2.12). We then take advantage of Lemma 7.1 in the proof of Theorem 2.8 by working locally in a concentrationcompactness method.

Extension to other non-linear operators

We have limited the presentation of the results to the simplest operator (1.1) in order to avoid some technicalities and the use of abstract existence Theorems for non-linear PDEs. However, the result of this paper extends to more general operators. We explain below the type of problems that we can address with the technique developed in this work.

The first direct extension concerns the equivalent of (1.1) when a is a matrix-valued coefficient. This corresponds to the following non-linear operator:

a(y, ξ) := A(y)ξ, ξ p-2 2 A(y)ξ, y ∈ R d , ξ ∈ R d , (2.30)
where A is of the form A = A per + A. We assume that the matrix A per is periodic and that A and A per are symmetric and positive definite, that is,

∃λ > 0, ∀y ∈ R d , λ -1 |ξ| 2 ≤ A(y)ξ, ξ ≤ λ|ξ| 2 and λ -1 |ξ| 2 ≤ A per (y)ξ, ξ ≤ λ|ξ| 2 .
The perturbation A satisfies A ∈ L p ∩ C 0,1 (R d ) d×d . The periodic correctors can be defined thanks to variational techniques by considering the minimization problem min

w per ξ ∈H 1,per (Q) 1 p Q A(y)(ξ + ∇w per ξ ), ξ + ∇w per ξ p/2
.

The non-periodic equation corresponding to (2.12) is

-div a(•, ξ + ∇w per ξ + ∇ w ξ ) -a(•, ξ + ∇w per ξ ) = div(h), (2.31)
where

h := a per (•, ξ + ∇w per ξ ) -a(•, ξ + ∇w per ξ ), (2.32)
where a per (•, ξ)

:= A per (•)ξ, ξ p-2 2 A per (•)ξ. It is easily proved that f ∈ L p (R d ) d
and that the method of proof of Section 4 extends to this case by studying the functional

F ξ (v) := 1 p R d A(y)(ξ + ∇w per ξ + ∇v), ξ + ∇w per ξ + ∇v p/2 -A(y)(ξ + ∇w per ξ ), ξ + ∇w per ξ p/2 -p A(y)(ξ + ∇w per ξ ), ξ + ∇w per ξ p-2 2 A(y)(ξ + ∇w per ξ ) • ∇v dy + R d h • ∇v.
Note that the inequalities given in Appendix B are valid for the matrix model (2.30). Concerning the continuity results for the application ξ -→ ∇ w ξ , the results proved in sections 5, 6 and 7 still hold true.

The second less direct extension corresponds to non-variational operators, that is, PDEs that cannot be written as a minimization problem. We consider operators a(y, ξ) that satisfy the following properties:

(1) for all ξ ∈ R d , a(•, ξ) is a measurable function and ξ -→ a(y,

•) for fixed y ∈ R d is of class C 1 (R d )
and of class C 2 (R d \ {0}).

(2) the application ξ -→ a(y, ξ) is homogeneous i.e. a(y, tξ) = t p-1 a(y, ξ) for t ∈ R and y, ξ ∈ R d . We also assume that a(•, ξ) is a uniformly in ξ Lipschitz continuous function: there exists λ > 0 such that ∀y,

y ∈ R d , ∀ξ ∈ R d , |a(y, ξ) -a(y , ξ)| ≤ λ|y -y ||ξ| p-1 . ∀y, y ∈ R d , ∀ξ ∈ R d , |∂ ξ a(y, ξ) -∂ ξ a(y , ξ)| ≤ λ|y -y ||ξ| p-2 .
(3) we have that a(y, ξ) = a per (y, ξ) + a(y, ξ) where a per (•, ξ) is a periodic function satisfying the same homogeneity and regularity properties as a. We assume that the perturbation a satisfies:

∃b ∈ L p ∩L ∞ (R d ), ∀ξ ∈ R d , ∀y ∈ R d , a(y, ξ) ≤ b(y)|ξ| p-1 and ∂ ξ a(y, ξ) ≤ b(y)|ξ| p-2 .
(4) There exists λ > 0 such that

{a(y, ξ) -a(y, ξ )} • {ξ -ξ } ≥ λ -1 |ξ| p-2 + |ξ | p-2 |ξ -ξ | 2 , {a per (y, ξ) -a per (y, ξ )} • {ξ -ξ } ≥ λ -1 |ξ| p-2 + |ξ | p-2 |ξ -ξ | 2 , and |a(y, ξ) -a(y, ξ )| ≤ λ |ξ| p-2 + |ξ | p-2 |ξ -ξ | |a per (y, ξ) -a per (y, ξ )| ≤ λ |ξ| p-2 + |ξ | p-2 |ξ -ξ |.
We also assume that sup

y∈R d sup |ξ|=1 ∂ 2 ξ a(y, ξ) ≤ λ. (2.33)
We define the operator

A :            W ξ+∇w per ξ -→ W ξ+∇w per ξ ∇v -→    W ξ+∇w per ξ -→ R ∇h -→ R d a(•, ξ + ∇w per ξ + ∇v) -a(•, ξ + ∇w per ξ ) • ∇h.
(2.34)

We can show that A is hemicontinuous, bounded, coercive and strictly monotone. By [22, Corollary 8.1], the PDE A(∇v) = F, where F := div a(•, ξ + ∇w per ξ ), admits a unique solution in W ξ+∇w per ξ . The results of Section 7, which are sufficient to prove the qualititative homogenization of Section 6 (which is in fact the main result of this paper), only use the PDE and are thus directly generalized. The results of Section 5 can be proved using the PDE instead of the minimization problem (4. [START_REF] Goudey | A periodic homogenization problem with defects rare at infinity[END_REF]). These extensions are detailed in [25,Chapter 5].

Remark 2.14. A simple example of a non-variational operator satisfting the above assumptions is a(y, ξ) = A(y)ξ |ξ| p-2 , where A is a positive definite and bounded symmetric matrix that can be written

under the form A = A per + A where A ∈ L p ∩ C 0,1 (R d ) d×d .
We check that a is not variational: assume by contradiction that there exists a function

F : R d × R d → R such that a(y, ξ) = ∂ ξ F (y, ξ).
In particular, thanks to Schwartz Theorem, we should have that for all i, j ∈ {1, ..., d},

∂ ξj [a(y, ξ) i ] = ∂ ξi [a(y, ξ) j ] .
Expanding each term gives, for ξ = 0,

A(i, j)|ξ| p-2 + (p -2) [A(y)ξ] i ξ j |ξ| p-4 = A(j, i)|ξ| p-2 + (p -2) [A(y)ξ] j ξ i |ξ| p-4
In particular, for all ξ = 0 and (i, j) ∈ {1, ..., d} 2 ,

[A(y)ξ] i ξ j = [A(y)ξ] j ξ i .
This shows that A is a scalar matrix i.e. proportional to the identity.

Remark 2.15. Assumption (2.33) is only needed in the proofs of Theorem 2.4 and Theorem 2.5. Note also that, together with homogeneity, this Assumption implies that for all δ > 0, sup

y∈R d sup |ξ|=δ ∂ 2 ξ a(y, ξ) ≤ λδ p-3 .

The one-dimensional setting

We consider the homogenization of (1.1) in the one-dimensional case. This equation reads as:

-a(./ε)u ε |u ε | p-2 = f u ε (-1 2 ) = u ε ( 1 2 ) = 0, (3.1)
where a is of the form a = a per + a with a ∈ L q ∩C 0,α (R), 1 < q < +∞ and a satisfies Assumption (A1).

In this section, we assume that f ∈ L p (-1 2 , 1 2 ). Direct computations show that

u ε = -F + C ε a(./ε) 1/(p-1) , F (x) = x -1 2 f, (3.2)
where

x 1 p-1 := sgn(x)|x| 1 p-1 for x ∈ R. The constant C ε is such that 1 2 -1 2 -F + C ε a(./ε) 1/(p-1) = 0. (3.3)
We note that the function F is bounded and thus the sequence (C ε ) ε>0 is bounded. Passing to the limit ε -→ 0 in (3.2) and (3.3), we get that u ε -

ε→0 u * in W 1,p (-1 2 , 1 2 ) and C ε -→ ε→0 C * , where (u * ) = -F + C * a * 1/(p-1) , 1 2 -1 2 (-F + C * ) 1/(p-1) = 0.
and the homogenized coefficient is defined by

a * := L p -weaklim ε→0 a • ε -1 p-1 -(p-1)
.

We easily show with the ingredients used in Remark 3.2 below that

a * =   1 2 -1 2 1 a per 1 p-1   -(p-1)
.

The homogenized equation solved by u * is

-a * (u * ) |(u * ) | p-2 = f u * -1 2 = u * 1 2 = 0.
The corrector equations (2.9) and (2.12)-(2.13) in the direction ξ ∈ R are easy to solve (see Remark 3.2 below):

ξ + w ξ = ξ a * a 1 p-1 and ξ + (w per ξ ) = ξ a * a per 1 p-1 . (3.4) Let R ε := u ε -(u * ) -w (u * ) (./ε
). be the remainder between u ε and its two scale expansion. When u * is regular enough, we have that

R ε = (u 1 ε ) -(u * ) (1 + w (./ε)) -εw(./ε)(u * ) = (-F + C ε ) 1/(p-1) -(-F + C * ) 1/(p-1) a(./ε) 1/(p-1) =:(u 1 ε ) -εw(./ε)(u * ) , (3.5)
where w := w 1 . We concentrate in the sequel on the first term of (3.5), the second one being related to the regularity of u * on the one hand (which is not related to homogenization) and to the sublinearity of w on the other hand. We prove briefly that w is sublinear: indeed, we can write w = (w per ) + w where, due to Remark 3.1 below, w ∈ L q (R d ). By Hölder (or Morrey) inequality, we get immediately that w is sublinear. Since w per is periodic and bounded, it is in particular also sublinear. This proves that w is sublinear. We use Lemma B.2 stated in Appendix B to obtain the bound

|u ε -(u * ) (1 + w (./ε))| ≤ λ|C ε -C * | 1/(p-1) -→ ε→0 0 uniformly. (3.6)
We have obtained the L ∞ -strong convergence of (u 1 ε ) to zero when we use the non-periodic corrector. Let us now introduce the "periodic" remainder R per ε which is defined by R per

ε := u ε -u * -εw per (u * ) (./ε). We have that (R per ε ) = (-F + C ε ) 1/(p-1) a(./ε) 1/(p-1) - (-F + C * ) 1/(p-1) a per (./ε) 1/(p-1) + εw per 1 (./ε)(u * ) = (u 1 ε ) + (-F + C * ) 1/(p-1) 1 a(./ε) 1/(p-1) - 1 a per (./ε) 1/(p-1) + εw per 1 (./ε)(u * ) . (3.7)
The first term tends uniformly to zero while the second one does not tend to zero in L ∞ unless a = 0 or C * = 0. Indeed, testing (3.7) at the microscale gives:

(-F + C * ) 1/(p-1) 1 a(./ε) 1/(p-1) - 1 a per (./ε) 1/(p-1) (εx) ≥ c(p, λ)| -F (εx) + C * | 1/(p-1) | a(x)| -→ ε→0 c(p,λ)|C * | 1/(p-1) | a(x)| =0
. This shows that the convergence of the remainder deteriorates when using w per ξ instead of w ξ . We close this section by commenting on the integrability of the correctors in the particular 1D setting. We show in Remark 3.1 that, in this case, the exponent given by Theorem 2.5 is optimal for q = p , see also Remark 2.6.

Remark 3.1. Suppose that a ∈ L q (R d ) ∩ C 0,α (R d ), 1 < q < +∞. An explicit calculation shows that w ξ = -ξ + (w per ξ ) + ξ + (w per ξ ) 1 - a a 1 p-1 , (3.8
)

and |ξ + (w per ξ ) | ≥ c|ξ|. Since a(x) -→ |x|-→+∞ 0, we have that w ξ ∼ x→±∞ - 1 p -1 a(ξ + (w per ξ ) ) a .
Thus w ξ ∈ L q (R d ), that is w ξ has the same integrability as a and this exponent is optimal.

Remark 3.2. We show below that there exists a unique solution w ξ to (2.9) that is sublinear at infinity. This justifies, in dimension one, to search w ξ under the form w per ξ + w ξ where w ξ ∈ L p (R).

Assume that w ξ is a sublinear solution to (3.4). Then, there exists a constant C such that ξ 1) . We have by sublinearity that

+ w ξ = (C/a) 1/(p-
ξ = lim x→+∞ - x 0 ξ + w ξ = lim x→+∞ - x 0 C a 1 p-1 = C 1 p-1 lim x→+∞ - x 0 1 a 1 p-1 .
However, by Lemma B.2, we have that

- x 0 1 a 1 p-1 -- x 0 1 a per 1 p-1 ≤ - x 0 a 1 p-1 -(a per ) 1 p-1 a 1 p-1 (a per ) 1 p-1 ≤ Cst. - x 0 | a| 1 p-1 ,
where Cst. denotes a constant depending only on p and λ. Since a ∈ L p (R d ), we get by Hölder inequality that

- x 0 | a| 1 p-1 -→ x→+∞ 0.
This shows that

lim x→+∞ - x 0 1 a 1 p-1 = lim x→+∞ - x 0 1 a per 1 p-1 = 1 a * 1 p-1
and gives that C = ξ|ξ| p-2 a * . This shows that w ξ is necessarily of the form (3.4).

Numerical experiments. We have implemented for p = 3 the solution to (1.1) in the 1D setting for f (x) = 2x and a(y) := a per (y) + a(y) = 2 + cos(2πy) + 10e -|y| on the domain Ω := (-1 2 , 1 2 ). The boundary conditions are homogeneous Dirichlet conditions i.e. u ε (- 1 2 ) = u ε ( 1 2 ) = 0. The coefficient a satisfies of course Assumptions (A1)-(A3). The results are plotted on Figure 1. We comment on these results. We have plotted for different values of ε the function u ε (which is labeled as 'exact solution'), the periodic two scale approximation (u * ) + (w per ) (./ε)(u * ) (which is labeled as 'periodic two-scale approx.') and the non-periodic two scale approximation (u * ) +w (./ε)(u * ) (which is labeled as 'non-periodic two-scale approx.'). Tables 1 and2 give numerical values for the periodic and non-periodic remainders in L 2 and L ∞ -norm for different values of ε. We see that on Figure 1, qualitatively, the non-periodic two-scale approximation fits efficiently the exact solution for each chosen value of ε. The periodic two-scale approximation corresponds to the exact solution far from the defect, which, as ε -→ 0, concentrates aroung the origin. We notice that the non-periodic corrector is useful to reconstruct the oscillations of the exact solution locally around the defect. Tables 1 and2 express the same idea: the L ∞ -norms of the periodic remainders remain unchanged as ε decreases whereas those of the non-periodic remainder decrase with ε. For the L 2 -norm, which is weaker than the L ∞ -norm, both norms decrease as ε gets closer to zero although the nonperiodic approximation is more accurate than the periodic approximation. This means that, depending on the precision we want (and also on the regularity on f and a), we may use the periodic corrector, which is much easier to compute, or the non-periodic corrector, if we seek for a fine approximation of the exact solution. This can also be seen theoretically since R per ε -R ε = ε w (./ε)(u * ) and, for all q ≤ p,

ε w (./ε)(u * ) L q (0,1) ≤ Cε d/p (u * ) L ∞ (0,1) w L p (R) .
In any case, we get that R per

ε -R ε -→ ε→0 0 in L q -norm, q ≤ p but not in L ∞ -norm.
Another way to reformulate the preceding remark is the following: the non-periodic corrector provides a better approximation at the microscale. 4 Existence of the non-periodic correctors: proof of Theorem 2.3

ε R per ε L ∞ R ε L ∞ 0.
We start this section with some preliminary results: (ii) Its topological dual space is

-div(g), g = g 1 + g 2 |ξ + ∇w per ξ | p-2 , g 1 ∈ L p (R d ), g 2 |ξ + ∇w per ξ | p-2 2 ∈ L 2 (R d ) .
(iii) Each bounded sequence in W ξ+∇w per ξ admits a weakly converging subsequence.

Proof. We refer to [25,Chapter 5] for the proof of this elementary Lemma.

We now fix ξ ∈ R d , h ∈ L p (R d ) d , a coefficient a satisfying Assumptions (A1)-(A2)-(A3). We introduce the functional F ξ defined by

F ξ (v) := 1 p R d ag ξ+∇w per ξ (∇v) + R d h • ∇v, (4.1)
where the function g ξ is defined ny (B.4):

g ξ (x) := |ξ + x| p -|ξ| p -pξ|ξ| p-2 • x.
Since g ξ (x) ≥ 0 over R d , we immediately have that F ξ is defined over

V := v ∈ W 1,1 loc (R d ), ∇v ∈ L p (R d ) /R (4.2)
and takes its values in R ∪ {+∞}. Note that since F ξ (v) only depends on ∇v, F ξ is well-defined on the space of equivalence classes V . For R > 0, we define the mapping

F R ξ :    V -→ R v -→ 1 p B R ag ξ+∇w per ξ (∇v) + R d h • ∇v. (4.3)
We gather in Lemmas 4.2 and 4.3 below the key properties satisfied by the functional

F ξ . Lemma 4.2. Let ξ ∈ R d , h ∈ L p (R d ) d ,
-h L p (R d ) ∇v L p (R d ) + cλ -1 R d |∇v| p + |ξ + ∇w per ξ | p-2 |∇v| 2 ≤ F ξ (v) ≤ h L p (R d ) ∇v L p (R d ) + Cλ R d |∇v| p + |ξ + ∇w per ξ | p-2 |∇v| 2 . (4.5)
Easy computations allow to deduce that

F ξ (v) ≤ 1 p h p L p (R d ) + (2Cλ + 1) v p W ξ+∇w per ξ .
This proves the right-most inequality of (4.4) after changing the constant C. For the left-most inequality, we write that, by Young inequality

- λ 2c h p L p (R d ) + cλ -1 2 R d |∇v| p + cλ -1 R d |ξ + ∇w per ξ | p-2 |∇v| 2 ≤ F ξ (v).
We deduce the lower bound

1 2 v 2 W ξ+∇w per ξ -1 ≤ R d |∇v| p + R d |ξ + ∇w per ξ | p-2 |∇v| 2 .
Thus,

- λ 2c h p L p (R d ) + cλ -1 2 + cλ -1 4 v 2 W ξ+∇w per ξ ≤ F ξ (v).
After changing the constant c, we get (4.4). This proves (i).

The point (ii) follows readily from the strict convexity of the application z → |z| p .

Lemma 4.3. Let ξ ∈ R d , h ∈ L p (R d ) d ,
F ξ (v) • u := R d a (ξ + ∇w per ξ + ∇v) ξ + ∇w per ξ + ∇v p-2 -(ξ + ∇w per ξ ) ξ + ∇w per ξ p-2 + h • ∇u. (4.6)
Proof of Lemma 4.3. We fix v ∈ W ξ+∇w per ξ and u ∈ W ξ+∇w per ξ . We have that

F ξ (v + u) -F ξ (v) = 1 p R d a g ξ+∇w per ξ (∇v + ∇u) -g ξ+∇w per ξ (∇v) + R d h • ∇u. (4.7)
We note that We note that, using the definition of g ξ (B.4),

g
A = g ξ+∇w per ξ +∇v (∇u) ≤ C ∇u p + ξ + ∇w per ξ + ∇v| p-2 ∇u 2 ,
where we have used the right-most part of inequality (B.5). Thus, applying the inequality (b This proves that B ∈ L 1 (R d ) and that

1 +b 2 ) p-2 ≤ C(p)(b p-2 1 + b p-2 2 ) for b 1 , b 2 ≥ 0,
R d |B| ≤ C v 2 W ξ+∇w per ξ + u 2 W ξ+∇w per ξ + v p W ξ+∇w per ξ + u p W ξ+∇w per ξ , (4.15)
where the constant C is independent of v and u. We can now conclude the proof of Lemma 4.3: using (4.7) and the notations (4.9) and (4.10), we have that

F ξ (v + u) -F ξ (v) - 1 p R d B + R d h • ∇u = 1 p R d A. (4.16)
Defining

L v (u) := 1 p R d B + R d h • ∇u = R d a (
F ξ (v + u) -F ξ (v) -L v (u) = O u→0 u 2 W ξ+∇w per ξ . Lemma 4.3 is proved.
Proof of Theorem 2.3. We prove below that, for h ∈ L p (R d ) d , the PDE , it is in particular weakly lower semi-continuous. Thus

-diva ξ + ∇w per ξ + ∇w ξ p-2 (ξ + ∇w per ξ + ∇ w ξ ) -ξ + ∇w per ξ p-2 (ξ + ∇w per ξ ) = div(h), ( 4
F ξ (v) ≤ lim inf n→+∞ F ξ (v n ) = inf W ξ+∇w per ξ F ξ .
This concludes the existence of a solution to (4.19). The uniqueness is given by the strict convexity of F ξ , see Lemma 4.2 (ii). We finally note that the convexity of F ξ together with its differentiability ensure that being a solution to Problem (4. [START_REF] Goudey | A periodic homogenization problem with defects rare at infinity[END_REF]) is equivalent to solve the PDE (4.17), since (4.6) is exactly the weak form of (4.17) in the sense of Definition 2.2. Theorem 2.3 is proved.

5 Properties of the non-periodic correctors: proof of Theorem 2.4

A useful Lemma

We begin by introducing the following function: for all ξ, η ∈ R d , the function G ξ,η is defined over

R d × R d by G ξ,η (X, Y ) := |ξ+X| p +|η+Y | p -ξ + X + Y 2 p -η + X + Y 2 p - p 2 ξ|ξ| p-2 -η|η| p-2 •(X -Y ). (5.1)
The following Lemma gives a lower bound for G ξ,η that will allow to prove Theorem 2.5 (iii).

, η ∈ R d , G ξ,η (X, Y ) ≥ γ p |X -Y | p -c p |ξ -η| p-2 |X -Y | + |ξ -η||X + Y | p-2 + (|ξ| + |η|) p-3 |ξ -η||X + Y | |X -Y |. (5.3)
Proof of Lemma 5.1. We first give the proof of Estimate (5.2). We have that ξ = 0 and η = 0. For all X, Y ∈ R d , we define Z := X-Y 2 and T := X+Y 2 . Inequality (5.2) is equivalent to the following inequality: for any Z, T ∈ R d ,

|ξ + T + Z| p + |η + T -Z| p -|ξ + T | p -|η + T | p -p(ξ|ξ| p-2 -η|η| p-2 ) • Z ≥ γ p |Z| p -c p |ξ -η| p-2 |Z| + δ p-3 |ξ -η||T | |Z|.
(5.4)

We prove (5.4) for any Z, T ∈ R d . We fix T ∈ R d and we introduce the function

Φ γp (Z) := |ξ + T + Z| p + |η + T -Z| p -γ p |Z| p ,
where γ p > 0 is to be chosen later. Since p ≥ 2, the function Φ γp is of class C 2 . Besides, denoting by I the identity matrix, we have that

Φ γ (Z) = p|ξ + T + Z| p-2 I + p(p -2)|ξ + Z + T | p-4 (ξ + Z + T ) ⊗ (ξ + Z + T ) + p|η + T -Z| p-2 I + p(p -2)|η + T -Z| p-4 (η + T -Z) ⊗ (η + T -Z) -γ p p|Z| p-2 I -γ p p(p -2)|Z| p-4 Z ⊗ Z.
Thus, for all

h ∈ R d , Φ γ (Z)(h, h) ≥ p|ξ + T + Z| p-2 |h| 2 + p|η + T -Z| p-2 |h| 2 -γ p p|Z| p-2 |h| 2 -γ p p(p -2)|Z| p-4 (Z • h) 2 ≥ p |ξ + T + Z| p-2 + |η + T -Z| p-2 -γ p (p -1)|Z| p-2 |h| 2 .
(5.5) We next note that

|Z| p-2 = 1 2 (Z + ξ + T ) + 1 2 (Z -η -T ) + 1 2 (η -ξ) p-2 ≤ C(p) |ξ + T + Z| p-2 + |η + T -Z| p-2 + |ξ -η| p-2 , (5.6)
where we have used the triangle inequality together with the fact that for all m ≥ 1 and p ≥ 2, there exists a constant C(p, m) such that ∀a 1 , ..., a m ≥ 0, (a

1 + • • • + a m ) p-2 ≤ C(p, m) a p-2 1 + • • • + a p-2 m .
Estimate (5.6) together with inequality (5.5) give that

∀h ∈ R d , Φ γp (Z)(h, h) ≥ -p|ξ -η| p-2 |h| 2 for γ p := 1 C(p)(p -1)
.

The function Φ γp + p 2 |ξ -η| p-2 | • | 2 is convex, hence ∀Z ∈ R d , Φ γp (Z) + p 2 |ξ -η| p-2 |Z| 2 ≥ Φ γp (0) + ∇Φ γp (0) • Z.
We have thus proved that

Φ γp (Z) ≥ |ξ + T | p + |η + T | p + p (ξ + T )|ξ + T | p-2 -(η + T )|η + T | p-2 • Z - p 2 |ξ -η| p-2 |Z| 2 .
This proves estimate (5.4) if T = 0. If T = 0, it remains to prove that

(ξ + T )|ξ + T | p-2 -(η + T )|η + T | p-2 -ξ|ξ| p-2 + η|η| p-2 ≤ c p δ p-3 |ξ -η||T |.
(5.7)

We want to apply the mean-value inequality to the function Ψ T defined by

Ψ T (x) := |x + T | p-2 (x + T ) -x|x| p-2 , x ∈ [ξ, η] ⊂ R d \ B(0, δ/2),
which is differentiable over R d . We have that

Ψ T (x) = |x + T | p-2 -|x| p-2 I + (x + T ) ⊗ (x + T )|x + T | p-4 -x ⊗ x|x| p-4 .
We now note that there exists a constant C p > 0 such that for all x ∈ R d \ B(0, δ/2),

|x + T | p-2 -|x| p-2 ≤ C p δ p-3 |T | + |x| p-3 |T | (5.8) and (x + T ) ⊗ (x + T )|x + T | p-4 -x ⊗ x|x| p-4 ≤ C p δ p-3 |T | + |x| p-3 |T | .
(5.9)

Noting that |x| p-3 ≤ ( 1 2 ) p-3 |δ| p-3 since p ≤ 3, we have proved (5.7). The proof of Lemma 5.1 is completed up to the justification of (5.8)-(5.9).

Proof of (5.8) and (5.9). We concentrate on the first inequality: assume first that

|T | ≥ 1 2 |x| ≥ 1 4 |δ|, then |x + T | p-2 -|x| p-2 ≤ C p |T | p-2 ≤ C p δ p-3 |T |.
(5.10)

We now treat the case |T | ≤ 1 2 |x|. In particular T |x| ≤ 1 2 and thus (5.11) since the function y → x |x| + y p-2 is regular on B(0, 3 4 ) with derivative uniformly bounded in x. Estimate (5.9) is proved the same way. We have concluded the proof.

|x + T | p-2 -|x| p-2 = |x| p-2 x |x| + T |x| p-2 - x |x| p-2 ≤ C p |x| p-2 T |x| = C p |T ||x| p-3 ,
Proof of (5.3). We assume that p ≥ 3. With the above variables T and Z, (5.3) is equivalent to proving that for all Z, T, ξ and η ∈ R d , the following inequality holds true:

|ξ + T + Z| p +|η + T -Z| p -|ξ + T | p -|η + T | p -p(ξ|ξ| p-2 -η|η| p-2 ) • Z ≥ γ p |Z| p -c p |ξ -η| p-2 |Z| + |ξ -η||T | p-2 + (|ξ| + |η) p-3 |ξ -η||T | |Z|.
(5.12)

Applying the same method as for the proof of (5.2), we only have to prove that

(ξ + T )|ξ + T | p-2 -(η + T )|η + T | p-2 -ξ|ξ| p-2 + η|η| p-2 ≤ c p |ξ -η||T | p-2 + (|ξ| + |η) p-3 |ξ -η||T | . (5.13)
We once again appeal to the mean-value inequality on Ψ T , noticing that, in this case, see (5.10) and (5.11), we have for all x ∈ R d ,

|Ψ T (x)| ≤ C p |T | p-2 + |x| p-3 |T | ≤ C p |T | p-2 + (|ξ| + |η|) p-3 |T | , x ∈ [ξ, η]. (5.14)
Note that, contrary to the case p < 3, estimate (5.14) does not depend on δ. This gives (5.13) and finally (5.12).

Proof of Theorem 2.5

We start this section with a Remark:

Remark 5.2. The proofs of Theorem 2.4 (i) and 2.4 (ii) below do not use Theorem 2.5. Consequently, we may use freely the results of Theorem 2.4 (i) and 2.4 (ii) in the following proof.

Proof of Theorem 2.5. By homogeneity, we can prove Theorem 2.5 for all ξ ∈ R d such that |ξ| = 1. We fix such a ξ ∈ R d . By (A4), there exists a constant c > 0 independent of ξ such that |ξ + ∇w per ξ | ≥ c. In the proof, we introduce the notations

C per ∞ := sup |ξ|=1 ξ + ∇w per ξ L ∞ (Q)
and

C ∞ := sup |ξ|=1 ∇ w ξ L ∞ (R d ) , (5.15)
where these quantities are well-defined owing to Proposition 2.1 (ii) and Theorem 2. (2,p-1) .

+ y p-2 -(ξ + ∇w per ξ ) ξ + ∇w per ξ p-2 -|ξ + ∇w per ξ | p-2 I + (p -2)|ξ + ∇w per ξ | p-4 (ξ + ∇w per ξ ) ⊗ (ξ + ∇w per ξ ) y ≤ C(p, c)|y| 2 1 {|y|≤c/2} + C(p) |ξ + ∇w per ξ | p-2 |y| + |y| p-1 1 {|y|≥c/2} ≤ C(p, c)|y| 2 1 {|y|≤c/2} + C(p, c) (C per ∞ ) p-2 |y| 2 + |y| max(2,p-1) 1 {|y|≥c/2} ≤ C(p, c, C per ∞ ) |y| 2 + |y| max
(5.16)

By (5.16) applied with y = ∇ w ξ , we can write

ξ + ∇w per ξ + ∇ w ξ ξ + ∇w per ξ + ∇ w ξ p-2 -(ξ + ∇w per ξ ) ξ + ∇w per ξ p-2 = |ξ + ∇w per ξ | p-2 I + (p -2)|ξ + ∇w per ξ | p-4 (ξ + ∇w per ξ ) ⊗ (ξ + ∇w per ξ ) ∇ w ξ + g ξ (∇ w ξ ),
(5.17)

where, using (5.15),

|g ξ (∇ w ξ )| ≤ C p, c, C per ∞ , C ∞ |∇ w ξ | 2 , (5.18)
Thus, collecting (5.17) and (4.17), we get that ∇ w ξ solves

-div a |ξ + ∇w per ξ | p-2 I + (p -2)|ξ + ∇w per ξ | p-4 (ξ + ∇w per ξ ) ⊗ (ξ + ∇w per ξ ) ∇ w ξ = div(h) + div(ag ξ (∇ w ξ )) (5.19)
in the distribution sense. Equation (5.19) is of the form

-div (A ξ ∇ w ξ ) = div(h) + div(ag ξ (∇ w ξ )), (5.20)
where

A ξ := a |ξ + ∇w per ξ | p-2 I + (p -2)|ξ + ∇w per ξ | p-4 (ξ + ∇w per ξ ) ⊗ (ξ + ∇w per ξ ) . (5.21)
We may write that A ξ = A per ξ + A ξ , where

A per ξ := a per |ξ + ∇w per ξ | p-2 I + (p -2)|ξ + ∇w per ξ | p-4 (ξ + ∇w per ξ ) ⊗ (ξ + ∇w per ξ )
and

A ξ := a |ξ + ∇w per ξ | p-2 I + (p -2)|ξ + ∇w per ξ | p-4 (ξ + ∇w per ξ ) ⊗ (ξ + ∇w per ξ ) .
The matrix A per ξ is symmetric, periodic, Hölder continuous, bounded and coercive while the matrix

A ξ ∈ L p ∩ L ∞ (R d ) d×d by Assumption (A3), in particular A ξ ∇ w ξ ∈ L p ∩ L ∞ (R d ) d
due to Proposition 2.1 (ii) and Theorem 2.4 (ii). We write equation (5.20) as

-div A per ξ ∇ w ξ = div(h + ag ξ (∇ w ξ ) + A ξ ∇ w ξ ). (5.22)
We have that h ∈ L p ∩ L ∞ (R d ) and, thanks to the estimate (5.18) and the fact that

∇ w ξ ∈ L p (R d ), that ag ξ (∇ w ξ ) ∈ L p/2 ∩ L ∞ (R d ). Thus h + ag ξ (∇ w ξ ) + A ξ ∇ w ξ ∈ L max(p ,p/2) ∩ L ∞ (R d ) d .
Applying [3, Theorem p. 247] and [2, Theorem A] to (5.22) gives ∇ w ξ ∈ L max(p ,p/2) (R d ) with the estimate

∇ w ξ L max(p ,p/2) (R d ) ≤ C(d, p, c, C per ∞ , α) h + ag ξ (∇ w ξ ) + A ξ ∇ w ξ L max(p ,p/2) (R d ) ≤ (5.18) C d, p, c, C per ∞ , C ∞ , λ a L max(p ,p/2) (R d ) (C per ∞ ) p-1 + |∇ w ξ | 2 L max(p ,p/2) (R d ) + a L max(p ,p/2) (R d ) (C per ∞ ) p-2 ∇ w ξ L ∞ (R d ) ≤ C a L p (R d ) , λ, d, p, α, c, C per ∞ , C ∞ , C p , (5.23) where
C p = sup |ξ|=1 ∇ w ξ L p (R d ) . If p ≥ p/2, Theorem 2.5 is proved. Otherwise, ∇ w ξ ∈ L p/2 (R d ) d
and we iterate the argument. We have, thanks to (5.18), that

h + ag ξ (∇ w ξ ) + A ξ ∇ w ξ ∈ L max(p ,p/4) ∩ L ∞ (R d ) d ,
thus by [START_REF] Avellaneda | Un théorème de liouville pour des équations elliptiques à coefficients périodiques[END_REF], we get that ∇ w ξ ∈ L max(p ,p/4) (R d ) d and we can prove, similarly to (5.23) that

∇ w ξ L max(p ,p/4) (R d ) ≤ C a L p (R d ) , λ, d, p, α, c, C per ∞ , C ∞ , C p , (5.24)
where the constant on the right-hand side of (5.24) is potentially greater than the one on the righthand side of (5.23) but the dependance on the data remains the same. If p ≥ p/4, the Theorem is proved. Otherwise, we iterate similarly. The procedure ends at step k for which p/2 k ≤ p : we thus obtain that ∇ w ξ ∈ L p (R d ) d and that there exists a constant

C f inal := C a, λ, d, p, α, c, C per ∞ , C ∞ , C p such that ∇ w ξ L p (R d ) ≤ C f inal .
Theorem 2.5 is proved.

Proof of Theorem 2.4

Proof of (i). This is due to Proposition 2.1 (i), to the form of the PDE (2.12)-(2.13) defining ∇ w ξ and the fact that this PDE is uniquely solvable in the sense of Definition 2.2. Note that we use that for t = 0, W ξ+∇w per ξ = W tξ+∇w per tξ .

Proof of (ii). This result is proved in [24, Lemma 2.2] but we reproduce the proof here for the sake of completeness. Let ξ ∈ R d . By Definition 2.2 with ∇φ = ∇ w ξ , the inequality (B.1), Hölder inequality together with (4.18), we have c

R d |∇ w ξ | p ≤ f L p (R d ) ∇ w ξ L p (R d ) ≤ a L p (R d ) ξ + ∇w per ξ p-1 L ∞ (Q) ∇ w ξ L p (R d ) . (5.25)
Thus, by Proposition 2.1 (ii) and (5.25), we obtain the first estimate of (2.16).

We show that there exists α > 0 independent of ξ such that ∇w ξ ∈ C 0,α (R d ). We introduce the function w ξ := ξ • x + w ξ , then ∇w ξ solves the standard homogeneous p-Laplace equation with varying coefficient a. Applying [21, Theorem 1], we get that ∇w ξ is continuous over R d . Besides, by [START_REF] Kuusi | A nonlinear stein theorem[END_REF]Theorem 4], there exists a constant c ≥ 1 and a radius r > 0 depending only on d, p, λ and the Lipschitz constant of a, denoted a Lip such that for all x ∈ R d ,

|∇w ξ (x)| ≤ c - B(x,r) |∇w ξ | p 1/p ≤ c - B(x,r) |∇w ξ | p 1/p . (5.26)
Due to the form of ∇w ξ , see also (2.14) and the first estimate of (2.16), we have that

|∇w ξ (x)| ≤ c|ξ| + cr -d/p ∇ w ξ L p (R d ) ≤ C(d, p, λ, a Lip )|ξ|. (5.27)
In particular, (5.27) proves that ∇w ξ is bounded and that ∇w ξ L ∞ (R d ) ≤ C(d, p, λ, a Lip )|ξ|. By Assumption (A2), the non-linear operator a(y, z) = a(y)z|z| p-2 falls into the scope of [START_REF] Dibenedetto | C (1+ alpha) local regularity of weak solutions of degenerate elliptic equations[END_REF]. Let x ∈ R d , up to subtracting of w ξ (x), we have by (5.27) that |w ξ | ≤ C(d, p, λ, a Lip )|ξ| on B(x, 2). Thus, applying [15, Theorem 2], there exist α > 0 and C 0 > 0 depending only on λ, a Lip , p, d, p, and C(d, p, λ, a Lip )|ξ| such that ∇w ξ ∈ C 0,α (B(x, 1)) and

[∇w ξ ] C 0,α (B(x,1)) ≤ C 0 .
(5.28)

To specify the dependence of C 0 in ξ, we first take |ξ| = 1 and we then apply the homogeneity, Theorem 2.4 (i). This gives that C 0 = C 0 (p, d, λ, a Lip )|ξ| and concludes the proof of (ii), gathering (5.27) and (5.28) and the fact that

∇ w ξ C 0,α (R d ) ≤ |ξ| + ∇w ξ C 0,α (R d ) .
Proof of (iii). We assume that 2 ≤ p < 3. Let us fix ξ ∈ R d such that |ξ| = 1. In the proof, c > 0 will denote a universal constant given by (A4). We consider η ∈ R d such that ξ = η. In the sequel, we fix δ 0 ∈ (0, 1) such that C(1 + 2 1-γ )δ γ 0 + δ 0 ≤ c/2, where C and γ are given by (2.8). Case 1. We assume that |ξ -η| ≥ δ 0 . Then, thanks to Theorem 2.4 (ii), we have that

∇ w ξ -∇ w η L p (R d ) ≤ C p + C p |η|.
(5.29)

We now note that for all 0 < β ≤ 1,

C p + C p |η| ≤      C p δ β 0 |ξ -η| β (1 + |η|) ≤ 2 β C p δ β 0 |ξ -η| β (1 + |η| 1-β ) if |η| ≤ 2. C p C( β) |η| -1 β (1 + |η| 1-β ) ≤ C p C( β)|ξ -η| β (1 + |η| 1-β ) if |η| > 2, (5.30)
where we used that the function x

→ 1+x |x-1| β (1+x 1-β ) is bounded on [2, +∞[. Thus ∇ w ξ -∇ w η L p (R d ) ≤ C(δ 0 , β, C p )|ξ -η| β (1 + |η| 1-β ).
(5.31)

This gives (2.17).

Case 2. We assume that |ξ -η| < δ 0 . Then, by the choice of δ 0 and Proposition 2.1 (iv), we have that

ξ + ∇w per ξ -η + ∇w per η L ∞ (Q) ≤ c 2 and |ξ + ∇w per ξ | ≥ c. (5.32)
Recalling the notation (4.1), we have that

F ξ (∇ w ξ ) + F η (∇ w η ) < F ξ ∇ w ξ + ∇ w η 2 + F η ∇ w ξ + ∇ w η 2 < +∞, (5.33)
where we have used that ξ = η, F z admits a unique minimizer for z ∈ R d and ∇

w ξ ∈ L 2 (R d ), ∇ w η ∈ L 2 (R d ). We recall that F R z (∇v) := B R ag z+∇w per z (∇v) + R d f z • ∇v, z ∈ R d , ∇v ∈ L p (R d ) (5.34)
and that R -→ F R z (∇v) is a non-decreasing function. Thus, for R large enough, we have the inequality

F R ξ (∇ w ξ ) + F R η (∇ w η ) -F R ξ ∇ w ξ + ∇ w η 2 -F R η ∇ w ξ + ∇ w η 2 ≤ 0. (5.35)
We now use Lemma 5.1 applied with δ = c. Taking into account (5.32), this gives

G ξ+∇w per ξ ,η+∇w per η (∇ w ξ ,∇ w η ) ≥ γ p |∇ w ξ -∇ w η | p -c p |ξ + ∇w per ξ -(η + ∇w per η )| p-2 ||∇ w ξ -∇ w η | + c p-3 |ξ + ∇w per ξ -(η + ∇w per η )||∇ w ξ + ∇ w η | |∇ w ξ -∇ w η |.
(5.36) For all R > 0, we can integrate (5.36) over the ball B R . Using the notation (5.34) and the form of the map G ξ,η (X, Y ), see (5.1), this yields

F R ξ (∇ w ξ ) + F R η (∇ w η ) -F R ξ ∇ w ξ + ∇ w η 2 -F R η ∇ w ξ + ∇ w η 2 - 1 2 R d (h ξ -h η ) • (∇ w ξ -∇ w η ) ≥ γ p B R a|∇ w ξ -∇ w η | p -c p B R a |ξ + ∇w per ξ -(η + ∇w per η )| p-2 ||∇ w ξ -∇ w η | + c p-3 |ξ + ∇w per ξ -(η + ∇w per η )||∇ w ξ + ∇ w η | |∇ w ξ -∇ w η |, (5.37)
where h z = a(z + ∇w per z )|z + ∇w per z | p-2 for z ∈ R d . For R large enough, we get because of (5.35) that

- 1 2 R d (h ξ -h η ) • (∇ w ξ -∇ w η ) ≥ γ p B R a|∇ w ξ -∇ w η | p -c p B R a |ξ + ∇w per ξ -(η + ∇w per η )| p-2 • |∇ w ξ -∇ w η | + c p-3 |ξ + ∇w per ξ -(η + ∇w per η )||∇ w ξ + ∇ w η | |∇ w ξ -∇ w η |,
(5.38) Letting R -→ +∞ in (5.38) and using Theorem 2.5, we get by the monotone convergence Theorem that

- 1 2 R d (f ξ -f η )•(∇ w ξ -∇ w η ) ≥ γ p R d |∇ w ξ -∇ w η | p -c p R d |ξ + ∇w per ξ -(η + ∇w per η )| p-2 • |∇ w ξ -∇ w η | + c p-3 |ξ + ∇w per ξ -(η + ∇w per η )||∇ w ξ + ∇ w η | |∇ w ξ -∇ w η |.
Thus, applying the Hölder inequality, Proposition 2.1 (iv) and Theorem 2.5 under the form 2) . This gives (2.17) when |ξ| = 1. The case |ξ| = 1 is treated by homogeneity.

∇ w z L p (R d ) ≤ C|z|, z ∈ R d , we get R d |∇ w ξ -∇ w η | p ≤ C a L p |ξ -η| γ ∇ w ξ -∇ w η L p (R d ) + c p C|ξ -η| γ(p-2) + c p-3 C|ξ -η| γ ∇ w ξ -∇ w η L p . Thus ∇ w ξ -∇ w η p-1 L p (R d ) ≤ C|ξ -η| γ(p-
Gathering Case 1 and Case 2, we have proved Theorem 2.4 (iii) for p ∈ [START_REF] Avellaneda | Lp bounds on singular integrals in homogenization[END_REF][START_REF] Avellaneda | Un théorème de liouville pour des équations elliptiques à coefficients périodiques[END_REF]. The proof of the case p ≥ 3 is performed using the same method and (5.3).

Remark 5.3. As suggested by (5.3), the assumptions of Theorem 2.4(iii) may be weakened when p ≥ 3. In this case, it is sufficient to assume, instead of (A4), that ∇ w ξ ∈ L p (R d ) , that ∇w η ∈ L p (R d ) and that we have an estimate of the form (2.19).

Proof of (iv). It is analogous to the proof of Proposition 2.1 (iv).

6 Qualitative Homogenization: proof of Theorem 2.7

The proof of Theorem 2.7 is an adaptation of [START_REF] Fusco | On the homogenization of quasilinear divergence structure operators[END_REF] and [START_REF] Dal | Correctors for the homogenization of monotone operators[END_REF]Theorem 2.1] to the present setting. We start with the following central Lemma: Lemma 6.1. For ξ ∈ R d , let us write ∇ w ξ the solution to (2.12)-(2.13) given by Theorem 2.3. Assume that the application

R d -→ L p unif (R d ) ξ -→ ∇ w ξ (6.1) is continuous. Then for all Ψ ∈ L p (Ω) d , lim sup ε→0 Ω ∇ w MεΨ • ε p = 0. (6.2)
Proof of Lemma 6.1. We first show the following assertion: We now turn to the proof of (6.2). By an immediate application of the Jensen inequality, we have that

∀δ > 0, ∃A > 0, ∀|x| ≥ A, ∀ξ ∈ R d , ∇ w ξ L p (
∀B ∈ N ∪ {+∞}, |k|<B, ε(Q+k)⊂Ω ε d Ψ k ε p ≤ Ω∩B∞(0,εB) |Ψ| p , Ψ k ε := - ε(Q+k) Ψ, (6.4)
where B ∞ (x, r) denotes the ball centered in x and of radius r > 0 for the | • | ∞ -norm on R d . Let δ > 0 and A be given by (6.3). We have that Since (6.6) is true for all δ > 0, we haved proved (6.2).

Ω ∇ w MεΨ • ε p = k∈Z d , ε(Q+k)⊂Ω ε d Q+k ∇ w Ψ k ε p ≤ |k|<A, ε(Q+k)⊂Ω ε d Q+k ∇ w Ψ k ε p + |k|≥A, ε(Q+k)⊂Ω ε d Q+k ∇ w Ψ k ε p ≤ (6.3),(2.16) C |k|<A, ε(Q+k)⊂Ω ε d |Ψ k ε | p + δ p |k|≥A, ε(Q+k)⊂Ω ε d |Ψ k ε | p ≤ (6.
We now state the analogous of [START_REF] Dal | Correctors for the homogenization of monotone operators[END_REF]Lemma 3.5] to the present non-periodic setting. Before that, we introduce for ξ, y ∈ R d the notations

lim sup ε→0 p(•/ε, M ε Ψ) -p(./ε, Φ) L p (Ω) ≤ C Ψ 1-β L p (Ω) + Φ 1-β L p (Ω) Ψ -Φ β L p (Ω) , (6.8)
where β is given by Proposition 2.1 (iii).

Proof of Lemma 6.2. We first notice that

Ω ∇ w Φ • ε p = m j=1 Ωj ∇ w ηj • ε p ≤ ε d m j=1 R d ∇ w ηj p -→ ε→0 0. (6.9)
With the notations (6.7), we have, applying [START_REF] Dal | Correctors for the homogenization of monotone operators[END_REF]Lemma 3.5], that lim sup

ε→0 p(•/ε, M ε Ψ) -p(./ε, Φ) L p (Ω) ≤ lim sup ε→0 p per (•/ε, M ε Ψ) -p per (./ε, Φ) L p (Ω)
≤RHS of (6.8)

+ lim sup ε→0 ∇ w MεΨ (./ε) L p (Ω) + ∇ w Φ (./ε) L p (Ω)
=0 by Lemma 6.1 and (6.9)

.

With these tools, we can prove Theorem 2.7. The first point (i) is not detailed here since it is mainly a rewriting of [START_REF] Fusco | On the homogenization of quasilinear divergence structure operators[END_REF]. Note that for this point, the continuity of ξ -→ ∇ w ξ is not neeeded. The only result on the non-periodic correctors ∇w ξ , ξ ∈ R d that is used is Theorem 2.4 (ii). The proof of Theorem 2.7 (ii) follows the proof of [START_REF] Dal | Correctors for the homogenization of monotone operators[END_REF]Theorem 2.1]. In the following, we sketch the proof of Theorem 2.7 (ii) by insisting on the points that differ from [START_REF] Dal | Correctors for the homogenization of monotone operators[END_REF]. The proof of Theorem 2.7 (iii) follows from Theorem 2.7 (ii) together with Lemma 6.1.

Sketch of proof of Theorem 2.7 (ii). Since M ε ∇u * converges to ∇u * when ε → 0 in L p (Ω), it is sufficient to show, using the notation (6.7) that

R ε := ∇u ε -p(./ε, M ε ∇u * ) -→ ε→0 0 in L p (Ω).
(6.10)

During the proof, we introduce a step function Φ as in Lemma 6.2 satisfying ∇u * -Φ L p (Ω) ≤ δ. By monotonicity of the p-Laplace operator, see (B.1), and Assumption (A1), we have that

λ -1 c R ε p L p (Ω) ≤ Ω a(./ε)|∇u ε | p-2 ∇u ε -a(./ε)|p(./ε, M ε ∇u * )| p-2 p(./ε, M ε ∇u * ), ∇u ε -p(./ε, M ε ∇u * ) = A ε -B ε -C ε + D ε , (6.11
) where

A ε := Ω a(./ε)|∇u ε | p , B ε := Ω a(./ε)|∇u ε | p-2 ∇u ε • p(./ε, M ε ∇u * ) C ε := Ω a(./ε)|p(./ε, M ε ∇u * )| p-2 p(./ε, M ε ∇u * ) • ∇u ε and D ε := Ω a(./ε)|p(./ε, M ε ∇u * )| p .
The term A ε is obviously treated by the L p -weak convergence u ε -ε→0 u * :

A ε = Ω f u ε -→ ε→0 Ω f u * = Ω a * (∇u * ) • ∇u * .
(6.12)

We study the term B ε when M ε ∇u * is replaced by Φ. This gives:

Ω a(./ε)|∇u ε | p-2 ∇u ε • p(./ε, Φ) = m j=1 Ωj a(./ε)|∇u ε | p-2 ∇u ε • p(./ε, η j ).
We then apply the standard div-curl Lemma, keeping in mind that a(./ε)|∇u ε | p-2 ∇u ε converges L p (Ω)-weakly to a * (∇u * ) when ε → 0, that p(./ε, η j ) converges L p -weakly to η j and that, thanks to Theorem 2.4 (ii), a(./ε)|∇u ε | p-2 ∇u ε • p(./ε, η j ) is bounded in L p (Ω), uniformly with respect to ε. Thus,

m j=1 Ωj a(./ε)|∇u ε | p-2 ∇u ε • p(./ε, η j ) -→ ε→0 m j=1 Ωj a * (∇u * ) • η j = Ω a * (∇u * ) • Φ.
In view of Lemma 6.

D ε -D per ε ≤ Ω a • ε p • ε , M ε ∇u * p + C Ω a per • ε p p • ε , M ε ∇u * p-1 + p per • ε , M ε ∇u * p-1 ∇ w Mε∇u * • ε . (6.16)
We show that each term of the RHS of (6.16) vanishes as ε -→ 0. We use Theorem 2.4 (ii) and (6.4) with B = +∞, which imply that there exists a constant C > 0 independent of ε such that

Ω p • ε , M ε ∇u * p + p per • ε , M ε ∇u * p ≤ C ∇u * p L p (Ω) . (6.17)
With the Hölder inequality and Lemma 6.1, we prove that the second term of the RHS of (6.16) tends to zero as ε -→ 0. As for the first term, we write that

Ω a • ε p • ε , M ε ∇u * p ≤ C Ω a • ε p • ε , Φ p + C Ω a • ε p • ε , M ε ∇u * -p • ε , Φ p ≤ (2.16),(6.17) C a • ε L 1 (Ω) =O(ε d/p ) Φ p L ∞ (R d ) + C a L ∞ (R d ) p • ε , M ε ∇u * -p • ε , Φ p L p (R d ) , (6.18)
where we used that a ∈ L ∞ (R d ) and the bound |p(y, ξ)| ≤ C|ξ| where C > 0 is independent of y and ξ. Collecting (6.15), (6.16), (6.18) and Lemma 6.2, we have proved that lim sup

ε→0 D ε - Ω a * (∇u * ) • ∇u * = O(δ pβ ). (6.19)
Finally, collecting (6.12), (6.13), (6.14), (6. [START_REF] Goudey | A periodic homogenization problem with defects rare at infinity[END_REF]) and (6.11), we obtain that

lim sup ε→0 R ε p L p (Ω) ≤ Ω |a * (∇u * ) • (∇u * -Φ)| + Ω |(a * (∇u * ) -a * (Φ)) • ∇u * | + O(δ β ). (6.20)
Using the following property of a * , see [13, Remark 1.3],

a * (∇u * ) -a * (Φ) L p (Ω) ≤ C ∇u * p-2 L p (Ω) + Φ p-2 L p (Ω) ∇u * -Φ L p (Ω) , we conclude that lim sup ε→0 R ε p L p (Ω) = O(δ β
) where the O is independent of δ. Since this is true for all δ > 0, we conclude that R ε L p (Ω) -→ ε→0 0. Remark 6.3. Using the same strategy as above, it is straightforward to show that Theorem 2.7 holds with the operator M ε replaced by M ε ν , 0 < ν < 1. In this case, the continuity of the application ξ → ∇ w ξ is not needed and we only use that ∇ w ξ ∈ L p (R d ).

7 Continuity of ξ → ∇w ξ : proof of Theorem 2.8

Preliminary Lemmas

We begin this section with the following lemma that defines weak solution of PDEs of the form (7.1):

)|ξ + ∇w per ξ | p-2 • ∇w = - R d h • ∇w. (7.2)
Proof of Lemma 7.1. We define u := ξ + ∇w per ξ in the proof. We fix w ∈ W u . In the following, χ will denote a smooth and compactly supported function with support in Q(0, 1) such that χ = 1 in Q(0, 1 2 ). We fix R > 0 and we introduce the function

Φ R := w -- Q R \Q R/2 w χ • R .
By the Poincaré-Wirtinger inequality, we have that Φ R ∈ W 1,p 0 (Q R ). By (B.3), we have the bound

a (u + ∇v)|u + ∇v| p-2 -u|u| p-2 ≤ λC |u + ∇v| p-2 + |u| p-2 |∇v| ≤ λC(p) |u| p-2 |∇v| + |∇v| p-1 , (7.3)
where we have used the inequality (b

1 + b 2 ) p-2 ≤ C(p)(b p-2 1 + b p-2 2 ) for b 1 , b 2 ≥ 0. Thus, since ∇v ∈ L p (Q R ) and p = p (p -1), div a (u + ∇v)|u + ∇v| p-2 -u|u| p-2 ∈ W -1,p (Q R ) and div(h) ∈ W -1,p (Q R ).
Consequently, we can test (7.1) against Φ R and obtain, after expanding ∇Φ R ,

Q R a (u + ∇v)|u + ∇v| p-2 -u|u| p-2 • χ • R ∇w + 1 R Q R a (u + ∇v)|u + ∇v| p-2 -u|u| p-2 • w -- Q R \Q R/2 w ∇χ • R = - Q R h • χ • R ∇w - 1 R Q R h • w -- Q R \Q R/2 w ∇χ • R . (7.4)
We now recall the following Poincaré-Wirtinger inequality:

w -- Q R \Q R/2 w L p (Q R \Q R/2 ) ≤ CR ∇w L p (Q R \Q R/2 ) (7.5)
which is simply a rescaled version of the L p inequality on Q\Q 1/2 . Besides, thanks to Assumption (A4)' (and its rescaled version), we have that

|u| p-2 2 w -- Q R \Q R/2 w L 2 (Q R \Q R/2 ) ≤ CR |u| p-2 2 ∇w L 2 (Q R \Q R/2 ) . 29
This yields, together with (7.3), Hölder inequality and the inclusion supp(∇χ

) ⊂ Q R \ Q R/2 , that Q R \Q R/2 a (u + ∇v)|u + ∇v| p-2 -u|u| p-2 • w -- Q R \Q R/2 w ∇χ • R ≤ λCR ∇χ L ∞ ∇v p-1 L p (Q R \Q R/2 ) ∇w L p (Q R \Q R/2 ) + |u| p-2 2 ∇v L 2 (Q R \Q R/2 ) |u| p-2 2 ∇w L 2 (Q R \Q R/2 ) (7.6) and Q R h • w -- Q R \Q R/2 w ∇χ • R ≤ CR ∇χ L ∞ h L p (Q R \Q R/2 ) ∇w L p (Q R \Q R/2 ) . (7.7)
Collecting (7.4), (7.6) and (7.7) and recalling that v, w ∈ W u , we have that

R d χ • R a (u + ∇v)|u + ∇v| p-2 -u|u| p-2 + h • ∇w -→ R→+∞ 0. (7.8)
On the other hand, by the dominated convergence Theorem, again since v, w ∈ W u , we have that

R d χ • R a (u + ∇v)|u + ∇v| p-2 -u|u| p-2 + h • ∇w -→ R→+∞ R d a (u + ∇v)|u + ∇v| p-2 -u|u| p-2 + h • ∇w. (7.9)
Thus (7.2) is satisfied.

The next lemma allows to pass to the limit in PDEs of the form (7.1).

Lemma 7.2. Let (∇φ n ) n∈N ⊂ L ∞ (R d ) d , (a n ) n∈N ⊂ L ∞ (R d ), (h n ) n∈N ⊂ L p (R d ) d and (v n ) n∈N ⊂ V (see (4. 2
)), such that v n ∈ W ∇φn for all n ∈ N. We assume that, for all n ∈ N:

1. The coefficient a n satisfies Assumption (A1) with λ uniformly bounded in n. By (i), we obtain that ∇v n -→ n→+∞ ∇v in L p (B) up to a subsequence. We easily show that the convergence in fact holds for the whole sequence. We consequently get the L p loc (R d )-convergence since B is arbitrary. We now pass to the limit n → +∞ in (7.10). Let Ψ ∈ D(R d ), we test (7.10) against Ψ. By (iv), it is clear that Step 1. We have that ∇ w ξn - The required convergences follow from (7.21) and Proposition 2.1 (ii) and (iv). We get that ∇ w ξn converges when n -→ +∞ to ∇v in L p loc (R d ) and that ∇v solves (2.12) in the distribution sense. Thus, ∇v solves (2.12) in the weak sense in W ξ+∇w per (iii) This is (7.28).

The function ∇v

R d h n • ∇Ψ -→ n→+∞ R d h • ∇Ψ. ( 7
(iv) By the same argument as in (ii), we have that h n -→ n→+∞ 0 in L p loc (R d ).

We have proved that, up to exacting a subsequence, ∇ w

+ ∇ V 1 )|ξ + ∇w per ξ + ∇ V 1 | p-2 -(ξ + ∇w per ξ )|ξ + ∇w per ξ | p-2 • ∇ V 1 = 0.
Applying (B.1) allows to conclude that ∇ V 1 = 0 in R d and thus ∇ v 1 = 0. This proves (7.26) and concludes the proof of Theorem 2.8.

Remark 7.4. From the above theorem, we can deduce that

R d -→ L ∞ (R d ) ξ -→ ∇ w ξ is continuous.
The continuity of ξ → ∇w per ξ is due to Proposition 2.1 (iv). We prove that ξ → ∇ w ξ is continuous for the L ∞ (R d ) topology. By contradiction, suppose that there exists ξ ∈ R d , two sequences (ξ n ) n∈N ⊂ R d and (x n ) n∈N ⊂ R d and a δ > 0 such that ξ n -→ n→+∞ ξ and ∀n ∈ N, ∇ w ξn (x n ) -∇ w ξ (x n ) ≥ δ.

By Theorem 2.4 (ii), there exists η independent of n such that ∀n ∈ N, ∀y ∈ B(x n , η), ∇ w ξn (y) -∇ w ξ (y) ≥ δ 2 .

Thus, for all n ∈ N, ∇ w ξn -∇ w ξ L p unif (R d ) ≥ |B(0, 1)| 1/p δη d/p > 0. which is a contradiction with Theorem 2.8.

We extend Φ by zero outside Q R . By Hölder inequality, we have that (∇w R -∇Φ)

L 2 (Q R ) ≤ w R -Φ W 1,p (Q R ) + C|ξ| p-2 2 w R -Φ H 1 (Q R ) ≤ C(d, p, a per )|ξ| p-2 2 + 1 ε.
As for the first term of (B.11), we write that

∇v -∇w R = ∇v(1 -χ R ) + 1 R v -- Q R \Q R/2
v ∇χ(./R) Thus, applying the L p Poincaré-Wirtinger inequality, we have that

∇v -∇w R L p (R d ) ≤ ∇v L p (Q c R/2 ) + C poinc ∇v L p (Q R \Q R/2 ) ≤ (B.

Lemma 4 . 1 .

 41 Let ξ ∈ R d and W ξ+∇w per ξ be defined by (2.10).

Figure 1 :

 1 Figure 1: Numerical simulation in the particular 1D case.

F,c - 1 + v 2 W. ( 4 . 4 )

 1244 ξ be defined by (4.1) over V and the space W ξ+∇w per ξ be defined by (2.10).(i) There exist two constants c, C > 0 such that for all v ∈ W ξ+∇w perξ ξ+∇w per ξ ≤ F ξ (v) ≤ C 1 + v pW ξ+∇w per ξ In particular, F ξ (v) is finite if and only if v ∈ W ξ+∇w per ξ . (ii) The function F ξ is convex over V and strictly convex over W ξ+∇w per ξ . Proof of Lemma 4.2. The point (i) is a simple application of (B.5). Let v ∈ W ξ+∇w per ξ , we have thanks to (B.5) together with Hölder inequality that

Lemma 5 . 1 .. 2 )

 512 Suppose that 2 ≤ p < 3. For all δ > 0, there exist constants γ p = γ(p) > 0 and c p = c(p) > 0 such that for all X, Y ∈ R d , for all ξ ∈ R d \ B(0, δ) and η ∈ B(ξ, δ/2), we have thatG ξ,η (X, Y ) ≥ γ p |X -Y | p -c p |ξ -η| p-2 |X -Y | + δ p-3 |ξ -η||X + Y | |X -Y |.(5Suppose that p ≥ 3. There exist constants γ p = γ(p) > 0 and c p = c(p) > 0 such that for all X, Y ∈ R d and all ξ

+ y ξ + ∇w per ξ + y p- 2

 2 4 (ii). We use the following Taylor inequality (5.16) for the function y -→ ξ + ∇w per ξ which is of class C 2 over B(0, 3c/4). For all y ∈ R d , we have, using also (B.3) when |y| ≥ c/2, ξ + ∇w per ξ + y ξ + ∇w per ξ

p. 7 ) 6 . 2 .j=1 η j 1

 7621 per (y, ξ) := ξ + ∇w per ξ (y) and p(y, ξ) := ξ + ∇w ξ (y) = p per (y, ξ) + ∇ w ξ (y). (6Lemma Assume that the Assumptions of Lemma 6.1 are satisfied. Let Ψ ∈ L p (Ω) and Φ ∈ L p (Ω) such that Φ = m Ωj where m j=1 Ω j ⊂⊂ Ω, Ω k ∩ Ω = ∅ for k = and |∂Ω j | = 0 for j ∈ {1, m}. Then there exists a constant C > 0 independent of ε, Ψ and Φ such that

2 2 ∇v n - n→+∞ |∇φ| p- 2 2

 22 n is solution, in the distribution sense, to-div a n (∇φ n + ∇v n )|∇φ n + ∇v n | p-2 -∇φ n |∇φ n | p-2 = div(h n ). (7.10)We also assume the following convergences:(i) ∇φ n -→ n→+∞ ∇φ in L ∞ (R d ); (ii) a n -→ n→+∞ a in L ∞ loc (R d ); (iii) ∇v n -n→+∞ ∇v in L p (R d ) and |∇φ n | p-∇v in L 2 (R d ) which v ∈ W ∇φ ; (iv) h n -→ n→+∞ h in L p loc (R d ) with h ∈ L p (R d ).Then∇v n -→ n→+∞ ∇v in L p loc (R d )and ∇v is solution in the distribution sense to -div a (∇φ + ∇v)|∇φ + ∇v| p-2 -∇φ|∇φ| p-2 = div(h). (7.11) Using (7.17), (B.1), that χ ≥ 0 and χ = 1 in B, we get B ∇φ n + ∇v n -∇φ -∇v p

n→+∞∇ 2 2 2 2 ∇ w ξn - n→+∞ |ξ + ∇w per ξ | p- 2 2

 222 w ξ in L p (R d ). Indeed, by (B.1), (B.2) and the form of h, see (2.13), we have the following a priori estimate: there exists a constant C = C(d, p, a, a per ) > 0 such that for all n ∈ N, w ξn W ξn +∇w per ξn ≤ C |ξ n | + |ξ n | p/2 . (7.20) In particular, there exists v ∈ V (see (4.2) for the definition of V ) and w ∈ L 2 (R d ) such that ∇ w ξn -n→+∞ ∇v in L p (R d ) and |ξ n + ∇w per ξn | p-∇ w ξn -n→+∞ w in L 2 (R d ). (7.21) Taking into account Remark 7.3, we have that v ∈ W ξ+∇w per ξ and |ξ n + ∇w per ξn | p-∇v in L 2 (R d ). n = a, v n = w ξn φ n = ξ n • x + w per ξn , φ = ξ • x + w per ξ h n = a|ξ n + ∇w per ξn | p-2 (ξ n + ∇w per ξn ), h = a|ξ + ∇w per ξ | p-2 (ξ + ∇w per ξ ).

ξ

  (see Definition 2.2) by Lemma 7.1, with h(ii) We have a n = a per (• + x n ) + a(• + x n ). Since x n -→ n→+∞ x in T d , we have by Assumption (A2) that a per (• + x n ) -→ n→+∞ a per (• + x) in L ∞ (Q). Let B be a bounded domain, then since a ∈ C 0,α ∩ L p (R d ), we have a -→ |x|→+∞ 0. Thus a(• + x n ) -→ n→+∞ 0 in L ∞ (B)and finally a n converges locally uniformly to a := a per (• + x) when n → +∞.

∇ v 1 2 -

 12 in L p loc (R d ) where v 1 ∈ W ξ+∇w per ξ (•+x) solves in the distribution sense the PDE -div a per (• + x) ξ + ∇w per ξ (• + x) + ∇ v 1 ξ + ∇w per ξ (• + x) + ∇ v 1 p-ξ + ∇w per ξ (• + x) ξ + ∇w per ξ (• + x) p-2 = 0. Introducing V 1 := v 1 (• -x), we get that V 1 ∈ W ξ+∇w per ξ and that ∇ V 1 solves in the distribution sense the PDE -diva per (ξ + ∇w per ξ + ∇ V 1 )|ξ + ∇w per ξ + ∇ V 1 | p-2 -(ξ + ∇w per ξ )|ξ + ∇w per ξ | p-2 = 0. (7.29) Applying Lemma 7.1 to (7.29) with w := V 1 gives now R d a per (• + x) (ξ + ∇w per ξ

w

  R -Φ H 1 0 (Q R ) ≤ ε. (B.9) We next show that v -Φ W ξ +∇w per ξ ≤ C(ξ, d, p, a per , C poinc )ε, (B.10)where C poinc denotes the maxmimum between the L p Poincaré-Wirtinger constant on Q \ Q 1/2 and the weighted L 2 Poincaré-Wirtinger constant, given by Assumption (A4)', on Q \ Q 1/2 . By the triangle inequality, we have thatv -Φ W ξ +∇w per ξ ≤ v -w R W ξ +∇w per ξ + w R -Φ W ξ +∇w per ξ . (B.11)We study separately each term of (B.11). By Proposition 2.1 (iv), (B.8) and (B.9), we have thatw R -Φ W ξ +∇w per ξ = ∇w R -∇Φ L p (Q R ) + |ξ + ∇w per ξ | p-22

7 )( 1 + 2 2L 2

 7122 C poinc )ε. (B.12) As for the L 2 |ξ + ∇w per ξ | pdλ norm, we use Assumption (A4)' to obtain that ∇v -∇w R L 2 |ξ+∇w per ξ | (Q R \Q R/2 ) ≤ (1 + C poinc ) ε. (B.13) Gathering together (B.11), (B.12) and (B.13), we get (B.10) and conclude the proof of the Lemma.

Table 1 :

 1 Numerical errors for different values of ε in L ∞ -norm.

	1	0.156		0.109
	0.05	0.163		0.137
	0.01	0.170		0.0657
	0.005	0.170		0.0288
	0.001	0.170		0.0245
	0.0005	0.171		0.0136
	ε	R per ε	L 2	R ε L 2
	0.1	6.39		3.85
	0.05	5.01		3.16
	0.01	2.13		0.740
	0.005	1.47		0.331
	0.001	0.654		0.108
	0.0005	0.46		0.0461

Table 2 :

 2 Numerical errors for different values of ε in L 2 -norm.

  F ξ be defined by (4.1) over V and the space W ξ+∇w per ξ be defined by (2.10). Then the application F ξ is Fréchet-differentiable over W ξ+∇w per

		ξ	. Its differential is
	given, for v ∈ W ξ+∇w per ξ	, by

  ξ+∇w per

ξ (∇v + ∇u) -g ξ+∇w per ξ (∇v) = ξ + ∇w per ξ + ∇v + ∇u p -ξ + ∇w per ξ + ∇v p -p(ξ + ∇w per ξ + ∇v)|ξ + ∇w per ξ + ∇v| p-2 • ∇u + p (ξ + ∇w per ξ + ∇v)|ξ + ∇w per ξ + ∇v| p-2 -(ξ + ∇w per ξ )|ξ + ∇w per ξ | p-2 • ∇u = A + B, (4.8) where A := ξ + ∇w per ξ + ∇v + ∇u p -ξ + ∇w per ξ + ∇v p -p(ξ + ∇w per ξ + ∇v)|ξ + ∇w per ξ + ∇v| p-2 • ∇u (4.9) and B := p (ξ + ∇w per ξ + ∇v)|ξ + ∇w per ξ + ∇v| p-2 -(ξ + ∇w per ξ )|ξ + ∇w per ξ | p-2 • ∇u. (4.10)

  we get that|A| ≤ C |∇u| p + |ξ + ∇w per ξ | p-2 |∇u| 2 + |∇v| p-2 |∇u| 2 . C |ξ + ∇w per ξ | p-2 |∇v| 2 + |ξ + ∇w per ξ | p-2 |∇u| 2 + |∇v| p + |∇u| p .

	we obtain					1-2/p		2/p
	|∇v| p-2 |∇u| 2 ≤		|∇v| p		|∇u| p	.	(4.12)
	R d			R d			R d
	Gathering (4.11), (4.12) and recalling the definition (2.11), we have proved that A ∈ L 1 (R d ) and that
	R d	|A| ≤ C u 2 W ξ+∇w per ξ	+ u p W ξ+∇w per ξ	,	(4.13)
	where the constant C does not depend on u. We now turn to estimating B, see (4.10). Using (B.3),
	Cauchy-Schwarz inequality and Young inequality, we have that
	|B| ≤ C |ξ + ∇w per ξ ≤ (4.14) + ∇v| p-2 + |ξ + ∇w per ξ | p-2 |∇v||∇u|
								(4.11)
	We now note that, due to Hölder inequality and the fact that
	p 2	p 2	-1 =	p 2	=⇒	p 2	p -2 = p,

  (iii), we get that the sequence (v n ) n∈N weakly converges, up to a subsequence, to some v in W ξ+∇w per ξ when n -→ +∞. Since by Lemma 4.2 (ii) and Lemma 4.3, F ξ is convex and continuous over W ξ+∇w per ξ

	This Problem admits a unique solution. The existence is guaranteed by the following procedure: let
	(v n ) n∈N ⊂ W ξ+∇w per ξ	be a minimizing sequence. Then, by the left-hand estimate of (4.4), we have
	that the sequence	v n W ξ+∇w per ξ	n∈N	ξ is bounded (see (2.11) for the definition of • W ξ+∇w per	). By
	Lemma 4.1
						.17)
	admits a unique solution w ξ ∈ W ξ+∇w per ξ	in the weak sense (see Definition 2.2). Theorem 2.3 is then
	proved by defining
			h := a(ξ + ∇w per ξ )|ξ + ∇w per ξ | p-2 .	(4.18)
	Because of Proposition 2.1 (ii) and Assumptions (A2)-(A3), it is clear that h ∈ L p (R d ) d . Since (4.17)
	is solvable for this choice of h, Theorem 2.3 is proved.
	We are thus left to study the PDE (4.17) for an abstract right-hand side h ∈ L p (R d ) d . With
	Lemma 4.2, Lemma 4.3 and Lemma 4.1, we prove in a standard way that Problem (4.17) admits a
	unique solution. Indeed, let us consider the minimization Problem:
					min v∈W ξ+∇w per ξ	F ξ (v).	(4.19)

  x+Q) ≤ δ|ξ|. (6.3) By contradiction, if (6.3) does not hold, there exists δ > 0 and two sequences (x n ) n∈N ⊂ R d and (ξ n ) n∈N ⊂ R d such that |x n | -→ n→+∞ +∞ and ∇ w ξn L p (xn+Q) ≥ δ|ξ n |. By Theorem 2.4 (i), we can assume that |ξ n | = 1. Thus, up to a subsequence, ξ n -→ large enough, we have that ∇ w ξn -∇ w ξ L p (xn+Q) ≤ δ/2. Thus, for n large enough, we have that ∇ w ξ L p (xn+Q) ≥ δ/2. Since |x n | -→

		n→+∞	ξ. However, by (6.1), for all n
	n→+∞	+∞, this contradicts that ∇ w ξ ∈ L p (R d ). Thus (6.3) is
	satisfied.

  ) • Φ = O(δ β ),(6.13) where the O is independent of δ. The term C ε is also treated by replacing M ε ∇u * by Φ and using the div-curl Lemma. Noticing that a(./ε)p(./ε, η j )|p(./ε, η j )| p-2 -Besides, since |x| p -|y| p ≤ C |x| p-1 + |y| p-1 |x -y| for all x, y ∈ R d , we get

		2, we obtain that
		lim sup	B ε -
		ε→0
					(6.14)
	We introduce D per ε	:= Ω a per (./ε) |p per (./ε, M ε ∇u * )| p . By [13, Step 1, pp.1161-1162], we have that
		D per ε	-→ ε→0 Ω	a * (∇u * ) • ∇u * .	(6.15)

Ω a * (∇u * ε→0 a * (η j ) in L p (Ω), we obtain that lim sup ε→0 C ε -Ω a * (Φ) • ∇u * = O(δ β ).

  Lemma 7.1. Let ξ ∈ R d , a coefficient a satisfying Assumption (A1) and h ∈ L p (R d ) d . Assume that Assumption (A4)' is satisfied. Let v ∈ W ξ+∇w perThen ∇v solves (7.1) in the weak sense of Definition 2.2: for all w ∈ W ξ+∇w per

				ξ	be solution in the distribution sense to the following
	PDE:
	-div a (ξ + ∇w per ξ	+ ∇v)|ξ + ∇w per ξ	+ ∇v| p-2 -(ξ + ∇w per ξ )|ξ + ∇w per ξ | p-2 = div(h).	(7.1)
					ξ	,
	R d	a (ξ + ∇w per ξ + ∇v)|ξ + ∇w per ξ + ∇v| p-2 -(ξ + ∇w per ξ

  .[START_REF] Goudey | A periodic homogenization problem with defects rare at infinity[END_REF] Besides, by (B.3), (i), (ii) and the L p loc (R d )-convergence of ∇v n , we have thata n (∇φ n + ∇v n )|∇φ n + ∇v n | p-2 -∇φ n |∇φ n | p-2 -→ n→+∞ a (∇φ + ∇v)|∇φ + ∇v| p-2 -∇φ|∇φ| p-2 in L p loc (R d ).

	This shows that

R d a (∇φ + ∇v)|∇φ + ∇v| p-2 -∇φ|∇φ| p-2 • ∇Ψ = -R d h • ∇Ψ,

and concludes the proof of the Lemma 7.2.

7.2 Proof of Theorem 2.8

Let (ξ n ) n∈N ⊂ R d such that ξ n -→ n→+∞ ξ for ξ ∈ R d . We aim at showing that ∇w ξn -→ n→+∞ ∇w ξ in L p unif (R d )

. By Proposition 2.1 (iii), it is sufficient to show that ∇ w ξn -→ n→+∞ ∇ w ξ in L p unif (R d ).

Acknowledgments

The author warmly thanks his PhD advisor Xavier Blanc for fruitful discussions and for reading many versions of this manuscript.

Proof of Lemma 7.2. We fix two bounded smooth domains B, B such that B ⊂⊂ B . Let χ ∈ D(B ) such that χ = 1 on B and 0 ≤ χ ≤ 1 in B . We introduce the function

We immediately check that, up to extracting a subsequence, we have by the Rellich compactness Theorem and (i), (iii) that

Since Ψ n ∈ W 1,p 0 (B ), we can test Ψ n against (7.10). We re-organize the terms and get that

which is bounded in L p (B ), uniformly with respect to n by (i) and (iii) and (7.12). The term B n vanishes as n -→ +∞ by (7.13) and since, by (i), (ii) and (iii):

The term C n vanishes by (7.13) and the L p loc -strong convergence of the sequence (h n ) n∈N . The term D n vanishes by (7.13) and the convergence of a n |∇φ n | p-2 ∇φ n to a|∇φ| p-2 ∇φ in L ∞ (B ). We have proved that

However, since (∇φ + ∇v)|∇φ + ∇v| p-2 ∈ L p (B ) and because of (7.12) and (i), we also have that .16) The difference between (7.15) and (7.16) gives that

given by (2.13). In addition, by Theorem 2.3, the solution of this PDE is unique in W ξ+∇w per ξ . Thus ∇v = ∇ w ξ and this concludes Step 1 since the sequence (∇ w ξn ) n∈N has one possible limit.

Step 2. Suppose by contradiction that ∇ w ξn does not converge to ∇ w ξ in L p unif (R d ) when n -→ +∞. Then there exists δ > 0, a subsequence of (ξ n ) n∈N that we de not relabel and a sequence (

)

In particular, (7.22) gives

We show in the sequel that, up to a subsequence, for i = 1, 2,

In particular, passing to the limit n -→ +∞ in (7.25) will lead to a contradiction.

Step 3. Proof of (7.26). We prove (7.26) for i = 1, the proof is standard for i = 2. We have that

We may apply Lemma 7.2 with

We check the required convergences.

A Proof of Proposition 2.1

Proof of (i). This point is obvious by the form of (2.9) and the uniqueness of w per ξ in W 1,p per (Q)/R.

Proof of (ii). The first estimate follows for example from (2.4) and in particular:

together with (A1). The proof of the second estimate is exactly the same as the one of Theorem 2.4 (ii) (see Subsection 5.3) with a replaced by a per and w ξ replaced by w per ξ .

Proof of (iii). Let ξ 1 , ξ 2 ∈ R d . Applying (2.3) with φ = w per ξ1 -w per ξ2 with ξ = ξ i , i = 1, 2 and making the difference between the two expressions gives:

Thus, adding the term

in the left and right-hand side of (A.1), applying (B.1) on the left-hand side and (B.3) on the right-hand side provides

We apply the Hölder inequality on the right-hand side with exponents p/(p -2), p and p find, using (2.16):

This yields (2.16) by taking the 1/(p -1)-th power of the above inequality.

Proof of (iv). We argue by contradiction. Suppose that there exist three sequences (

B Some technical inequalities

We gather in this subsection some useful inequalities. We first have

In the above inequalities (B.1)-(B.3), c and C refer to universal constants that only depend on p. For a proof of these inequalities, we refer to [START_REF] Iwaniec | Projections onto gradient fields and l p -estimates for degenerated elliptic operators[END_REF]. For ξ, x ∈ R d , we introduce the function

We have the following lemma:

Lemma B.1. There exist two constants c, C > 0 depending only on p such that

Proof. This proof is elementary and will be omitted here.

Lemma B.2.

There exists a constant C > 0 such that for all x, h ∈ R,

Proof. This proof is elementary and will be omitted here.