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Research article

Tick based clustering methodologies establishing support and resistance levels in the currency exchange market

  • Received: 20 August 2020 Accepted: 19 October 2020 Published: 23 October 2020
  • JEL Codes: C45, C63, C81, C83

  • We establish support and resistance levels from data in intraday currency exchange market activity based on machine learning methods. Specifically we design two semi-supervised classification neural networks. The first one is based on a variant of the K-means method while the second is based on a Gaussian mixture model with expectation maximisation. Each performs classification from tick data on very short time windows and produces the corresponding support and resistance price levels. We test the methodology on actual market data for the EUR-USD currency exchange. As a sanity check we also perform mock trades based on actual market data. We evaluate the results for statistical significance using a number of performance metrics while also comparing against traditional methods.

    Citation: Karl Tengelin, Alexandros Sopasakis. Tick based clustering methodologies establishing support and resistance levels in the currency exchange market[J]. National Accounting Review, 2020, 2(4): 354-366. doi: 10.3934/NAR.2020021

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  • We establish support and resistance levels from data in intraday currency exchange market activity based on machine learning methods. Specifically we design two semi-supervised classification neural networks. The first one is based on a variant of the K-means method while the second is based on a Gaussian mixture model with expectation maximisation. Each performs classification from tick data on very short time windows and produces the corresponding support and resistance price levels. We test the methodology on actual market data for the EUR-USD currency exchange. As a sanity check we also perform mock trades based on actual market data. We evaluate the results for statistical significance using a number of performance metrics while also comparing against traditional methods.


    The purpose of this paper is to address the homogenization problem for a second order elliptic equation in divergence form with a certain class of oscillating coefficients:

    {div(a(x/ε)uε)=finΩ,uε(x)=0inΩ, (1)

    where Ω is a bounded domain of Rd (d1) sufficiently regular (the regularity will be made precise later on) and f is a function in L2(Ω). The class of (matrix-valued) coefficients a considered is that of the form

    aper+˜a, (2)

    which describes a periodic geometry encoded in the coefficient aper and perturbed by a coefficient ˜a that represents a non-local perturbation (a "defect") that, although it does not vanish at infinity, becomes rare at infinity. More specifically, we consider coefficients ˜a that locally behave like L2(Rd) functions in the neighborhood of a set of points localized at an exponentially increasing distance from the origin. Formally, the coefficient ˜a is an infinite sum of localized perturbations, increasingly distant from one another. A prototypical one-dimensional example of such a defect reads as kZϕ(xsign(k)2|k|) for some fixed ϕD(R), where |k| denotes the absolute value of k and sign(k) denotes its sign. It is depicted in Figure 1.

    Figure 1. 

    Prototype perturbation in dimension d=1

    .

    Homogenization theory for the unperturbed periodic problem (1)-(2) when ˜a=0 is well-known (see for instance [5,19]). The solution uε converges strongly in L2(Ω) and weakly in H1(Ω) to u, solution to the homogenized problem:

    {div(au)=finΩ,u(x)=0inΩ, (3)

    where a is a constant matrix. The convergence in the H1(Ω) norm is obtained upon introducing a corrector wper,p defined for all p in Rd as the periodic solution (unique up to the addition of a constant) to:

    div(aper(wper,p+p))=0inRd. (4)

    This corrector allows to both make explicit the homogenized coefficient

    (a)i,j=QeTiaper(y)(ej+wper,ej(y))dy, (5)

    (where Q denotes the d-dimensional unit cube, (ei) the canonical basis of Rd) and define the approximation

    uε,1=u(.)+εdi=1iu(.)wper,ei(./ε), (6)

    such that uε,1uε strongly converges to 0 in H1(Ω) (see [1] for more details). In addition, convergence rates can be made precise, with in particular:

    uεuε,1L2(Ω)CεfL2(Ω),uεuε,1L2(Ω1)CεfL2(Ω)foreveryΩ1⊂⊂Ω,

    for some constants independent of f.

    Our purpose here is to extend the above results to the setting of the perturbed problem (1)-(2). The main difficulty is that the corrector equation

    div((aper+˜a)(wp+p))=0,

    (formally obtained by a two-scale expansion (see again [1] for the details) and analogous to (4) in the periodic case) is defined on the whole space Rd and cannot be reduced to an equation posed on a bounded domain, as is the case in periodic context in particular. This prevents us from using classical techniques. The present work follows up on some previous works [6,8,9,10] where the authors have developed an homogenization theory in the case where ˜aLp(Rd) for p]1,[. The existence and uniqueness (again up to an additive constant) of a corrector, the gradient of which shares the same structure "periodic + Lp" as the coefficient a, is established. Convergence rates are also made precise. Similarly to [6,8,9,10], we aim to show here, in a context of a perturbation rare at infinity, there also exists a corrector (unique up to the addition of a constant), and such that its gradient has the structure (2) of the diffusion coefficient: it can be decomposed as a sum of the gradient of a periodic corrector and a gradient that becomes rare at infinity (in a sense similar to that for ˜a, and made precise below).

    We introduce here a suitable functional setting to describe the class of defects we consider.

    In order to formalize our mathematical setting, we first define a generic infinite discrete set of points denoted by G={xp}pZd. In the sequel, each point xp actually models the presence of a defect in the periodic background modeled by aper and our aim is to ensure these defects are sufficiently rare at infinity.

    We next introduce the Voronoi diagram associated with our set of points. For xpG, we denote by Vxp the Voronoi cell containing the point xp and defined by

    Vxp=xqG{xp}{xRd||xxp||xxq|}. (7)

    We now consider three geometric assumptions that ensure an appropriate distribution of the points in the space. The set G is required to satisfy the following three conditions :

    xpG,|Vxp|<, (H1)
    C1>0, C2>0, xpG,C11+|xp|D(xp,G{xp})C2, (H2)
    C3>0,xpG,Diam(Vxp)D(xp,G{xp})C3, (H3)

    where |A| denotes the volume of a subset ARd, Diam(A) the diameter of A and D(.,.) the euclidean distance.

    Assumption (H2) is the most significant assumption in our case since it implies that the points are increasingly distant from one another far from the origin. It in particular implies

    limxpG, |xp|D(xp,G{xp})=+.

    More precisely, it ensures the distance between a point xp and the others has the same growth as the norm |xp| and, therefore, requires the Voronoi cell Vxp (which contains a ball of radius D(xp,G{xp})2 as a consequence of its definition) to be sufficiently large. This assumption actually ensures that the defects modeled by the points xp are sufficiently rare at infinity. In particular, we show in Section 2 that Assumption (H2) implies that the number of points xp contained in a ball BR of radius R>0 is bounded by the logarithm of R. This property is an essential element for the methods used in the proof of this article.

    In contrast to (H2), Assumptions (H1) and (H3) are only technical and not very restrictive. They limit the size of the Voronoi cells. In the case where these assumptions are not satisfied, our main results of Theorems 1.1 and 1.2 stated below still hold. Their proofs have to be adapted, upon splitting the Voronoi cells in several subsets such that each subset satisfies geometric constraints similar to (H1), (H2) and (H3). To some extent, our assumptions (H1) and (H3) ensure we consider the worst case scenario, where the set G contains as many points as possible while satisfying (H2).

    In addition, although we establish in Section 2 all the geometric properties satisfied by the Voronoi cells Vxp which are required in our approach to study the homogenization problem (1) with the whole generality of Assumptions (H1), (H2) and (H3), we choose, for the sake of illustration and for pedagogic purposes, to work with a particular set of points (for which the coordinates are powers of 2) and to establish our main results of homogenization in this specific setting. There are, of course, many alternative sets that satisfy (H1), (H2) and (H3) but our specific choice is convenient. To define our specific set of points, we first introduce a constant C0>1 and a set of indices PC0 defined by:

    PC0={pZd | maxpi0{|pi|}C0+minpi0{|pi|}}. (8)

    Our specific set of points (see Figure 2) is then defined by:

    GC0={xp=(sign(pi)2|pi|)i{1,...d} | (p1,...,pd)PC0}. (9)
    Figure 2. 

    Example of points in ambient dimension 2 that satisfy our assumptions along with their associated Voronoi diagram

    .

    We use here the convention sign(0)=0. The set of indices (8) contains only the points with integer coordinates on the axes Span(ei) and the points close to each diagonal of the form Span(ei1+...+eik) for k{2,..,d} and (i1,...,ik){1,...,d}k. In this way, the points of GC0 are exponentially distant from each other with respect to the norm of p. In Section 2, we show that the set GC0 defined by (9) indeed satisfies Assumptions (H1), (H2) and (H3).

    In the sequel, we use the following notation:

    BR: the ball of radius R>0 centered at the origin; BR(x): the ball of radius R>0 and center xRd; AR,R: the set BRBR for R>R>0.

    QR(x): the set {yRd | maxi|yixi|R} for R>0 and xRd; QR: the set QR(0).

    B: the cardinality of a discrete set B.

    2p: the point xpGC0 for pPC0; τp: the translation τ2p where we denote τxf=f(.+x) for xRd; Vp: the Voronoi cell V2p.

    |p|: the norm defined by maxi{1,...,d}|pi| for pPC0.

    In addition, for a normed vector space (X,.X) and a matrix-valued function fXn, nN, we use the notation fXfXn when the context is clear.

    We associate to (8)-(9) the following functional space:

    B2(Rd)={fL2unif(Rd) | fL2(Rd),lim|p|fτpfL2(Vp)=0}, (10)

    equipped with the norm

    fB2(Rd)=fL2(Rd)+fL2unif(Rd)+suppPC0fτpfL2(Vp). (11)

    In (10), (11) we have denoted by:

    L2unif(Rd)={fL2loc(Rd), supxRdfL2(B1(x))<},

    and

    fL2unif(Rd)=supxRdfL2(B1(x)).

    Intuitively, a function in B2(Rd) behaves, locally at the "vicinity" of each point xp, as a fixed L2 function truncated over the domain Vp. We show several properties of the functional space B2(Rd) in Section 3.

    As specified above, in the sequel we focus on homogenization problem (1) with non-local perturbations induced by the particular setting (8)-(9)-(10). We note, however, that the definition of B2(Rd) can be naturally adapted to the generality of Assumptions (H1)-(H2)-(H3) and the homogenization results established in the present study can of course be extended to this general setting. More precisely, most of our proofs only involve the general structure of the functional space B2(Rd) and several geometric properties related to the rarity of the points xp that are established under our general assumptions in Section 2. The specific geometric properties of the set (9) are only explicitly used to study the equation div(aperu)=div(f) when fB2(Rd), particularly to establish the convergence of several sums involving the asymptotic behavior of the Green function of the Laplacian operator (see Lemmas 4.3, 4.4 and 5.2). However, these results are not specific to the set (9). We explain how to adapt their proofs under our general assumptions in Remarks 4 and 8.

    We henceforth assume that the ambient dimension d is equal to or larger than 3. The one-dimensional and two-dimensional contexts are specific. Some results or proofs must be adapted in these particular cases but we will not proceed in that direction in all details. This is due to the asymptotic behavior of the Green function of the Laplacian operator in these two dimensions. In these two particular cases, we claim that it is still possible to show the existence of the corrector defined by Theorem 1.1 below. However, the method used in Lemmas 4.3 and 4.4, both useful for the proof of Theorem 1.1, need to be adapted. The one-dimensional context can be addressed easily because the solution to (14) is explicit. The two-dimensional case requires more work. We explain how to adapt our proof in Remark 5. In contrast, in dimensions d=1 and d=2, the convergence rates of Theorem 1.2 no longer hold. Indeed, the corrector wp is then not necessarily bounded (see Lemma 5.2 for details). We are only able to prove weaker results in these cases. Additional details about these cases may be found in Remarks 5, 7, and 9.

    For α]0,1[, we denote by C0,α(Rd) the space of uniformly Hölder continuous and bounded functions with exponent α, that is:

    C0,α(Rd)={fL1loc(Rd) | fC0,α(Rd)<},

    where

    fC0,α(Rd)=fL(Rd)+supx,yRd, xy|f(x)f(y)||xy|α.

    We consider a matrix-valued coefficient of the form (2) with aperL2per(Rd)d×d and ˜aB2(Rd)d×d. We denote by ˜a the matrix-valued limit L2-function associated with ˜a, where each coefficient (˜a)i,j is the limit L2-function associated with (˜a)i,jB2(Rd) and defined in (10). We assume that aper, ˜a and ˜a satisfy:

    λ>0, xξRdλ|ξ|2a(x)ξ,ξ,λ|ξ|2aper(x)ξ,ξ, (12)

    and

    aper, ˜a, ˜aC0,α(Rd)d×d, α]0,1[. (13)

    The coercivity (12) and the L bound on a ensure that the sequence of solutions (uε)ε>0 to (1) converges in weakH1(Ω) and strongL2(Ω) up to an extraction when ε0. Classical results of homogenization show the limit u is a solution to a diffusion equation of the form (3) for some matrix-valued coefficient a to be determined. The questions that we examine in this paper are: What is the diffusion coefficient a of the homogenized equation? Is it possible to define an approximate sequence of solutions uε,1 as in (6)? For which topologies does this approximation correctly describe the behavior of uε? What is the convergence rate?

    In answer to our first question, we prove in Proposition 13 that the homogenized coefficient a is constant and is the same as in the periodic case. This result is a direct consequence of Proposition 10 which ensures that the perturbations of B2(Rd) have a zero average in a strong sense. Consequently, our perturbations are "small" at the macroscopic scale and do not affect the homogenization that occurs in the periodic case associated with the periodic coefficient aper. In reply to the other questions, our main results are contained in the following two theorems:

    Theorem 1.1. For every pRd, there exists a unique (up to an additive constant)function wpH1loc(Rd) such that wp(L2per(Rd)+B2(Rd))dC0,α(Rd)d, solution to:

    {div((aper+˜a)(p+wp))=0in Rd,lim|x||wp(x)|1+|x|=0. (14)

    Theorem 1.2. Assume Ω is a C2,1-bounded domain. Let Ω1⊂⊂Ω. We define uε,1=u+εdi=1iuwei(./ε) where wei is defined by Theorem 1.1 for p=ei and u is the solution to (3). Then Rε=uεuε,1 satisfies the following estimates:

    RεL2(Ω)C1εfL2(Ω), (15)
    RεL2(Ω1)C2εfL2(Ω), (16)

    where C1 and C2 are two positive constants independent of f and ε.

    Our article is organized as follows. In Section 2 we prove some geometric properties satisfied by our set of points GC0, in particular we show that it satisfies Assumptions (H1), (H2) and (H3). In section 3 we study the properties of B2(Rd) and its elements. In Section 4 we prove Theorem 1.1. Finally, in Section 5 we obtain the expected homogenization convergences stated in Theorem 1.2. We conclude this introduction section with some comments.

    A first possible extension of the above results, which is studied in [16,Appendix A], consists in considering the functional spaces Br for r2, 1<r<, defined similarly to B2, but using the Lr topology. In this case the convergence rates of Theorem 1.2 are modified and depend upon the value of r and the ambient dimension d. Indeed, some results related to the strict sub-linearity of the corrector allow to show that the convergence rate of Rε is εdr|log(ε)|1r if r>d and ε else.

    In addition, although we have not pursued in these directions, we believe it is possible to extend the above results in several other manners.

    1) First, under additional assumptions satisfied by the function f, we expect the estimates of Theorem 1.2 to hold, with possibly different rates, in other norms than L2 such as Lq, for 1<q< or C0,α, for α]0,1[. It seems that such questions could be addressed by adapting the proofs of Section 5 and consider the methods employed in [6] using the behavior of the Green function associated with problem (1).

    2) We also believe that it is possible to show results analogous to that of Theorems 1.1 and 1.2 in the case of equations not in divergence form, instead of (1),

    aijiju=f,

    where a is a periodic coefficients perturbed by a defect in B2(Rd) of the form (2). One way to address this question could be to adapt the methods of [8,Section 3] in the case of local perturbations, that is, to show the existence of an invariant measure m=mper+˜m in L2per+B2(Rd) solution to:

    i,j(ai,jmi,j)=0in Rd,

    such that infm>0. Indeed, using the method presented in [3], this study could be then reduced to a problem of divergence form operator as soon as such a measure m exists and the results established in this article could allow to conclude.

    3) In the same way, another possible generalization concerns advection-diffusion equation in the form:

    aijiju+bjju=fin Rd,

    where a and b are two periodic coefficients perturbed by a defect in B2(Rd). The method [7] is likely to be adapted to this case, showing the existence of an invariant measure m in L2per+B2(Rd) solution to

    i(j(ai,jmi,j)+bimi,j)=0in Rd.

    We start by studying the geometric properties of the Voronoi cells associated to every sets of points G satisfying the general Assumptions (H1), (H2) and (H3). In particular, we show these assumptions ensure the rarity of the points xp in the space proving, in Proposition 3 and Corollary 1, that the number of points of G contained in a ball of radius R>0 is bounded by the logarithm of R. In Propositions 2 and 4, we also show two technical properties regarding the size and the structure of the cells. All these properties are actually fundamental for the rest of our work since they allow us to prove several results regarding the existence and uniqueness of solutions to the class (31) of diffusion equations div(au)=div(f) studied in Section 4. In particular, as we shall see in the proof of Lemma 4.3, we use these geometric properties to bound several integrals in order to define a solution to equation (35), that is (31) with a=aper, using the associated Green function. To conclude this section, we also show that our specific set of points GC0, defined by (9), satisfies (H1), (H2) and (H3).

    In this subsection only, we proceed with the whole generality of Assumptions (H1), (H2) and (H3) and we introduce several useful geometric properties satisfied by every sets of points G satisfying these assumptions. These properties relate to the size of the Voronoi cells, their volume and their distribution in the space Rd.

    To start with, we show two properties regarding the volume of the Voronoi cells.

    Proposition 1. There exist C1>0 and C2>0 such that for every xG, we have the following bounds:

    C1|x|d|Vx|C2|x|d.

    Proof. For every xG, using the definition of the Voronoï diagram, we have the following inclusion:

    BD(x,G{x})/2(x)Vx.

    Therefore, there exists a constant C(d)>0 such that:

    C(d)D(x,G{x})d=|BD(x,G{x})/2(x)||Vx|Diam(Vx)d.

    We conclude using (H2) and (H3).

    Proposition 2. There exists a sequence (xn)nNGN such that (Vxnxn) is an increasing sequence of sets and:

    nN(Vxnxn)=Rd.

    Proof. We consider a sequence (xn)nNGN such that the sequence |xn| is increasing and limn|xn|= (such a choice is always possible according to Assumptions (H1) and (H2)). Since we have assumed that G satisfies (H2), there exists C>0 such that for all nN:

    D(xn,G{xn})C|xn|.

    Therefore, as a consequence of the definition of the Voronoi cells, the ball BC|xn|/2(xn) is included in Vxn and, by translation, the ball BC|xn|/2 is included in Vxnxn. Since (xn)nN is an increasing sequence such that limn|xn|=, we use (H1) and we obtain, up to an extraction, that Vxn is included in BC|xn+1|/2(xn). Thus

    nN,VxnxnBC|xn+1|/2Vxn+1xn+1.

    The sequence (Vxnxn) is therefore an increasing sequence of sets and, in addition,

    Rd=nNBC|xn|/2nN(Vxnxn).

    We directly deduce that Rd=nN(Vxnxn).

    The next results ensure a certain distribution of the Voronoi cells in the space. In particular, we prove that the number of cells contained in a ball of radius R>0 increases at most as the logarithm of this radius. This property reflects the rarity of our points far from the origin and is essential in our approach.

    Proposition 3. There exists a constant C(d)>0 that depends only of the ambient dimension d such that:

    {xG|xA2n,2n+1}C(d).

    Proof. Let xG such that xA2n,2n+1. The definition of the Voronoi cells ensures that the distance D(x,Vx) is equal to D(x,G{x})2. Property (H2) gives the existence of a constant C1>0 independent of x such that:

    D(x,G{x})2C1|x|2C12n1.

    Then, the ball BC12n1(x) is contained in Vx, that is x is the only element of G in this ball. In addition, since |x|2n+1, we obtain the following inclusion using a triangle inequality :

    BC12n1(x)B(C1+4)2n1.

    Since this inclusion is valid for every xGA2n,2n+1 we obtain:

    xGA2n,2n+1BC12n1(x)B(C1+4)2n1.

    Therefore, there exists C2(d)>0 such that:

    |xGA2n,2n+1BC12n1(x)||B(C1+4)2n1|C2(d)2d(n1). (17)

    Next, we know that the Voronoi cells are disjoint and, therefore, the collection of balls (BC12n1(x))xGA2n,2n+1 is also disjoint. Thus, there exists C3(d)>0 such that:

    |xGA2n,2n+1BC12n1(x)|={xG|xA2n,2n+1}|BC12n1|
    ={xG|xA2n,2n+1}C3(d)2d(n1). (18)

    With (17) and (18), we conclude that:

    {xG|xA2n,2n+1}C2(d)C3(d).

    Corollary 1. There exists C>0 such that for every R>0 and x0Rd:

    {xG|VxBR(x0)}<Clog(R). (19)

    Proof. We start by proving the result if R=2n for nN. Without loss of generality, we can assume that n is sufficiently large to ensure there exists x in GB2n(x0). Using a triangle inequality, we remark that if yB2n(x0) we have:

    |xy||xx0|+|yx0|2n+1D(y,RdB2n+3(x0)).

    That is, if ˜xG is such that ˜xB2n+3(x0), every point yB2n(x0) is closer to x than to ˜x, that is V˜xB2n(x0)=. Therefore, we have

    {xG|VxB2n(x0)}{xG|xB2n+3(x0)}.

    Next, if |x0|2n+4, we have B2n+3(x0)B22n+7 and we use Proposition 3 to obtain the existence of a constant C>0 independent of n such that:

    {xG|xB2n+3(x0)}{xG|xB22n+7}=2n+6k=0{xG|xA2k,2k+1}+{xG|xB1}Cn.

    If |x0|>2n+4, we denote by mn+4 the unique integer such that 2m<|x0| and |x0|2m+1. In this case, we use a triangle inequality and we have

    B2n+3(x0)A|x0|+2n+3,|x0|2n+3A2m+2,2m1.

    Proposition 3 gives the existence of C>0 independent of x0 and n such that:

    {xG|xB2n+3(x0)}{xG|xA2m+2,2m1}=1k=1{xG|xA2m+k,2m+k+1}C.

    Finally, we have estimate (19) in the particular case R=2n.

    Next, for any R>0, we have:

    R=2log2(R)2[log2(R)]+1,

    where [.] denotes the integer part. Thus, we obtain the following upper bound:

    {xG|VxBR(x0)}{xG|VxB2[log2(R)]+1(x0)}C([log2(R)]+1),

    and we can conclude.

    To conclude this section, we now introduce a particular set (denoted by Wx in the proposition below) containing a point xG which is both bigger than the cell Vx and far from all the others points of G. As we shall see in Lemmas 4.3 and 4.4, this set is actually a technical tool that allows us to show the existence of the corrector stated in Theorem 1.1.

    Proposition 4. For every xG, there exists a convex open set Wx of Rd and C1, C2, C3, C4 and C5 five positive constants independent of x such that:

    VxWx, (i)
    Diam(Wx)C1|x| and D(Vx,Wx)C2|x|, (ii)
    yG{x}, D(y,Wx)C3|x|, (iii)
    {yG|VyWx}C4, (iv)
    yG{x}, D(VyWx,Vx)C5|y|. (v)

    Proof. Let x be in G. In the sequel, we denote Ix,y={zRd | |zx||zy|} and φx the homothety of center x and ratio 32. For yG{x}, we denote by Hx,y the set defined by:

    Hx,y=φx(Ix,y).

    The set Hx,y can be easily determined, it is the half-space defined by:

    Hx,y={zRd | |zx||zy|}+14xy=Ix,y+14xy.

    We finally consider:

    Wx=yG{x}Hx,y,

    which is actually the image of the cell Vx by the homothety φx (see figure 3).

    Figure 3. 

    Example for the choice of the open subset Wx when d=2

    .

    We next prove that Wx satisfies (ⅰ), (ⅱ), (ⅲ), (ⅳ) and (ⅴ).

    (i): For every yG{x} we have Ix,yHx,y and therefore, we obtain using definition (7) of Vx:

    Vx=yG{x}Ix,yyG{x}Hx,y=Wx,

    and we have the first inclusion.

    (ii): Wx is a 32-dilation of Vx, thus we have Diam(Wx)=32Diam(Vx). We use (H2) and (H3) to obtain the first estimate. Next, the definitions of the sets Hx,y and Wx give:

    D(Vx,Wx)=14infyG{x}|xy|=14D(x,G{x}).

    We conclude using (H2).

    (iii): Let y be in G{x}. By definition, for every vWx, there exists uIx,y such that v=u+14xy. Therefore, we use the triangle inequality and we have:

    |vy|D(y,Ix,y)14|xy|=12|xy|14|xy|=14|xy|.

    Taking the infimum over all vWx in the above inequality and using (H2), we finally obtain:

    D(y,Wx)14|xy|14D(x,G{x})C14|x|,

    where C>0 is independent of x and y.

    (iv): First, we have proved there exists a constant C11 independent of x such that Diam(Wx)C1|x|. Second, using Assumption (H2), we know there exists a constant C2>0 such that for every yG we have D(y,G{y})C2|y|. Let k>2 be an integer such that:

    C22k21>4C1. (20)

    We denote nN, the unique integer such that xA2n,2n+1. Here, it is sufficient to establish a bound for x sufficiently large, thus without loss of generality, we can assume that n>k. We next show that if yG satisfies |y|2k1|x|2nk or |y|2k|x|2n+k, then WxVy=.

    We start by assuming that yG(RdB2n+k). Since

    Diam(Wx)C1|x|C12n+1,

    we have WxBC12n+1(x). Therefore, using a triangle inequality we obtain WxBC12n+2. Our aim here is to prove that Iy,xBC12n+2= in order to deduce Iy,xWx=. For every zIy,x:

    |z||zx||x|D(x,Ix,y)|x|.

    In addition, for every yG{y}, we have D(x,Ix,y)=12|xy|12D(y,G{y}) and we deduce that:

    |z|D(y,G{y})|x|C22|y||x|C22n+k12n+12n+1(C22k21)C12n+3.

    Therefore, Ix,y(RdBC12n+3) and we obtain WxIx,y=. Since, Vy=zG{y}Iz,y, we deduce that VyWx=.

    Next we assume that yB2nk and we want to prove that VyHx,y=. As above, we can show that VyBC12nk+1 and for every zHx,y:

    |z|14|xy||y|14C2|x||y|2nk(C22k21)C12nk+2.

    Therefore Hx,yBC12nk+2 and we have VyHx,y=. We deduce that VyWx=.

    To conclude, we use Proposition 3 and we obtain the existence of a constant C3>0 independent of n such that:

    {xG|xA2n,2n+1}C3,

    and therefore:

    {yG|VyWx}km=k{xG|xA2m,2m+1}n+km=nkC3=(2k+1)C3.

    We have finally proved (ⅳ).

    (v): Let y be in G{x}. We first assume that 2k1|y|>|x|, where k is defined as in (20) and is independent of x. In the proof of (ⅳ) above, we have shown that WyVx=. Therefore, using Properties (ⅰ) and (ⅱ) of Wy we easily obtain that there exists a constant M1>0 independent of x and y such that D(Vx,Vy)>M1|y| and we can conclude. Next, we assume that 2k1|y||x|. Using again Properties (ⅰ) and (ⅱ) of Wx, we obtain the existence of M2>0 independent of x and y such that D(Vx,VyWx)M2|x|M22k1|y|. Finally, we have proved (ⅴ) with C5=min(M1,M22k1).

    We next prove that the set GC0 defined by (9) satisfies Assumptions (H1), (H2) and (H3). In order to avoid many unnecessary technical details, we study here the Voronoi diagram only for d=3 and, in the sequel, we admit that these properties still hold in higher dimension. We also consider the cell Vp only for p=(p1,p2,p3)(R+)3. Since the distribution of the points 2p is symmetric with respect to the origin, the other cases are similar and we omit them.

    Proof of (H1). Let p=(p1,p2,p3) be in PC0(R+)3. We first prove the following inclusion:

    Vp3i=1[2pi1,2|p|+3]. (21)

    To this aim, we want to show that if (x,y,z)3i=1[2pi1,2|p|+3], then there exists xqPC0{xp} such that the point (x,y,z) is closer to xq than to xp and therefore (x,y,z)Vp. We consider (x,y,z)(R+)3 and we start by assuming that x<2p11. We have

    D((x,y,z),xp)2=|x2p1|2+|y2p2|2+|z2p3|2,

    and

    D((x,y,z),(0,2p2,2p3))2=|x|2+|y2p2|2+|z2p3|2.

    Since x<2p11, we use a triangle inequality and

    |x2p1|>2p12p11=2p11>|x|.

    We obtain that D((x,y,z),xp)2>D((x,y,z),(0,2p2,2p3))2. That is, (x,y,z) is closer to (0,2p2,2p3)GC0 than to xp and we deduce that (x,y,z)Vp. We can therefore conclude that Vp is included in {(x,y,z)R3 | 2p11x}.

    We next assume that x>2|p|+3. Since |p|p1, we have:

    |x2p1|2=(x2|p|+1+2|p|+12p1)2(x2|p|+1+2|p|+12|p|)2=|x2|p|+1|2+2|p|+1(x2|p|+1)+22|p|.

    Using x>2|p|+3, it follows:

    |x2p1|2>|x2|p|+1|2+13×22|p|. (22)

    On the other hand, we have

    |y2p2|2=|y|22p2+1y+22p2,
    |y2p2+1|2=|y|22p2+2y+22p2+2.

    We obtain

    |y2p2|2|y2p2+1|23×2p2|y2p2+1|23×22|p|.

    Similarly, we can show that |z2p3|2|z2p3+1|23×2|p| and, using (22), we have

    D((x,y,z),xp)2>|x2|p|+1|2+|y2p2+1|2+|z2p3+1|2+7×2|p|>D((x,y,z),(2|p|+1,2p2+1,2p3+1))2.

    Now we claim that (2|p|+1,2p2+1,2p3+1)GC0. Indeed, since (p1,p2,p3)PC0, we have using (8) :

    max{|p|+1,p2+1,p3+1}=max{p1+1,p2+1,p3+1}min{p1+1,p2+1,p3+1}+C0min{|p|+1,p2+1,p3+1}+C0.

    Since D((x,y,z),xp)2>D((x,y,z),(2|p|+1,2p2+1,2p3+1))2, we therefore conclude that (x,y,z) is closer to (2|p|+1,2p2+1,2p3+1) than to xp and that (x,y,z)Vp.

    Using the symmetry of the distribution, we can use exactly the same argumentation to treat the cases y<2p21, y>2|p|+3, z<2p31 and z>2|p|+3. We have finally established inclusion (21). Since the volume of the cube 3i=1[2pi1,2|p|+3] is bounded by 83.23|p|, we can deduce that:

    |Vp|83.23|p|.

    (H1) is proved.

    Proof of (H2). Let p be in PC0(R+)3. We have:

    D(xp,GC0{xp})D(xp,0)=|xp|,

    and therefore:

    11+|xp|D(xp,GC0{xp}).

    To show the upper bound, we consider xqGC0{xp}. Without loss of generality, we can assume |p1|=|p| and there are three cases:

    ● If |q1||p1|, then:

    D(xp,xq)|sign(p1)2|p1|sign(q1)2|q1|||2|p1|2|q1||=2|p1||12|q1||p1||2|p1|12=2|p|1.

    ● If p1=q1, since pPC0, we have max(|p2|,|p3|)|p|C0. Since xqxp, we obtain as above:

    D(xp,xq)max(2|p2||12|q2||p2||,2|p3||12|q3||p3||)2|p|C01.

    ● If p1=q1, we have:

    D(xp,xq)|sign(p1)2|p1|sign(q1)2|q1||=2|p|+1.

    In the three cases we conclude there exists C>0 independent of q such that D(xp,xq)C2|p|. Finally, since |xp|=(22p1+22p2+22p3)1/23.2|p|, we obtain the existence of a constant C1>0 independent of p such that:

    1+|xp|D(xp,GC0{xp})C1.

    Proof of (H3). Let p=(p1,p2,p3) be in PC0(R+)3. We use (21) to bound the diameter of Vp by the diameter of the cube [0,2|p|+3]3, that is:

    Diam(Vp)3.2|p|+3.

    In addition, (H2) shows the existence of C>0 such that for every xpG, we have :

    D(xp,GC0{xp})C|xp|C2|p|,

    and we obtain (H3).

    We finally conclude this section establishing an estimate regarding the norm of each element xp of GC0. Using Proposition 1, the next property shall be useful to estimate the volume of the Voronoi cells in our particular case.

    Proposition 5. There exists C1>0 and C2>0 such that for every p in PC0, we have:

    C12|p||xp|C22|p|. (23)

    Proof. For pPC0, we have:

    |xp|=(i1,...,d22|pi|)1/2.

    We first use the inequality |pi||p| to obtain the upper bound. That is:

    |xp|(i1,...,d22|p|)1/2d2|p|.

    For the lower bound, we denote j=argmaxi{1,...,d}|pi| and we have:

    |xp|2|pj|=2|p|.

    We have established the norm estimate (23).

    In the sequel of this work, we only consider the specific set GC0, defined by (9), for a fixed arbitrary constant C0>1. Therefore, for the sake of clarity and without loss of generality, we will denote G and P instead of GC0 and PC0.

    In this section we prove some properties satisfied by the functional space B2(Rd). The following results are heavily based upon the geometric distribution of the xp. They are key for the understanding of the structure of B2 and to establish the homogenization of problem (1).

    To start with, we show the uniqueness of a limit L2-function f in L2(Rd) defined in (10) and characterizing each element of B2(Rd). This result ensures that the definition of the function space B2(Rd) is consistent.

    Proposition 6. Let f be a function of B2(Rd). Then, the limit function f of L2(Rd) defined in (10) is unique.

    Proof. We assume there exist two functions f and g in L2(Rd) such that

    lim|p|fτpfL2(Vp)=lim|p|fτpgL2(Vp)=0.

    By a triangle inequality, we obtain for every pP:

    τpfτpgL2(Vp)fτpfL2(Vp)+fτpgL2(Vp)|p|+0.

    In addition, we have τpfτpgL2(Vp)=fgL2(Vp2p). According to Proposition 2, we can find a sequence (pn)nNP such that limn|pn|= and:

    nN(Vpn2pn)=Rd.

    We can finally conclude that fgL2(Rd)=0, that is f=g in L2(Rd).

    We next study the structure of the space B2(Rd) showing two essential properties that shall allow us to establish the existence of the corrector in Section 4. In particular, we prove in Proposition 7 that B2(Rd) is a Banach space.

    Proposition 7. The space B2(Rd) equipped with the norm defined by (11), is a Banach space.

    Proof. Let (fn)nN be a Cauchy sequence in B2(Rd). Definitions (10) and (11) ensure the existence of a Cauchy sequence fn, in L2(Rd) such that for every nN,

    lim|p|fnτpfn,L2(Vp)=0.

    Then, for any ε>0, there exists NN such that for all n>N, k>0:

    fn+kfnL2unifε,fn+k,fn,L2(Rd)ε,suppP(fn+kτpfn+k,)(fnτpfn,)L2(Vp)ε2. (24)

    Since L2 and L2unif are Banach spaces, there exist fL2unif(Rd) and fL2(Rd) such that fnn+f in L2unif(Rd) and fn,n+f in L2(Rd). We consider the limit in (24) when k and we obtain:

    suppP(fτpf)(fnτpfn,)L2(Vp)ε2.

    Since ε can be chosen arbitrary small, we deduce:

    limnsuppP(fτpf)(fnτpfn,)L2(Vp)=0.

    The function f is therefore the limit of fn for the norm (11). We just have to show that fB2(Rd) to conclude. Indeed, for a fixed n>N and for p sufficiently large, we have:

    fnτpfn,L2(Vp)ε2.

    Using a triangle inequality, it follows:

    fτpfL2(Vp)fnτpfn,L2(Vp)+suppP(fτpf)(fnτpfn,)L2(Vp)ε.

    Finally, we obtain lim|p|fτpfL2(Vp)=0.

    Proposition 8. Let α]0,1[, then C0,α(Rd)B2(Rd) is dense in (B2(Rd),.B2(Rd)).

    Proof. We consider fB2(Rd) and fL2(Rd) the associated limit function defined by (10). First, for any ε>0, there exists ϕD(Rd) such that ϕfL2(Rd)<ε3, thus τpϕτpfL2(Vp)ε3 for all pP. Second, since fB2(Rd):

    PN, pP, |p|>P fτpfL2(Vp)<ε3.

    Since ϕ is compactly supported there also exists P, which we can always assume larger than P, such that for every |p|>P and for all qp, we have (τqϕ)|Vp=0.

    The finite sum |q|P1Vqf (where 1A denotes the indicator function of A) is compactly supported and then belongs to L2(Rd). Again, we can find ψD(Rd) such that ψ|q|P1VqfL2(Rd)ε3. We fix g=ψ+|p|>Pτpϕ and we want to show that g is a good approximation of f in B2(Rd)C0,α(Rd), that is g is close to f on each Vp, uniformly in p. First, we have:

    g|Vp={ψif |p|P,ψ+τpϕelse.

    Therefore g is bounded and we can easily prove that gB2(Rd)C(Rd) where the associated limit function in L2(Rd) is given by g=ϕ. Furthermore, g is in C0,α(Rd) since it is a C function and all of its derivatives are bounded. Indeed, for every k in Nd, we denote k=k1x1k2x2...kdxd and we have:

    (kg)|Vp={kψif |p|P,kψ+τpkϕelse.

    and kg is clearly bounded.

    Let pP, we consider two cases. If |p|P, then:

    gfL2(Vp)=ψ|q|P1VqfL2(Vp)ε.

    Else, if |p|>P, using that |q|P1Vqf has support in |q|PVq we have:

    ψL2(Vp)=ψ|q|P1VqfL2(Vp)ψ|q|P1VqfL2(Rd)ε3.

    We obtain:

    gfL2(Vp)=ψ+τpϕfL2(Vp)ψL2(Vp)+τpϕτpfL2(Vp)+τpffL2(Vp)ε,

    and we can conclude.

    We now establish a property regarding multiplication of elements of B2(Rd).

    Proposition 9. Let g and h be in B2(Rd)L(Rd). We assume the associated L2 function of g, denoted by g, is in L(Rd), then hgB2(Rd).

    Proof. Since gL(Rd), we clearly have ghL2(Rd). Using that for all pP:

    ghτp(gh)=(hτph)τpg+(gτpg)h.

    We have by the triangle inequality:

    ghτp(gh)L2(Vp)hτphL2(Vp)gL(Rd)+gτpgL2(Vp)hL(Rd).

    It follows, taking the limit for |p|, that ghB2(Rd) and that (gh)=gh.

    Our next result is one of the most important properties for the sequel. As we shall see in section 5, it first implies that the homogenized coefficient in our setting is the same as the homogenized coefficient in the periodic case, that is, without perturbation. In addition, it gives some information about the growth of the corrector defined in Theorem 1.1 (in particular, we give a proof in Proposition 11 of the strict sublinearity of the corrector). We will use all of these properties to prove the convergence stated in Theorem 1.2 in our case.

    Proposition 10. Let uB2(Rd). Then, for every x0Rd:

    limR1|BR|BR(x0)|u(x)|dx=0, (25)

    with the following convergence rate:

    1|BR|BR(x0)|u(x)|dxC(logRRd)12, (26)

    where C>0 is independent of R and x0.

    Proof. We fix R>0. Using the Cauchy-Schwarz inequality, we have:

    1|BR|BR(x0)|u(x)|dx1|BR|(BR(x0)|u(x)|2dx)12=1|BR|(pPVpBR(x0)|u(x)|2dx)12.

    Since the number of Vp such that BR(x0)Vp is bounded by log(R) according to Corollary 1, we obtain:

    1|BR|BR(x0)|u(x)|dx(logR)12|BR|suppuL2(Vp)C(d)(log(R)Rd)12suppuL2(Vp).

    Here, C(d) depends only on the ambient dimension d. The last inequality yields (26) and conclude the proof.

    Corollary 2. Let uB2(Rd)L(Rd), then |u(./ε)| is convergent to 0 in the weak*-L topology when ε0.

    Proof. We fix R>0 and we first consider φ=1BR. For any ε>0, we have:

    |Rd|u(x/ε)|φ(x)dx|BR|u(x/ε)|dxy=x/ε=εdBR/ε|u(y)|dy=|BR|εd|BR|BR/ε|u(y)|dy=φL1(Rd)εd|BR|BR/ε|u(y)|dy.

    We next use (26) in the right-hand term and we obtain the existence of C>0 independent of ε and φ such that:

    |Rdu(x/ε)φ(x)dx|CφL1(Rd)(εdlog(1/ε))12ε00.

    We conclude using the density of simple functions in L1(Rd).

    We next introduce the notion of sub-linearity which is actually a fundamental property in homogenization. Indeed, in order to precise the convergence of the approximated sequence of solutions (6), we have to study the behavior of the sequences εwei(./ε) when ε0. The convergence to zero of these sequences and the understanding of the rate of convergence are key for establishing estimates (15) and (16) stated in Theorem 1.2. In the sequel, we therefore study this phenomenon for the functions with a gradient in B2(Rd).

    Definition 3.1. A function u is strictly sub-linear at infinity if:

    lim|x||u(x)|1+|x|=0.

    In the next proposition we prove the sub-linearity of all the functions u such that u(B2(Rd)L(Rd))d. We assume, for this general property only, that d2.

    Proposition 11. Assume d2. Let uH1loc(Rd) with u(B2(Rd)L(Rd))d. Then u is strictly sub-linear at infinity and for all s>d, there exists C>0 such that for every x,yRd with xy:

    |u(x)u(y)|C|log(|xy|)|1s|xy|1ds. (27)

    Proof. Let x,yRd with xy and fix r=|xy|. Since uL(Rd)d, we have uLsloc(Rd)d for every s1. We next fix s>d. We know there exists a constant C>0, depending only on d, such that:

    |u(x)u(y)|Cr(1rdBr(x)|u(z)|sdz)1s.

    This estimate is established for instance in [13,Remark p.268] as corollary of the Morrey's inequality ([13,Theorem 4 p.266]). Since s>d2, we use the boundedness of u to obtain:

    |u(x)u(y)|Cu(s2)/sL(Rd)r(1rdBr(x)|u(z)|2dz)1s. (28)

    We next split the integral of (28) on each Vp such that VpBr(x) and we have :

    |u(x)u(y)|Cu(s2)/sL(Rd)r(1rdpPBr(x)Vp|u(z)|2dz)1s.

    We finally use Corollary 1 and we obtain the existence of a constant C1>0 such that:

    |u(x)u(y)|C1u(s2)/sL(Rd)u2/sB2(Rd)|log(r)|1sr1ds.

    This inequality is true for all s>d, which allows us to conclude. In addition, the sub-linearity of u is obtained fixing y=0 and letting |x| go to the infinity in estimate (27).

    Remark 1. In the case d=1, since s2, the above proof gives:

    |u(x)u(y)|C|log|xy||12|xy|12.

    The last proposition of this section gives an uniform estimate of the integral remainders of the functions of B2(Rd). The idea here is that the functions of B2(Rd) behave like a fixed L2-functions at the vicinity of the points of G and therefore, have to be small in a L2 sense far from these points. This property will be used in the proof of Lemma 4.3 in next section to establish an estimate in B2(Rd) satisfied by the solutions to diffusion equation (35).

    Proposition 12. Let f be in B2(Rd) and f the associated limit function in L2(Rd). For any ε>0, there exists R>0 such that for every R>R and every p,qP:

    (VqBR(2q)c|fτpf|2)1/2<ε,

    where BR(2q)c denotes the set RdBR(2q). Therefore, we have the following limit:

    limRsup(p,q)P2pq(VqBR(2q)c|fτpf|2)1/2=0.

    Proof. Let ε>0. First, for every R>0, p,qP we use a triangle inequality and we obtain the following upper bound:

    (VqBR(2q)c|fτpf|2)1/2(VqBR(2q)c|fτqf|2)1/2+(VqBR(2q)c|τqf|2)1/2+(VqBR(2q)c|τpf|2)1/2=Ip,q1(R)+Ip,q2(R)+Ip,q3(R).

    We want to bound the three terms Ip,q1(R), Ip,q2(R) and Ip,q3(R) by ε uniformly in p,q.

    We start by considering Ip,q1. We have assumed that fB2(Rd), then, by definition, there exists P>0 such that for every qP satisfying |q|>P, we have:

    (Vq|fτqf|2)1/2<ε3.

    In addition, since the volume of each Vq is finite according to assumption (H1), there exists R1>0 such that for every |q|P, BR1(2q)cVq=. Therefore, as soon as |q|P and RR1, we have Ip,q(R)=0. Finally, considering successively the case |q|P and the case |q|>P, we obtain for every q,pP and RR1:

    Ip,q1(R)<ε3.

    We next study the second term Ip,q2. Since f is in L2(Rd), there exists R2>0, which we can always assume larger than R1, such that for every qP:

    (BR2(2q)c|τqf(y)|2dy)1/2x=y2q=(BcR2|f(x)|2dx)1/2<ε3. (29)

    And we directly obtain, for every RR2:

    Ip,q2(R)<ε3.

    Finally, in order to bound the last term, we know that lim|l|D(2l,G{2l})=+ as a consequence of Assumption (H2). Therefore, there exists a finite number of indices l such that:

    D(Vl,G{2l})R2. (30)

    Thus, we deduce the existence of a positive radius R3 independent of q,p such that for every l satisfying (30) we have BR3(2l)cVl=. Again we can always assume R3 larger than R2. There are two cases depending on the value of q:

    1) If q satisfies (30), we have BR3(2q)cVq= and we obtain Ip,q3(R3)=0

    2) Else, for every yVq, we have |y2p|>R2. Therefore:

    Ip,q3(R3)(Vq|τpf|2)1/2x=y2p=(Vq2p|f|2)1/2(BcR2|f|2)1/2.

    Using (29), we have for every RR3, Ip,q3(R)ε3.

    In the two cases, we obtain for RR3:

    Ip,q3(R)ε3.

    Since the values of R3 is independent of p and q we can conclude the proof for R=R3.

    This section is devoted to the proof of Theorem 1.1. Equation (14) being posed on the whole space Rd, we need to use here the geometric distribution of the 2p and introduce some constructive techniques involving the fundamental solution of the operator div(a.) to solve it. To start with, we establish some general results on equations

    div(au)=div(f)in Rd, (31)

    for coercive coefficients a of the form (2) and right hand side f in B2(Rd)d in order to deduce the existence of the corrector stated in Theorem 1.1. For this purpose, we consider the following strategy adapted from [8]: we first study diffusion problem (35) in the periodic context, that is, when the diffusion coefficient a=aper is periodic. Secondly we show in Lemma 4.5 the continuity of the associated reciprocal linear operator (diva)1div from B2(Rd) to B2(Rd). Finally, we use this continuity in order to generalize the existence results of the periodic context to the general context when a is a perturbed coefficient of the form (2). To this end, we apply a method based on the connexity of the set I=[0,1] as we shall see in the proof of Lemma 4.6.

    We begin by establishing the uniqueness of a solution u to (31) such that uB2(Rd)d. This result is actually essential in the proof of Theorem 1.1 since it both ensures the uniqueness of the corrector solution to (14) and also allows us to establish the continuity estimate of Lemma 4.5 which is key in our approach to show the existence of a solution to (31).

    Lemma 4.1. Let a be an elliptic and bounded coefficient, and uH1loc(Rd), such that suppPVp|u|2<, be a solution to:

    div(au)=0inRd, (32)

    in the sense of distribution. Then u = 0.

    Proof. we consider uH1loc(Rd) solution to (32). Since u is a solution to (32), there exists C>0 such that for every R>0, we have the following estimate (for details see for instance [14,Proposition 2.1 p.76] and [14,Remark 2.1 p.77]):

    BR|u|2CR2AR,2R|uuAR,2R|2,

    where:

    uAR,2R=1|AR,2R|AR,2Ru(x)dx.

    We use the Poincaré-Wirtinger inequality on the right-hand side and we obtain:

    BR|u|2CAR,2R|u|2.

    Furthermore, we can write this inequality in the following form:

    BR|u|2C1+CB2R|u|2. (33)

    In addition, using Corollary (1), we know there exists a constant C1>0 independent of R such that:

    B2R|u|2=VpB2RVpB2R|u|2C1log(2R)suppVp|u|2. (34)

    Next, we define F(R)=BR|u|2. The inequalities (33) and (34) yield for all R>0 and for every nN, we obtain

    F(R)(C1+C)nF(2nR)C1(C1+C)nlog(2nR)suppVp|u|2.

    Since C1+C<1, we have:

    limn(C1+C)nlog(2nR)=0,

    and it therefore follows, letting n go to infinity, that F(R)=0 for all R>0, thus u=0.

    Corollary 3. Let fB2(Rd)d, then a solution u to (31) with uB2(Rd)d is unique up to an additive constant.

    Remark 2. Here the restriction made on the dimension is actually not necessary. The result and the proof of Lemma 4.1 of uniqueness still hold if we assume d=1 or d=2.

    Remark 3. We remark that Assumptions (2) and (13) regarding the structure and the regularity of the coefficient a are not required to establish the uniqueness result of Lemma 4.1. In the proof, we only use the "Hilbert" structure of L2, induced by the assumptions satisfied by u, and the fact that a is elliptic and bounded.

    Now that uniqueness has been dealt with, we turn to the existence of the solution to (31). We need to first establish it for a periodic coefficient considering the equation:

    div(aperu)=div(f)in D(Rd). (35)

    We start by introducing the Green function Gper:Rd×RdR associated with the operator div(aper.) on Rd. That is, the unique solution to

    {divx(aper(x)xGper(x,y))=δy(x)in D(Rd),lim|xy|Gper(x,y)=0.

    According to the results established in [4,Section 2] about the asymptotic growth of the Green function (see also [2,Theorem 13,proof of Lemma 17] and [18] for bounded domain or [11, Proposition 8] for additional details), there exists C1>0, C2>0 and C3>0 such that for every x,yRd with xy:

    |yGper(x,y)|C11|xy|d1, (36)
    |xGper(x,y)|C21|xy|d1, (37)
    |xyGper(x,y)|C31|xy|d. (38)

    We first introduce a result of existence in the L2(Rd) case. The following lemma allows us to define a solution to (35) using the Green function when f belongs to L2(Rd)d. The proof of this result is established in [4].

    Lemma 4.2. Let f be in L2(Rd)d, then the function:

    u=RdyGper(.,y).f(y)dy, (39)

    is a solution in H1loc(Rd) to (35) such that uL2(Rd)d.

    Our aim is now to generalize the above result to our case and, in particular, to give a sense to the function u define by (39) when fB2(Rd)d. The idea here is to split the function f into a sum of L2-functions fp compactly supported in each Vp for pP. Using Lemma 4.2, we shall obtain the existence of a collection up of solution to (35) when f=fp. The main difficulty here is to show that the function u defined as the sum of the up is bounded.

    Lemma 4.3. Let fL2loc(Rd)d such that suppPfL2(Vp)<, then the function u defined by

    u=RdyGper(.,y)f(y)dy (40)

    is a solution in H1loc(Rd) to (35).In addition, u is the unique solution to (35) which satisfies suppPuL2(Vp)< and there exists C>0 independent of f and u such that we have the following estimate:

    suppPuL2(Vp)CsuppPfL2(Vp). (41)

    Proof. Step 1: u is well defined

    We start by proving that definition (40) makes sense and, in particular, that the above integral defines a function u solution to (35) in H1loc(Rd). In the sequel the letter C denotes a generic constant that may change for one line to another. For every qP, we first introduce a set Wq and five constants C1, C2, C3, C4 and C5 independent of q and defined by Proposition 4 such that:

    VqWq, (i)
    Diam(Wq)C12|q| and D(Vq,Wq)C22|q|, (ii)
    rP{q}, Dist(2r,Wq)C32|q|, (iii)
    {rP|VrWq}C4, (iv)
    rP{q}, D(Vq,VrWq)C52|r|. (v)

    To start with, we define for each qP the function:

    uq=RdyGper(.,y)f(y)1Vq(y)dy. (42)

    Lemma 4.2 ensures this function is a solution in H1loc(Rd) to:

    {div(aperuq)=div(f1Vp)in Rd,uqL2(Rd)d.

    Considering the gradient of (42), we have for every xRdVq:

    uq(x)=RdxyGper(x,y)f(y)1Vq(y)dy.

    Next, for every NN, we define:

    UN=qP, |q|Nuq,

    and

    SN=UN=qP, |q|Nuq. (43)

    We next show that the two series UN ans SN are convergent in L2loc(Rd). To this aim, since the collection (Vp)pP is a partition of Rd, it is sufficient to prove that they normally converge in L2(Vp) for every pP. We fix pP and for every qP such that VqWp=, we use the Cauchy-Schwarz inequality to obtain:

    uqL2(Vp)=(Vp|VqyGper(x,y)f(y)dy|2dx)1/2(VpVq|yGper(x,y)|2dyVq|f(y)|2dy dx)1/2.

    Next, estimate (36) gives:

    uqL2(Vp)CsuprPfL2(Vr)(VpVq1|xy|2d2dy dx)1/2. (44)

    Since VqWp=, Property (ⅴ) gives the existence of C>0 such that for every xVp and yVq, we have |xy|C2|q|. We next use Propositions 1 and 5 to obtain the existence of a constant C>0 independent of p and q such that |Vq|C2d|q|. Finally:

    uqL2(Vp)CsuprPfL2(Vr)(VpVq12(2d2)|q|dy dx)1/2CsuprPfL2(Vr)(Vp|Vq|2(2d2)|q|dx)1/2CsuprPfL2(Vr)|Vp|1/22|q|(d/21).

    We thus obtain the following upper bound:

    qPuqL2(Vp)=qP,VqWpuqL2(Vp)+qP,VqWp=uqL2(Vp)qP,VqWpuqL2(Vp)+CqP12|q|(d/21).

    The first sum is finite according to Property (ⅳ) and we only have to prove the convergence of the second one. We have assumed d>2 and consequently d/21>0. In addition, since the number of qP such that |q|=nN is bounded independently of n (as a consequence of Proposition 3), we have:

    qP12|q|(d/21)CnN12n(d/21)<. (45)

    Therefore, for every pP, the absolute convergence of UN to u in L2(Vp) is proved. That is, since the sequence of the sets Vq defines a partition of Rd, UN converges to u in L2loc(Rd). Using asymptotic estimate (38) for xyGper we can conclude with the same arguments to prove the convergence of SN in L2loc(Rd). In addition, the gradient operator being continuous in D(Rd), we have:

    qPuq=limNSN=limNUN=u.

    To complete the proof, we have to show that u is a solution to (35). Let N be in N. By linearity of the operator div(aper.), UN is a solution in H1loc(Rd) to:

    div(aperUN)=div(qP, |q|N1Vqf)in Rd. (46)

    We take the L2loc-limit when N in (46) and we obtain:

    div(aperu)=div(f)in Rd.

    Therefore, u is a solution to (35) in D(Rd).

    Step 2: Proof of estimate (41)

    Let p be in P, we want to split u in two parts. For every xVp, we write:

    u(x)=WpyGper(x,y)f(y)dy+RdWpyGper(x,y)f(y)dy=I1,p(x)+I2,p(x).

    I1,p and I2,p are two distributions (they are in L2loc(Rd)), so we can consider their gradients in a distribution sense. In addition, I2,p is a differentiable function on Vp and

    I2,p(x)=RdWpxyGper(x,y)f(y)dy.

    We start by establishing a bound for I1,pL2(Vp). First, we use estimate (36) for yGper and we obtain:

    I1,p2L2(Wp)CWp(Wp1|xy|d1|f(y)|dy)2dx.

    We next apply the Cauchy-Schwarz inequality:

    I1,p2L2(Wp)CWp(Wp1|xy|d1dy)(Wp1|xy|d1|f(y)|2dy)dx.

    Property (ⅱ) implies that WpQC12|p|(2p). Therefore, for every xWp and yWp, we have by a triangle inequality that xyQC2|p|+1 and then:

    Wp1|xy|d1dyQC2|p|+11|y|d1dyC2|p|. (47)

    Using (47) and the Fubini theorem, we finally obtain:

    I1,p2L2(Wp)C2|p|Wp|f(y)|2QC12|p|+1(2p)1|xy|d1dxdyC22|p|f2L2(Wp). (48)

    Lemma 4.2 ensures that I1,p is a solution in D(Rd) to:

    div(aperI1,p)=div(f1Wp). (49)

    Since Property (ⅱ) ensures D(Vp,Wp)C2|p|, we can apply a classical inequality of elliptic regularity (see for instance [15,Theorem 4.4 p.63]) to equation (49) in order to establish the following estimate:

    I1,p2L2(Vp)C(122|p|I1,p2L2(Wp)+f2L2(Wp)), (50)

    and we deduce from previous inequalities (48) and (50) that:

    I1,p2L2(Vp)Cf2L2(Wp). (51)

    In addition, we have:

    f2L2(Wp)qP,VqWpf2L2(Vq)qP,VqWpsuprPf2L2(Vr).

    Next, we use a triangle inequality and Property (ⅳ) of Wp to obtain:

    f2L2(Wp)CsuprPf2L2(Vr).

    We apply this inequality in (51) and we finally obtain:

    I1,pL2(Vp)CsuprPfL2(Vr), (52)

    where C>0 is independent of p.

    We next prove a similar bound for I2,pL2(Vp). To start with, we want to show there exists a constant C>0 such that:

    I2,pL(Vp)C12d|p|/2suprPfL2(Vr). (53)

    To this aim, we fix xVp and we use estimate (38) for xyGper to obtain:

    |I2,p(x)|CqpVqWp1|xy|d|f(y)|dyCqp(VqWp1|xy|2ddy)1/2(VqWp|f(y)|2dy)1/2Cqp(VqWp1|xy|2ddy)1/2suprPfL2(Vr).

    Next, using Property (ⅱ) of Wp, there exists C>0 such that for every qp,

    D(VqWp,Vp)>C2|p|,

    and it follows:

    |q|<|p|(VqWp1|xy|2ddy)1/2C|q|<|p|(VqWp12|p|2ddy)1/2C|q|<|p|(|Vq|2|p|2d)1/2C|q|<|p|2|q|d/22|p|d.

    The last inequality is actually a direct consequence of Propositions 1 and 5. In addition, we have proved in Proposition 3 there exists a constant C>0 such that for every nN, the number of qP such that |q|=n is bounded by C. Therefore we have:

    |q|<|p|2|q|d/22|p|d=|p|n=0qP,|q|=n2|q|d/22|p|dC|p|n=02nd/22|p|d=C2|p|d/22|p|d=C12|p|d/2.

    And finally :

    |q|<|p|(VqWp1|xy|2ddy)1/2C12|p|d/2. (54)

    Furthermore, we have with similar arguments:

    |q||p|(VqWp1|xy|2ddy)1/2C|q||p|(VqWp12|q|2ddy)1/2C|q||p|(|Vq|2|q|2d)1/2C|q||p|12|q|d/2.

    And we obtain again:

    |q||p|12|q|d/2Cn|p|12nd/2=C12|p|d/2.

    That is:

    |q||p|(VqWp1|xy|2ddy)1/2C12|p|d/2. (55)

    Using estimates (54) and (55), we have finally proved (53) and it follows:

    I2,pL2(Vp)|Vp|1/2I2,pL(Vp)C2|p|d/212|p|d/2suprPfL2(Vr).

    Therefore we have the existence of a constant C>0 independent of p such that:

    I2,pL2(Vp)CsuprPfL2(Vr). (56)

    For every pP, using estimates (52) and (56) and a triangle inequality, we conclude that:

    uL2(Vp)I1,pL2(Vp)+I2,pL2(Vp)CsuprPfL2(Vr).

    We finally obtain expected estimate (41) taking the supremum over all pP in the above inequality.

    To conclude the study of problem (35) with a periodic coefficient, we next show that the solution to (40) given in Lemma 4.3 has a gradient in B2(Rd).

    Lemma 4.4. Let fB2(Rd)d, then the function u defined by (40) is the unique solution to (35) such that uB2(Rd)d.

    Proof. We want to prove there exists a function gL2(Rd)d such that

    lim|p|uτpgL2(Vp)=0.

    In this proof, the letter C also denotes a generic constant independent of p, u and f that may change from one line to another. Using the result of Lemma 4.2, we can define a function uL2loc(Rd) by:

    u(x)=RdyGper(x,y)f(y)dy

    solution in D(Rd) to:

    div(aperu)=div(f)in Rd, (57)

    such that uL2(Rd)d. For every pP, by subtracting a 2p-translation of (57) from (35), the periodicity of aper implies:

    div(aper(uτpu))=div(fτpf).

    For every pP, in the sequel we denote up=uτpu and fp=fτpf. In order to prove uB2(Rd)d, the idea is to show that lim|p|Vp|up|2dx=0. We start by fixing ε>0. Since fB2(Rd)d, Proposition 12 gives the existence of a radius R>0, such that for every p,qP,

    (VqBR(2q)c|fτpf|2)1/2<ε. (58)

    In the sequel, the idea is to repeat step by step the method used in the proof of Lemma 4.3. For pP, we thus introduce the set Wp as in the previous proof and we split up in two parts. For every xVp, we can write:

    up(x)=WpyGper(x,y)fp(y)dy+RdWpyGper(x,y)fp(y)dy=I1,p(x)+I2,p(x).

    In the sequel, we denote Ap the set WpVp. As in the previous proof (see the details of the proof of estimate (51)) we can show that:

    I1,p2L2(Vp)Cfp2L2(Wp),

    and we next prove that lim|p|fp2L2(Wp)=0. First, since fB(Rd)d, we already know that lim|p|Vp|fp|2=0 and we only have to treat the integration term on Ap. Using Property (ⅲ) of Wp, we know that the distance D(2q,Wp), for qp, is bounded from below by 2|p|. Therefore, if 2|p|>R, we obtains:

    Ap=qP{p}VqWpVqWpqP{p}VqWpVqBR(2q)c.

    In addition, Property (ⅳ) of Wp gives the existence of a constant C>0 such that the cardinality of the set of q satisfying VqWp is bounded by C. Estimate (58) therefore implies that

    Ap|fp|2qP{p}VqWpVqBR(2q)c|fp|2Cε.

    Since ε can be chosen arbitrarily small, we finally obtain lim|p|Ap|fp|2=0, that is

    lim|p|I1,p2L2(Vp)=0.

    We next prove that lim|p|I2,p2L2(Vp)=0. We split I2,p in two parts such that for every xVp:

    I2,p(x)=qPqp(VqWp)BR(2q)cxyGper(x,y)fp(y)dy+qPqp(VqWp)BR(2q)xyGper(x,y)fp(y)dy=J1,p(x)+J2,p(x).

    We want to estimate J1,pL2(Vp) and J1,pL2(Vp). We proceed exactly in the same way as in the previous proof (see the details of estimate (53)) and, using estimate (58), we obtain the following inequalities:

    J1,pL(Vp)C12d|p|/2supqPfpL2(VqBR(2q)c)C12d|p|/2ε, (59)

    and

    J2,pL(Vp)CRd|p|2|p|dsupqPfpL2(Vq). (60)

    To conclude, we consider P>0 such that for every pP satisfying |p|>P, we have:

    Rd|p|2|p|d/2<ε.

    Therefore, for every |p|>P, we use (59) and (60) and we obtain:

    I2,pL2(Vp)J1,pL2(Vp)+J2,pL2(Vp)|Vp|1/2(J1,pL(Vp)+J2,pL(Vp))C2|p|d/2(12|p|d/2ε+Rd|p|2|p|d)Cε.

    Since we can choose ε arbitrarily small, we conclude that lim|p|I2,pL2(Vp)=0. Finally, by a triangle inequality we have lim|p|upL2(Vp)=0, that is uB2(Rd)d.

    Remark 4. It is important to note that the essential point of the two above proofs is the convergence of the sums of the form qPVq1|xy|df(y)dy given in estimates (44), (54) and (55). Although we use here the particular distribution of the 2p, these convergence results are not specific to the set (9) considered in this study. They are actually ensured by Assumptions (H1), (H2) and (H3), particularly by the logarithmic bound given in Proposition 3 and Corollary 1. Indeed, with the notations of Section 1.2 and given Assumptions (H1)-(H2)-(H3), we could similarly argue to obtain estimates such as in (54)-(55) by splitting the sums over each annulus An:=A2n,2n+1 and studying nNxqGAnVxq1|xy|df(y)dy. The results of existence stated in this section therefore still hold if we consider a generic set G satisfying our general assumptions.

    Remark 5. In the two-dimensional context, the results of Lemmas 4.2, 4.3 and 4.4 remain true since estimates (36), (37) and (36) still hold. However the proof requires some additional technicalities, in particular to prove that the function u defined by (40) makes sense. In this case the series (45) does not actually converge but it is still possible to prove that the series of the gradients (43) converges. Here, the difficulty is to show that the limit of (43), denoted by T here, is the gradient in a distribution sense of a solution to (35). To this end, it is actually sufficient to show that iTj=jTi for every i,j{1,...,d}. This result is obtained considering the property of the limit of (43) in D(Rd).

    Our aim is now to generalize the results established in the case of periodic coefficients to our original problem (31). Here, our approach is to prove in Lemma 4.5 the continuity of the linear operator (diva)1div from B2(Rd)d to B2(Rd)d in order to apply a method adapted from [8] and based on the connexity of the set [0,1]. This method is used in the proof of existence of Lemma 4.6. Finally, this result allows us to prove the existence of a corrector stated in Theorem 1.1.

    Actually, we could have proved Lemmas 4.5 and 4.6 simultaneously but, in the interest of clarity, we first prove a priori estimate (61) and next, we establish the existence result in the general case.

    Lemma 4.5. There exists a constant C>0 such that for every f in B2(Rd)d and u solution in D(Rd) to (31) with u in B2(Rd)d, we have the following estimate:

    uB2(Rd)CfB2(Rd). (61)

    Proof. We give here a proof by contradiction using a compactness-concentration method. We assume that there exists a sequence fn in B2(Rd)d and an associated sequence of solutions un such that un is in B2(Rd)d and:

    div((aper+˜a)un)=div(fn), (62)
    limnfnB2(Rd)=0, (63)
    nNunB2(Rd)=1. (64)

    First of all, a property of the supremum bound ensures that for every nN, there exists xnRd such that:

    unL2unifunL2(B1(xn))unL2unif1n.

    Next, in the spirit of the method of concentration-compactness [20], we denote ˉun=τxnun, ˉfn=τxnfn, ˉan=τxna and ˉ˜an=τxn˜a and we have for every nN:

    unL2unifˉunL2(B1)unL2unif1n. (65)

    Next, for every nN, ˉun is a solution to:

    div(ˉanˉun)=div(ˉfn)in Rd.

    Since the norm of L2unif is invariant by translation, (63) and (64) ensure that ˉfn strongly converges to 0 in L2unif(Rd) and that the sequence (ˉun)nN is bounded in L2unif(Rd). Therefore, up to an extraction, ˉun weakly converges to a function ˉu in L2loc(Rd).

    The idea is now to study the limit of ˉan. To start with, we denote

    xn=(xn,imod(1))i{1,...,d}.

    Since aper is periodic, we have τxnaper=τxnaper. In addition, the sequence xn belongs to the unit cube of Rd and, therefore, it converges (up to an extraction) to xRd. Since aper is Holder continuous, τxnaper converges uniformly to τxaper, which also belongs to (L2per(Rd)C0,α(Rd))d×d.

    In order to study the convergence of ˉ˜an, we consider several cases depending on xn:

    1. If xn is bounded, it converges (up to an extraction) to xlimRd. Then, since ˜a is Holder-continuous, ˉ˜an strongly converges in L2loc(Rd) to τxlim˜aB2(Rd)d×d.

    2. If xn is not bounded, since (Vp)pP is a partition of Rd, there exists an unbounded sequence (pn)n in P such that xn=2pn+tn with tnVpn2pn.

    – If tn is bounded, it converges (up to an extraction) to tlimRd. In this case, for any compact subset K of Rd, we have

    ˉ˜an˜a(.+tlim)L2(K)˜a(.+2pn+tn)˜a(.+tn)L2(K)+˜a(.+tn)˜a(.+tlim)L2(K)=˜aτpn˜aL2(K+2pn+tn)+˜a(.+tn)˜a(.+tlim)L2(K).

    First, since tn is bounded and pn is unbounded, we have K+2pn+tn is included in Vpn for n sufficiently large. Therefore, ˜aτpn˜aL2(K+2pn+tn) converges to 0 when n. Second, ˜a is Holder-continuous and tn converges to tlim. Thus, ˜a(.+tn) converges uniformly to ˜a(.+tlim) and ˜a(.+tn)˜a(.+tlim)L2(K) converges to 0. Finally, ˉ˜an converges to ˜a(.+tlim) in L2(K) for every compact subset K.

    – If tn is unbounded, we can always assume that |tn| up to an extraction. We have for every K compact of Rd,

    ˉ˜anL2(K)˜a(.+2pn+tn)˜a(.+tn)L2(K)+˜a(.+tn)L2(K)=˜aτpn˜aL2(K+2pn+tn)+˜aL2(K+tn).

    First, since ˜a belongs to L2(Rd)d×d and tn is unbounded we have that ˜aL2(K+tn) converges to 0 when n. Secondly, we introduce the set W2pn defined as in Proposition 4. For every R>0, the properties of W2pn allow to show that there exists NN such that for all n>N, we have K+2pn+tnW2pn and:

    K+2pn+tnqPVqW2pnVqBR(2q).

    Using Proposition 4, we know that the number of q such that VqW2pn is not empty, is uniformly bounded with respect to n. Proposition 12 finally ensures that ˜aτpn˜aL2(K+2pn+tn)0. Therefore, ˉ˜an strongly converges to 0 in L2loc(Rd).

    In any case, the sequence aper+ˉ˜an therefore converges to a coefficient A=τxaper+˜A, where ˜A is of the form

    ˜A={τxlim˜aB2(Rd)d×d, if xn is bounded,τtlim˜aL2(Rd)d×d,if xn=2pn+tn, pn is not bounded, tn is bounded,0,if xn=2pn+tn, pn and tn are not bounded.

    In the three cases, as a consequence of Assumptions (12) and (13), the coefficient A is clearly bounded, elliptic and belongs to (C0,α(Rd))d×d. Moreover, as a consequence of the uniform Holder-continuity (with respect to n) of ˉanA, the convergence of ˉan to A is also valid in Lloc(Rd).

    The next step of the proof is to study the limit ˉu of ˉun in these three cases. First, since ˉan strongly converges to A in L2loc(Rd), considering the weak limit in (62) when n, we obtain

    div(Aˉu)=0in Rd. (66)

    We now state that ˉu=0. Indeed,

    1. if xn is bounded, assumption (64) ensures that there exists a constant C>0 such that for all nN and pP, we have:

    ˉunL2(Vp)=unL2(Vp+xn)C.

    Therefore, the property of lower semi-continuity satisfied by the norm .L2 implies

    pP,ˉuL2(Vp)lim infnˉunL2(Vp)<C.

    And we obtain suppˉuL2(Vp)<. Finally, since A is elliptic and bounded and ˉu is solution to (66), the uniqueness results of Lemma 4.1 gives ˉu=0 on Rd.

    2. if xn is not bounded, we know that xn=2pn+tn where |pn|. For every nN:

    ˉunL2(Vpnxn)=unL2(Vpn)1.

    Up to an extraction, the sequence Vpnxn is an increasing sequence of sets, and we can show that nN(Vpnxn)=Rd (see the proof of Proposition 2). Consequently, for every R>0, there exists NN such that BR(VpNxN) and

    n>N,ˉunL2(BR)1.

    Using again lower semi-continuity, we have for every R>0:

    ˉuL2(BR)lim infnˉunL2(BR)1.

    We obtain that ˉuL2(Rd). Since A is bounded and elliptic, a result of uniqueness established in [10,Lemma 1] finally ensures that ˉu=0.

    We are now able to show that ˉun strongly converges to 0 in L2(B1). To this aim, we note that, for every n, the addition of a constant to ˉun does not affect ˉun. Then, without loss of generality, we can always assume that B2ˉun = 0 and the Poincaré-Wirtinger inequality gives the existence of a constant C>0 independent of n such that:

    ˉunL2(B2)CˉunL2(B2).

    ˉun is therefore bounded in H1(B2) according to Assumption (64). The Rellich theorem ensures that, up to an extraction, ˉun strongly converges to ˉu, that is to 0, in L2(B2). Since ˉun is solution to (62), a classical inequality of elliptic regularity gives the following estimate (see for instance [15,Theorem 4.4 p.63]):

    B1|ˉun|2C(B2|ˉun|2+B2|ˉfn|2),

    where C depends only of a and the ambient dimension d. We therefore consider the limit when n to conclude that ˉun strongly converges to 0 in L2(B1). We next use (65) and the strong convergence of ˉun to 0 in L2(B1) to conclude that

    limnunL2unif(Rd)=0.

    That is, un strongly converges to 0 in L2unif(Rd).

    In order to conclude this proof, we will show that un actually converges to 0 in B2(Rd) and obtain a contradiction.

    First of all, we study the behavior of the sequence un,. For pP, we consider the 2p-translation of (62) and we have

    div((aper+τp˜a)τpun)=div(τpfn).

    Letting |p| go to the infinity, for every nN, we obtain that un, is a solution to:

    div((aper+˜a)un,)=div(fn,)in Rd.

    An estimate established in [8,Proposition 2.1], gives the existence of a constant C>0 independent of n such that:

    un,L2(Rd)Cfn,L2(Rd).

    By assumption, we have limnfn,L2(Rd)=0 and we deduce that un, strongly converges to 0 in L2(Rd), that is:

    limnun,L2(Rd)=0.

    The last step is to establish that:

    limnsuppunL2(Vp)=0.

    Let ε>0. Since ˜a belongs to (B2(Rd))d×d and is uniformly continuous, a direct consequence of Proposition 12 gives the existence of R>0 such that:

    qP,˜aL(VqBR(2q)c)<ε2.

    In addition, since un strongly converges to 0 in L2unif, there exists NN such that:

    n>N,unL2unif(Rd)<ε2|BR|˜aL(Rd).

    Using the last two inequalities, we obtain for every qP:

    Vq|˜a(x)un(x)|2dxVqBR(2q)c|˜a(x)un(x)|2dx+VqBR(2q)|˜a(x)un(x)|2dx˜aL(VqBR(2q)c)VqBR(2q)c|un(x)|2dx+˜aL(Rd)VpBR(2q)|un(x)|2dx˜aL(VqBR(2q)c)suppunL2(Vp)+˜aL(Rd)|BR|unL2unif(Rd)ε2+ε2=ε.

    Therefore:

    limnsuppVp|˜a(x)un(x)|2dx=0.

    We next consider equation (62) and we use Lemma 4.1 to ensure that, up to the addition of a constant, un is the unique solution to:

    div(aperun)=div(fn+˜aun)in Rd.

    such that suppunL2(Vp)<. Then, estimate (41) established in Lemma 4.3 gives the existence of a constant C>0 independent of n such that:

    suppunL2(Vp)C(suppfnL2(Vp)+supp˜aunL2(Vp)).

    Letting n go to the infinity, we deduce that limnsuppunL2(Vp)=0. We can finally conclude that

    limnunB2(Rd)=0,

    and, since un satisfies (64), we have a contradiction.

    Lemma 4.6. Let fB2(Rd)d, there exists uH1loc(Rd) solution to (31) such that uB2(Rd)d.

    Proof. First of all, we remark that it is sufficient to prove this existence result when f(B2(Rd)C0,α(Rd))d. Indeed, if we denote Φ=(diva)1div the reciprocal linear operator from (B2(Rd)C0,α(Rd))d to (B2(Rd))d associated with equation (31) and we assume that Φ is well defined, Lemma 4.5 ensures it is continuous with respect to the norm of B2(Rd). Then, we are able to conclude in the general case using the density result stated in Proposition 8. In the sequel of this proof, we therefore assume that f belongs to C0,α(Rd)d.

    To start with, we show a preliminary result of regularity satisfied by the solutions to (31). Assuming f(B2(Rd)C0,α(Rd))d, we want to prove that a solution u to (31) such that uB2(Rd)d also satisfies uC0,α(Rd)d. Indeed, if u is such a solution to (31), a consequence of a regularity result established in [15,Theorem 5.19 p.87] (see also [14,Theorem 3.2 p.88]) gives the existence of C>0 such that for all xRd:

    uC0,α(B1(x))C(uL2unif(Rd)+fC0,α(Rd)). (67)

    Therefore, u belongs to (C0,α(Rd)B2(Rd))d.

    In the sequel of the proof, we use an argument of connexity adapted from [8]. Let P(a) the following assertion: "There exists a solution uD(Rd) to:

    div(au)=div(f)in Rd

    such that u(B2(Rd)C0,α(Rd))d."

    For t[0,1], we denote at=aper+t˜a and we define the following set:

    I={t[0,1] | s[0,t],P(as) is true}.

    Our aim is to show that P(a1)=P(a) is true. To this end, we will prove that I is non empty, closed and open for the topology of [0,1] and conclude that I=[0,1].

    I is not empty

    For t=0, the existence of a solution u such that uB2(Rd)d is a direct consequence of Lemma 4.4. We just have to use (67) to show the uniform Holder continuity of the gradient of the solution.

    I is open

    We assume there exists tI and we will find ε>0 such that [t,t+ε]I. For f(B2(Rd)C0,α(Rd))d, we want to solve:

    div((at+ε˜a)u)=div(f)in Rd, (68)

    where u(B2(Rd)C0,α(Rd))d. According to Proposition 9, for such a solution, we have ε˜au(B2(Rd)C0,α(Rd))d. Next, we remark that equation (68) is equivalent to:

    u=Φt(ε˜au+f), (69)

    where Φt is the reciprocal linear operator associated with the equation when a=at. Lemma 4.5 and estimate (67) imply the continuity of Φt from (B2(Rd)C0,α(Rd))d to (B2(Rd)C0,α(Rd))d for the norm \|.\|_{\mathcal{B}^2(\mathbb{R}^d)} + \|.\|_{\mathcal{C}^{0,\alpha}(\mathbb{R}^d)} . We fix \varepsilon such that:

    \varepsilon \left(\|{\tilde a}\|_{L^{\infty}(\mathbb{R}^d)} + \|{\tilde a}_{\infty}\|_{L^{\infty}(\mathbb{R}^d)}\right) \|\Phi_t\|_{\mathcal{L}\left(\left(\mathcal{B}^2(\mathbb{R}^d) \cap \mathcal{C}^{0,\alpha}(\mathbb{R}^d)\right)^d\right)} < 1.

    Therefore g \rightarrow \Phi_t(\varepsilon{\tilde a}g + f) \in \mathcal{L}\left(\left(\mathcal{B}^2(\mathbb{R}^d) \cap \mathcal{C}^{0,\alpha}(\mathbb{R}^d)\right)^d \right) is a contraction in a Banach space. Finally, we can apply the Banach fixed-point theorem to obtain the existence and the uniqueness of a solution to (69) and we deduce that [t,t+ \varepsilon] \subset \mathcal{I} .

    \mathcal{I} is closed

    We assume there exist a sequence (t_n) \in \mathcal{I}^{\mathbb{N}} and t\in [0,1] such that \lim_{n \rightarrow \infty} t_n = t and t_n<t . For every t_n , there exists u_n solution to:

    \begin{equation*} \label{eqsuite} -\operatorname{div}(a_{t_n}\nabla u_n) = f \quad \text{in $\mathbb{R}^d$}, \end{equation*}

    such that \nabla u_n \in \mathcal{B}^2(\mathbb{R}^d)^d . For every n \in \mathbb{N} , Lemma 4.5 gives the existence of a constant C_n such that:

    \|\nabla u_n\|_{B^2(\mathbb{R}^d)} \leq C_n \|f\|_{B^2(\mathbb{R}^d)}.

    We first assume that C_n is bounded independently of n by a constant C>0 . Therefore, up to an extraction, \nabla u_n weakly converges to a gradient \nabla u in L^2_{loc}(\mathbb{R}^d) and, using the lower semi-continuity of the L^2 -norm, we have

    \begin{align*} \|\nabla u\|_{L^2_{unif}(\mathbb{R}^d)} + \sup\limits_p \|\nabla u \|_{L^2(V_p)} & \leq \liminf\limits_{n \to \infty} \|\nabla u_n\|_{L^2_{unif}(\mathbb{R}^d)} + \sup\limits_p \|\nabla u_n \|_{L^2(V_p)} \\ & \leq C \|f\|_{\mathcal{B}^2(\mathbb{R}^d)}. \end{align*}

    In addition, for every n\in \mathbb{N} , u_n is a solution to the equivalent equation:

    \begin{equation} -\operatorname{div}(a_t \nabla u_n) = \operatorname{div}(f + (a_{t_n} - a_{t})\nabla u_n). \end{equation} (70)

    Next, since t_n converges to t , we directly obtain that a_{t_n} converges to a_t in \mathcal{B}^2(\mathbb{R}^d) . In addition, since \nabla u_n is bounded by a constant independent of n in L^2_{unif}(\mathbb{R}^d) , the sequence (a_{t_n} - a_{t})\nabla u_n strongly converges to 0 in L^2_{loc}(\mathbb{R}^d) . We can therefore consider the limit in (70) when n \to \infty and deduce that u is a solution to:

    \begin{equation*} \label{eqlimite} -\operatorname{div}(a_t \nabla u) = \operatorname{div}(f). \end{equation*}

    We have to prove that \nabla u \in \mathcal{B}^2(\mathbb{R}^d) . For every m, \ n \in \mathbb{N} , u_n - u_m is a solution to:

    -\operatorname{div}(a_t (\nabla u_n -\nabla u_m)) = \operatorname{div}((a_{t_n} - a_{t})\nabla u_n - (a_{t_m} - a_{t})\nabla u_m),

    and we have the following estimate:

    \|\nabla u_n - \nabla u_m\|_{\mathcal{B}^2(\mathbb{R}^d)} \leq C \|(a_{t_n} - a_{t})\nabla u_n - (a_{t_m} - a_{t})\nabla u_m\|_{\mathcal{B}^2(\mathbb{R}^d)}.

    Therefore, \nabla u_n is a Cauchy-sequence in \mathcal{B}^2(\mathbb{R}^d)^d and since this space is a Banach space, we directly obtain that \nabla u belongs to \mathcal{B}^2(\mathbb{R}^d)^d and we have the expected result.

    Now, we want to prove that C_n is bounded independently of n using a proof by contradiction. We assume there exist two sequences f_n and u_n such that \nabla u_n belongs to \mathcal{B}^2(\mathbb{R}^d)^d and:

    \begin{equation*} -\operatorname{div}(a_{t_n}\nabla u_n) = \operatorname{div}(f_n) \quad \text{in $\mathbb{R}^d$}, \end{equation*}
    \begin{equation*} \lim\limits_{n \rightarrow \infty}\|f_n\|_{\mathcal{B}^2(\mathbb{R}^d)} = 0, \end{equation*}
    \begin{equation*} \forall n \in \mathbb{N} \quad \|\nabla u_n\|_{\mathcal{B}^2(\mathbb{R}^d)} = 1. \end{equation*}

    For every n \in \mathbb{N} , the above equation is equivalent to:

    -\operatorname{div}(a_t \nabla u_n) = \operatorname{div}(f_n + (a_{t_n} - a_{t})\nabla u_n).

    We can next remark that the boundedness of \nabla u_n in \mathcal{B}^2(\mathbb{R}^d) ensures that the sequence (a_{t_n} - a_{t})\nabla u_n is strongly convergent to 0 in \mathcal{B}^2(\mathbb{R}^d) . Finally, we can conclude exactly as in the proof of Lemma 4.5.

    Since [0,1] is a connected space, we can finally conclude that \mathcal{I} = [0,1] . In addition, if u \in \mathcal{D}'(\mathbb{R}^d) is such that \nabla u \in \mathcal{B}^2(\mathbb{R}^d)^d\subset{L^2_{loc}(\mathbb{R}^d)^d} , the result of [12,Corollary 2.1] finally ensures that u \in L^2_{loc}(\mathbb{R}^d) .

    In the above proof, we have proved the following result:

    Corollary 4. Let f \in \left(\mathcal{B}^2(\mathbb{R}^d) \cap \mathcal{C}^{0,\alpha}(\mathbb{R}^d)\right)^d and u \in H^1_{loc}(\mathbb{R}^d) solution to (31) such that \nabla u \in \mathcal{B}^2(\mathbb{R}^d)^d . Then \nabla u \in \mathcal{C}^{0,\alpha}(\mathbb{R}^d)^d .

    Remark 6. Again, we do not need the restriction that we did on the dimension to prove the existence results stated in this section and we can easily generalize the existence of a solution to (31) in a two-dimensional context.

    To conclude this section, we finally give a proof of Theorem 1.1 and, therefore, we obtain the existence of a unique corrector solution to (14) such its gradient belongs to L^2_{per}(\mathbb{R}^d) + \mathcal{B}^2(\mathbb{R}^d) . To this end, we remark that corrector equation (14) is equivalent to a an equation of the form (31) and we use the preliminary results of uniqueness and existence proved in this section.

    Proof of theorem 1.1. Existence

    Let p be in \mathbb{R}^d . We want to find a solution to (14) of the form w_{per,p} + \tilde{w}_p where w_{per,p} is the unique periodic corrector (that is the unique periodic solution to the corrector equation (14) when {\tilde a} = 0 ) and such that \nabla \tilde{w}_p\in \mathcal{B}^2(\mathbb{R}^d)^d . First of all, we remark that equation (14) is equivalent to:

    \begin{equation*} \label{correcteur2} -\operatorname{div}((a_{per} + {\tilde a})\nabla \tilde{w}_p) = \operatorname{div}({\tilde a}(p + \nabla w_{per,p})) \quad \text{in $\mathbb{R}^d$}. \end{equation*}

    It is well known that \nabla w_{per,p} \in \left(L^2_{per}(\mathbb{R}^d) \cap \mathcal{C}^{0,\alpha}(\mathbb{R}^d)\right)^d and therefore, using the periodicity of \nabla w_{per,p} , we can easily show that {\tilde a}(p + \nabla w_{per,p}) \in \left(\mathcal{B}^2(\mathbb{R}^d)\cap \mathcal{C}^{0,\alpha}(\mathbb{R}^d)\right)^d . Then, the existence of \tilde{w}_p such that \nabla \tilde{w}_p \in \left(\mathcal{B}^2(\mathbb{R}^d)\cap \mathcal{C}^{0,\alpha}(\mathbb{R}^d)\right)^d is given by Lemma 4.6 and Corollary 4. Since \nabla \tilde{w}_p \in \left(\mathcal{B}^2(\mathbb{R}^d)\cap \ L^{\infty}(\mathbb{R}^d)\right)^d , the sub-linearity at infinity of \tilde{w}_p is a direct consequence of Proposition 11.

    Uniqueness

    We assume there exist two solutions u_1 and u_2 to (14) such that \nabla u_1 and \nabla u_2 belong to \left(L^2_{per}(\mathbb{R}^d) + \mathcal{B}^2(\mathbb{R}^d)\right)^d . We denote v = u_1-u_2 and we have \nabla v = g_{per} + \tilde{g} where g_{per} \in L^2_{per}(\mathbb{R}^d)^d and \tilde{g}\in\mathcal{B}^2(\mathbb{R}^d)^d . For every q\in \mathcal{P} , we have \tau_{q}\nabla v = g_{per} + \tau_{q}\tilde{g} by periodicity of g_{per} . Since \tilde{g} belongs to \mathcal{B} ^2(\mathbb{R}^d)^d , there exists \tilde{g}_{\infty} \in L^2(\mathbb{R}^d)^d such that \tau_q\nabla v converges in \mathcal{D}'(\mathbb{R}^d) to \nabla v_{\infty} = g_{per} + \tilde{ g}_{\infty} when |q| \rightarrow \infty . In addition, considering the limit in equation (14), we obtain that v_{\infty} is a solution to:

    -\operatorname{div}((a_{per}+ {\tilde a}_{\infty})\nabla v_{\infty}) = 0 \quad \text{in } \mathbb{R}^d.

    Since a satisfies assumption (12) and (13), the coefficient a_{per} + {\tilde a}_{\infty} is a bounded and elliptic matrix-valued coefficient. Therefore, the result established in [10,Lemma 1] allows us to conclude that g_{per} = 0 and finally, that \nabla v = \tilde{g} \in \mathcal{B}^2(\mathbb{R}^d)^d . Since v is a solution to:

    -\operatorname{div}((a_{per} + {\tilde a})\nabla v) = 0 \quad \text{in } \mathbb{R}^d,

    we use Lemma 4.1 to obtain that \nabla v = 0 and the uniqueness is proved.

    In this section we use the corrector, solution to (14) and defined in Theorem 1.1, to establish an homogenization theory similar to that established in [6] for the periodic case with local perturbations. In Proposition 13 we first study the homogenized equation associated with (1) and we conclude showing estimates (15) and (16) stated in Theorem 1.2.

    To start with, we determine here the limit of the sequence u^{\varepsilon} of solutions to (1). In Proposition 13 below we show the homogenized equation is actually the diffusion equation (3) where the diffusion coefficient a^* is defined by (5), that is the homogenized coefficient is the same as in the periodic case when a = a_{per} . This phenomenon is similar to the results established in [8] in the case of localized defects of L^p(\mathbb{R}^d) . It is a direct consequence of Proposition 10 regarding the average of the functions in \mathcal{B}^2(\mathbb{R}^d) which is satisfied by our perturbations. The idea is that, on average, the perturbations belonging to \mathcal{B}^2(\mathbb{R}^d) therefore do not impact the periodic background.

    Proposition 13. Assume \Omega is an open bounded set of \mathbb{R}^d , let f \in L^2(\Omega) and consider the sequence u^{\varepsilon} of solutions in H^1_0(\Omega) to (1).Then the homogenized (weak- H^1(\Omega) and strong- L^2(\Omega) ) limit u^* obtained when \varepsilon \rightarrow 0 is the solution to (3) where the homogenized coefficient is identical to the periodic homogenized coefficient (5).

    Proof. We denote w = (w_{e_i})_{i \in \{1,...,d\}} , the correctors given by Theorem 1.1 for p = e_i . The general homogenization theory of equations in divergence form (see for instance [21,Chapter 6,Chapter 13]), gives the convergence, up to an extraction, of u^{\varepsilon} to a function u^* solution to an equation in the form (3). In addition, for all 1 \leq i,j \leq d , the homogenized matrix a^* associated with a is given by:

    \left[ a^* \right]_{i,j} = weak \lim\limits_{\varepsilon \rightarrow 0} a(./\varepsilon)(I_d + \nabla w(./\varepsilon)),

    where the weak limit is taken in L^{2}(\Omega)^{d \times d} . By assumption, we have a = a_{per} + {\tilde a} and we know that \nabla w_{e_i} = \nabla w_{per,e_i} + \nabla \tilde{w}_{e_i} where {\tilde a} \in \left(\mathcal{B}^2(\mathbb{R}^d) \cap L^{\infty}(\mathbb{R}^d)\right)^{d\times d} and \nabla \tilde{w}_{e_i}\; \in \left(\mathcal{B}^2(\mathbb{R}^d) \cap L^{\infty}(\mathbb{R}^d)\right)^{d} . Therefore, Corollary 2 ensures that |{\tilde a}(./\varepsilon)| and |\nabla \tilde{w}_{e_i}(./ \varepsilon)| converge to 0 in weak^*-L^{\infty} and, since a_{per} and \nabla w_{per} are bounded, we can deduce that:

    weak \lim\limits_{\varepsilon \rightarrow 0} a_{per}(./\varepsilon) \nabla \tilde{w}(./\varepsilon) + {\tilde a}(./\varepsilon)(I_d + \nabla w(./\varepsilon)) = 0.

    Consequently, we have

    \left[ a^* \right]_{i,j} = weak \lim\limits_{\varepsilon \rightarrow 0} a_{per}(./\varepsilon)(I_d + \nabla w_{per}(./\varepsilon)) = \left[ a_{per}^* \right]_{i,j}.

    This limit being independent of the extraction, all the sequence u^{\varepsilon} converges to u^* and we have the equality a^* = a^*_{per} .

    The existence of the corrector established in Theorem 1.1 allows to consider a sequence of approximated solutions defined by u^{\varepsilon,1} = u^* + \varepsilon \sum_{i = 1}^d \partial_{i}u^* w_{i}(./ \varepsilon) where for every i in \{1,...,d\} , we have denoted w_i = w_{e_i} . Our aim here is to estimate the accuracy of this approximation for the topology of H^1(\Omega) . In particular, we want to prove the convergence to 0 of the sequence R^{\varepsilon} defined by:

    \begin{equation*} \label{defR} R^{\varepsilon}(x) = u^{\varepsilon}(x) - u^*(x) - \varepsilon \sum\limits_{j = 1}^d w_j\left(\dfrac{x}{\varepsilon}\right)\partial_j u^*(x), \end{equation*}

    and specify the convergence rate in H^1(\Omega) .

    A classical method in homogenization used to obtain some expected quantitative estimates consists in defining a divergence-free matrix (as a consequence of corrector equation (14)) by

    M_k^i(x) = a_{i,k}^* - \sum\limits_{j = 1}^d a_{i,j}(x)(\delta_{j,k} + \partial_j w_k(x)),

    and to find a potential B which formally solves M = \operatorname{curl}(B) . Knowing that both the coefficient a and \nabla w belong to L^2_{per} + \mathcal{B}^2(\mathbb{R}^d) , we can split M in two terms and obtain M = M_{per} + \tilde{M} \in \left(L^2_{per}(\mathbb{R}^d) + \mathcal{B}^2(\mathbb{R}^d)\right)^{d\times d} . Therefore, we expect to find a potential of the same form, that is B = B_{per} + \tilde{B} . Rigorously, for every i,j \in \{1,...,d\} , we want to solve the equation:

    \begin{equation} - \Delta B^{i,j}_k = \partial_j M_k^i - \partial_i M_k^j \quad \text{in } \mathbb{R}^d. \end{equation} (71)

    The existence of a periodic potential B_{per} solution to M_{per} = \operatorname{curl}(B_{per}) is well known since, component by component, M_{per} is divergence-free. Here, the main difficulty is to show the existence of the potential \tilde{B} associated with the \mathcal{B}^2 - perturbation. This result is given by the following lemma.

    Lemma 5.1. Let \tilde{M} = \big( \tilde{M}^i_k\big)_{1 \leq i,k\leq d} \in \left(\mathcal{B}^2(\mathbb{R}^d)\right)^{d \times d} such that \operatorname{div}(\tilde{M}_k) = 0 for every k \in \{1, ...,d\} . Then, the potential \tilde{B}^{i,j}_k defined by:

    \begin{equation} \tilde{B}^{i,j}_k(x) = C(d) \int_{\mathbb{R}^d} \left( \dfrac{x_i - y_i}{|x-y|^d} \tilde{M}^j_k(y) - \dfrac{x_j - y_j}{|x-y|^d}\tilde{M}^i_k(y) \right) dy, \end{equation} (72)

    where C(d)>0 is a constant associated with the unit ball surface of \mathbb{R}^d , satisfies \nabla \tilde{B}^{i,j}_k \in \mathcal{B}^2(\mathbb{R}^d)^d and for all i,j,k \in \{1,...,d\} :

    \begin{align} - \Delta &\tilde{B}^{i,j}_k = \partial_j\tilde{M}_k^i - \partial_i \tilde{M}_k^j, \end{align} (73)
    \begin{align} & \tilde{B}_k^{i,j} = - \tilde{B}_k^{j,i}, \end{align} (74)
    \begin{align} & \sum\limits_{i = 1}^{d} \partial_i \tilde{B}_k^{i,j} = \tilde{M}_k^j. \end{align} (75)

    In addition, there exists a constant C_1>0 which only depends of the ambient dimension d such that:

    \begin{equation} \|\nabla \tilde{B}\|_{\mathcal{B}^2(\mathbb{R}^d)} \leq C_1 \|\tilde{M}\|_{\mathcal{B}^2(\mathbb{R}^d)}. \end{equation} (76)

    Proof. First, for every i,j,k\in \{1,...,d\} , equation (73) is equivalent to an equation of the following form:

    - \Delta \tilde{B}^{i,j}_k = \operatorname{div}\left(\mathcal{M}^{i,j}_k \right),

    where \mathcal{M}^{i,j}_k is a vector function defined by:

    \left(\mathcal{M}^{i,j}_k \right)_l = \left\{ \begin{array}{cc} \tilde{M}_k^i & \text{if } l = j, \\ -\tilde{M}_k^j & \text{if } l = i, \\ 0 & \text{else. } \\ \end{array} \right.

    Since \mathcal{M}^{i,j}_k \in \mathcal{B}^2(\mathbb{R}^d)^d , the existence of \tilde{B} and estimate (76) are given by Lemmas 4.3, 4.4 and 4.5 (here a_{per} \equiv 1 ). Equality (74) is a direct consequence of the definition of \tilde{B} . Property (75) can be easily obtained applying the divergence operator to (72).

    Corollary 5. The potential B = B_{per} + \tilde{B} , where \tilde{B} is given by Lemma 5.1, is the expected potential solution to (71). In addition, the couple (M,B) satisfies the following equalities:

    \begin{align*} \label{antisymetrie2} & B_k^{i,j} = - B_k^{j,i}, \\ \label{divergencepot2} & \sum\limits_{i = 1}^{d} \partial_i B_k^{i,j} = M_k^j. \end{align*}

    Now that existence of the potential B has been dealt with, we can remark that R^{\varepsilon} is a solution to the following equation:

    \begin{equation} -\operatorname{div}\left(a\left(\dfrac{x}{\varepsilon}\right)\nabla R^{\varepsilon}\right) = \operatorname{div}(H^{\varepsilon}) \quad \text{in } \Omega, \end{equation} (77)

    where:

    \begin{equation} H_i^{\varepsilon}(x) = \varepsilon \sum\limits_{j,k = 1}^d a_{i,j}\left(\dfrac{x}{\varepsilon}\right)w_{k}\left(\dfrac{x}{\varepsilon}\right) \partial_j \partial_k u^*(x) - \varepsilon \sum\limits_{j,k = 1}^d B_k^{i,j}\left(\dfrac{x}{\varepsilon}\right)\partial_j \partial_k u^*(x). \end{equation} (78)

    For a complete proof of equality (77), we refer to [6,Proposition 2.5].

    To conclude, we have to study the properties of H^{\varepsilon} . In particular, we next prove that both the corrector \tilde{w} and the potential \tilde{B} are bounded. This property is essential for establishing the estimates of Theorem 1.2.

    Lemma 5.2. The corrector w = \left(w_i\right)_{i \in \{1,...,d\}} defined by Theorem 1.1 and the potential B solution to (71) are in L^{\infty}(\mathbb{R}^d) .

    Proof. First, it is well known that both w_{per} and B_{per} belong to L^{\infty}(\mathbb{R}^d) . Next, for all i \in \{1,...,d\} , \tilde{w}_i solves:

    -\operatorname{div} \left(a_{per} \nabla \tilde{w}_i \right) = \operatorname{div}\left({\tilde a}\left(e_i + \nabla w_{per,i} + \nabla \tilde{w}_i \right)\right).

    We know the gradient of the corrector defined in Theorem 1.1 is in \mathcal{C}^{0,\alpha}(\mathbb{R}^d)^d . A direct consequence of Assumption (13) and Proposition 9 ensures that f = {\tilde a}\left(e_i + \nabla w_{per,i} + \nabla \tilde{w}_i \right) belongs to \left(L^{\infty}(\mathbb{R}^d) \cap \mathcal{B}^2(\mathbb{R}^d)\right)^d and the results of uniqueness and existence established in Lemmas 4.1 and 4.4 imply we have the following representation:

    \begin{equation*} \tilde{w}_i(x) = \int_{\mathbb{R}^d} \nabla_y G_{per}(x,y)f(y) dy. \end{equation*}

    We want to prove that the integral is bounded independently of x . We take x\in \mathbb{R}^d and denote p_x the unique element of \mathcal{P} such that x \in V_{p_x} . We define W_{p_{x}} = W_{2^{p_x}} such as in Proposition 4 and we split the integral in three parts:

    \begin{align*} \int_{\mathbb{R}^d} \nabla_y G_{per}(x,y)f(y) dy & = \int_{B_1(x)} \nabla_y G_{per}(x,y)f(y) dy \\ & + \int_{W_{p_x} \setminus B_1(x)} \nabla_y G_{per}(x,y)f(y) dy \\ & + \int_{ \mathbb{R}^d \setminus W_{p_x}} \nabla_y G_{per}(x,y)f(y) dy = I_1(x) + I_2(x) + I_3(x). \end{align*}

    We start by finding a bound for I_1(x) . To this end, we use estimate (36) for the Green function and we obtain

    \begin{align*} \left| I_1(x) \right| & \leq \|f\|_{ L^{\infty}(\mathbb{R}^d)} \int_{B_1(x)} \left| \nabla_y G_{per}(x,y)\right|dy \\ & \leq C \|f\|_{ L^{\infty}(\mathbb{R}^d)} \int_{B_1(x)} \dfrac{1}{\left|x-y\right|^{d-1}}dy \leq C \|f\|_{ L^{\infty}(\mathbb{R}^d)}. \end{align*}

    Where C denotes a positive constant independent of x . Indeed, the value of the integral in the last inequality depends only of the dimension d .

    Next, using Proposition 4, we know there exists C_1>0 and C_2>0 independent of x such that W_{p_x} \subset\; B_{C_12^{p_x}(x)} and the number of q \in \mathcal{P} such that V_q \cap W_{p_x} \neq \emptyset is bounded by C_2 . Therefore, using the Cauchy-Schwarz inequality, we have:

    \begin{align*} |I_2(x)| & \leq \int_{W_{p_x}\setminus{B_1(x)}} \dfrac{1}{|x-y|^{(d-1)}} |f(y)|dy \\ & \leq \left(\int_{W_{p_x}\setminus{B_1(x)}} \dfrac{1}{|x-y|^{2(d-1)}} dy\right)^{1/2}\left(\int_{W_{p_x}\setminus{B_1(x)}} |f(y)|^2dy\right)^{1/2} \\ & \leq C_2 \left(\int_{ B_{C_1 2^{p_x}(x)} \setminus{B_1(x)}} \dfrac{1}{|x-y|^{2(d-1)}} dy\right)^{1/2} \sup\limits_{p \in \mathcal{P}} \|f\|_{L^2(V_q)}. \end{align*}

    In addition since d>2 , we have:

    \begin{align*} \int_{ B_{C_1 2^{p_x}(x)} \setminus{B_1(x)}} \frac{1}{\left|x-y\right|^{2(d-1)}} dy & = \int_{ B_{C_1 2^{p_x}(0)} \setminus{B_1(0)}} \dfrac{1}{\left|y\right|^{2(d-1)}} dy \\ & \leq C\left(1 - \dfrac{1}{2^{|p_x|(d-2)}}\right), \end{align*}

    and therefore:

    I_2(x) \leq C \left(1 - \dfrac{1}{2^{|p_x|(d-2)}}\right)^{1/2} \leq C.

    Finally, to bound I_3(x) we split the integral on each cell V_q for q \in \mathcal{P} . Using the Cauchy-Schwarz inequality, we obtain:

    \begin{align*} \left|I_3(x)\right| & \leq \sum\limits_{q \in \mathcal{P}} \int_{ V_q \setminus{W_{p_x}}} \left|\nabla_y G_{per}(x,y)f(y) \right| dy \\ & \leq \sum\limits_{q \in \mathcal{P}} \left( \int_{V_q \setminus{W_{p_x}}} \left|\nabla_y G_{per}(x,y)\right| ^2 dy \right)^{\frac{1}{2}} \left(\int_{V_q \setminus{W_{p_x}}}\left|f(y)\right| ^2 dy\right)^{\frac{1}{2}} \\ & \leq \|f\|_{\mathcal{B}^2(\mathbb{R}^d)} \sum\limits_{q \in \mathcal{P} } \left( \int_{V_q \setminus{W_{p_x}}} \left|\nabla_y G_{per}(x,y)\right| ^2 dy \right)^{\frac{1}{2}}. \end{align*}

    We proceed exactly as in the proof of Lemma 4.3 (see the proof of estimate (53) for details) to obtain:

    \begin{align*} \sum\limits_{q \in \mathcal{P}} \left( \int_{V_q \setminus{W_{p_x}}} \left| \nabla_y G_{per}(x,y) \right|^2 dy \right)^{\frac{1}{2}} & \leq C \sum\limits_{q \in \mathcal{P}} \left( \int_{V_q \setminus{W_{p_x}}} \dfrac{1}{\left|x-y\right|^{2(d-1)}} dy \right)^{\frac{1}{2}} \\ & \leq C \sum\limits_{q \in \mathcal{P}} \frac{1}{2^{|q|(d-2)/2}} < \infty. \end{align*}

    Finally we have bounded the integral independently of x and we deduce that \tilde{w}_i \in \; L^{\infty}(\mathbb{R}^d) . With the same method we obtain the same result for B = B_{per} + \tilde{B} which allows us to conclude.

    Remark 7. The assumption d>2 is essential in the above proof and the boundedness of \tilde{w} in L^{\infty}(\mathbb{R}^d) may be false if d = 1 or d = 2 . If d = 1 we give a counter-example in Remark 9. The case d = 2 is a critical case and we are not able to conclude. This phenomenon is closely linked to the critical integrability of the function |x|^{-1} in L^2(\mathbb{R}^2) .

    Remark 8. As in the proofs of Lemmas 4.3 and 4.4, the above proof strongly uses the specific behavior of the Green function G_{per} and our approach consists in showing the convergence of a sum of the form \sum\limits_{q \in \mathcal{P}} \int_{V_q} \dfrac{1}{|x-y|^{d-1}}f(y) dy . Here, we explicitly use the geometric properties satisfied by the 2^p but, once again, this convergence is not specific to the set (9) and also holds under the generality of (H1), (H2) and (H3). We refer to Remark 4 for additional details.

    We are now able to give a complete proof of Theorem 1.2.

    Proof of Theorem 1.2. First, we use the explicit definition of H^{\varepsilon} given by (78) and a triangle inequality to obtain the following estimate:

    \|H^{\varepsilon}\|_{L^2(\Omega)} \leq \left( 1 + \|a\|_{L^{\infty}(\mathbb{R}^d)}\right) \| D^2 u^*\|_{L^2(\Omega)} \left(\| \varepsilon w(./\varepsilon) \|_{L^{\infty}(\Omega)} + \| \varepsilon B(./\varepsilon) \|_{L^{\infty}(\Omega)} \right).

    Applying Lemma 5.2, we obtain the existence of C>0 independent of \varepsilon such that

    \begin{equation} \|H^{\varepsilon}\|_{L^2(\Omega)} \leq C \varepsilon \| D^2 u^*\|_{L^2(\Omega)}. \end{equation} (79)

    Next, we use the following two estimates satisfied by R^{\varepsilon} :

    \begin{equation} \begin{split} \| R^{\varepsilon}\|_{L^2(\Omega)} \leq &C_1 \varepsilon (\| w(./\varepsilon) \|_{L^{\infty}(\Omega)} + \| B(./\varepsilon) \|_{L^{\infty}(\Omega)} ) \|f\|_{L^2(\Omega)} \\ &+ C_1\|H^{\epsilon}\|_{L^2(\Omega)}, \end{split} \end{equation} (80)

    and for every \Omega_1 \subset \subset \Omega :

    \begin{equation} \|\nabla R^{\varepsilon} \|_{L^2(\Omega_1)} \leq C_2 \left( \| H^{\varepsilon} \|_{L^2(\Omega)} + \| R^{\varepsilon} \|_{L^2(\Omega)} \right), \end{equation} (81)

    where C_1>0 and C_2>0 are independent of \varepsilon . These estimates are established for instance in [6] where the authors use the elliptic regularity associated with equation (77) and the properties of the Green function associated with the operator -\operatorname{div}(a^*\nabla.) on \Omega with homogeneous Dirichlet boundary condition. The first estimate is established in the proof of [6,Lemma 4.8] and the second estimate is a classical inequality of elliptic regularity proved in [6,Proposition 4.2] and applied to equation (77).

    In addition, an application of elliptical regularity to equation (3) provides the existence of C_3>0 such that:

    \begin{equation} \|u^*\|_{H^2(\Omega)} \leq C_3 \|f\|_{L^2(\Omega)}. \end{equation} (82)

    To conclude we use Lemma 5.2 to bound w and B and estimates (79), (80), (81) and (82). We finally obtain:

    \begin{equation*} \|R^{\varepsilon}\|_{L^2(\Omega)} \leq C \varepsilon \|f\|_{L^2(\Omega)}, \end{equation*}

    and

    \begin{equation*} \|\nabla R^{\varepsilon}\|_{L^2(\Omega_1)} \leq \tilde{C} \varepsilon \|f\|_{L^2(\Omega)}, \end{equation*}

    where C and \tilde{C} are independent of \varepsilon . We have proved Theorem 2.

    Remark 9. In the one-dimensional case, that is when d = 1 , we are not able to conclude in the same way. With an explicit calculation, we obtain:

    \begin{align*} &(u^{\varepsilon})'(x) = \left(a_{per} + {\tilde a}\right)^{-1} (x/\varepsilon)\left( F(x) + C^{\varepsilon}\right), \\ &(u^{*})'(x) = \left(a^*\right)^{-1} \left( F(x) + C^{*}\right),\\ &w(x) = - x + a^* \int_{0}^{x} \dfrac{1}{a_{per}(y)}dy - a^* \int_{0}^{x} \dfrac{{\tilde a}}{a_{per} \left(a_{per} + {\tilde a}\right)}(y) dy, \end{align*}

    where:

    \begin{align*} F(x) & = \int_{0}^x f(y)dy, \\ C^{\varepsilon} & = - \left( \int_0^1 (a_{per} + {\tilde a})^{-1}(y/\varepsilon) \right)^{-1} \int_0^1 (a_{per} + {\tilde a})^{-1}(y / \varepsilon) F(y) dy, \\ C^* & = - \int_0^1 F(y) dy. \end{align*}

    In this case,

    w_{per}(x) = - x + a^* \int_{0}^{x} \dfrac{1}{a_{per}(y)}dy,

    and

    \tilde{w}(x) = - a^* \int_{0}^{x} \dfrac{{\tilde a}}{a_{per} \left(a_{per} + {\tilde a}\right)}(y) dy,

    and we can show the corrector w is not necessarily bounded. However, the results of Proposition 10, allow us to obtain the following estimate:

    \begin{equation*} \label{estimate_one_dimensionnal} \|\left(R^{\varepsilon}\right)'\|_{L^2(\Omega)} \leq C \varepsilon^{\frac{1}{2}}\left|\log(\varepsilon)\right|^{\frac{1}{2}}. \end{equation*}

    As an illustration, we can consider \Omega = ]0,1[ , a_{per} = 1 and {\tilde a} = \sum\limits_{p \in \mathbb{Z}} \tau_{-2^p}\varphi , where \varphi is a positive function of \mathcal{D}(\mathbb{R}) , \|\varphi\|_{L^{\infty}} = 1 , \int_{\mathbb{R}} \varphi> 0 and Supp(\varphi) \in [0,1/2] . With these parameters, for every x\in \Omega , we have:

    |\tilde{w}(x/\varepsilon)| = \int_{0}^{x/\varepsilon} \dfrac{{\tilde a}}{1 + {\tilde a}}(y) dy \geq \frac{1}{2} \sum\limits_{0\leq p < [\log_2(x/\varepsilon)]} \int_{0}^{1/2} \varphi \stackrel{\varepsilon \rightarrow 0}{\longrightarrow} + \infty.

    And therefore, the corrector is actually not bounded.

    Remark 10. The result of Theorem 1.2 ensures that the corrector introduced in Theorem 1.1 allows to precisely describe the behavior of the sequence u^{\varepsilon} in H^1 using the approximation defined by u^{\varepsilon,1} = u^* + \varepsilon \sum\limits_{i = 1}^d \partial_i u^* w_i(./\varepsilon) . However, since the perturbations of \mathcal{B}^2(\mathbb{R}^d) are "small" at the macroscopic scale (in the sense of average given by (25)), we can naturally expect that it is also possible to approximate u^{\varepsilon} in H^1 considering the sequence u^{\varepsilon,1}_{per} : = u^* + \varepsilon \sum\limits_{i = 1}^d \partial_i u^* w_{per,i}(./\varepsilon) which only uses the periodic part w_{per} of our corrector. To this aim, we can show that u^{\varepsilon} - u^{\varepsilon,1}_{per} is solution to

    -\operatorname{div}\left(a\left(\frac{.}{\varepsilon}\right) \nabla (u^{\varepsilon} - u^{\varepsilon,1}_{per})\right) = \operatorname{div}\left(H^{\varepsilon}_{per}\right) \quad \text{on } \Omega,

    where the right-hand side

    \begin{equation} H^{\varepsilon}_{per} : = - a\left(\dfrac{.}{\varepsilon}\right)\left( \nabla \left( u^{\varepsilon} - u^{\varepsilon,1}\right) + \varepsilon \sum\limits_{i = 1}^d \nabla \partial_i u^* \tilde{w}_i(./\varepsilon) + \sum\limits_{i = 1}^d \partial_i u^* \nabla \tilde{w}_i(./\varepsilon)\right), \end{equation} (83)

    strongly converges to 0 in L^2 when \varepsilon \to 0 . A method similar to that used in the proof of Theorem 1.2 therefore allows to show the convergence to 0 of u^{\varepsilon} - u^{\varepsilon,1}_{per} in H^1 . It follows, at the macroscopic scale, that the choice of our adapted corrector instead of the periodic corrector seems to be not necessarily relevant in order to calculate an approximation of u^{\varepsilon} in H^1 . However, the choice of the periodic corrector is not adapted if the idea is to approximate the behavior of u^{\varepsilon} at the microscopic scale \varepsilon . Indeed, at this scale, the perturbations of the periodic background can be possibly large and it is necessary to consider a corrector that take these perturbations into account. Particularly, if H^{\varepsilon}_{per} is the function defined by (83), we can easily show that H^{\varepsilon}_{per}(\varepsilon.) does not converge to 0 in any ball B_R such that \varepsilon B_R \subset \Omega , which formally reflects a poor quality of the approximation of u^{\varepsilon} by u^{\varepsilon,1}_{per} at the scale \varepsilon . This fact particularly motivates the choice of our adapted corrector in order to approximate u^{\varepsilon} . We refer to [17] for a rigorous formalization of the above argument.

    The author thanks the anonymous referee for many constructive comments. The author also thanks Claude Le Bris and Xavier Blanc for suggesting this problem, for their support, and for many fruitful discussions.



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