Homogenization at different linear scales, bounded martingales and the two-scale shuffle limit

DOI:10.1051/cocv/2012039 www.esaim-cocv.org

HOMOGENIZATION AT DIFFERENT LINEAR SCALES, BOUNDED MARTINGALES AND THE TWO-SCALE SHUFFLE LIMIT

K´ evin Santugini

Abstract.In this paper, we consider two-scale limits obtained with increasing homogenization periods, each period being an entire multiple of the previous one. We establish that, up to a measure preserving rearrangement, these two-scale limits form a martingale which is bounded: the rearranged two-scale limits themselves converge both strongly in L² and almost everywhere when the period tends to +∞. This limit, called the Two-Scale Shuffle limit, contains all the information present in all the two-scale limits in the sequence.

Mathematics Subject Classification. 35B27.

Received September 15, 2011. Revised June 7, 2012 Published online July 4, 2013.

1. Introduction

Homogenization is used to study the solutions to equations when there are multiple scales of interest, usually a microscopic one and a macroscopic one. In particular, one may consider the solutionsu_εto a partial differential equation with locally ε-periodic coefficients and study their behavior as the small periodε tends to 0. Two- scale convergence, introduced by Nguetseng [10] and Allaire [1], is suited to study this particular subset of homogenization problems called periodic homogenization. It was later extended to the case of periodic surfaces by Neuss Radu [8,9] and Allaire, Damlamian and Hornung [2]. It can also be used in the presence of periodic holes in the geometry, see [5,6] or to homogenize multilayers [12,13].

Intuitively, two-scale convergence introduces the concept of two-scale limit u₀which is a function of both a macroscopic variablex – also called slow variable – and a microscopicp-periodic variable y – also called fast variable – such that, in some “meaning”,x→u₀(x,x/ε) is a good approximation ofu_ε.

As indicated by its name, two-scale convergence captures the behavior at two scales: the macroscopic one and the pε-periodic one. However, two-scale convergence does not capture all phenomena that happens at a scale linear in ε but only those whose length scale is pε/m where m is an integer. The two-scale limit of a sequence depends not only on the asymptotic scale, but also on the precise value of the chosen period. For example, any phenomena happening at the length scale of 2εwill not be fully apparent in the two-scale limit computed with periodε. The two-scale limit computed with period 2εwill contain no less – and might actually

Keywords and phrases.Homogenization, two-scale convergence.

1 University Bordeaux, IMB, UMR 5251, 33400 Talence, France.Kevin.Santugini@math.u-bordeaux1.fr

2 CNRS, IMB, UMR 5251, 33400 Talence, France.

3 INRIA, 33400 Talence, France.

Article published by EDP Sciences c EDP Sciences, SMAI 2013

contain more – information than the two-scale limit computed with periodε. For example, the homogenization of sin(2πx/ε) + sin(πx/ε) gives a two-scale limit ofu₀: (x, y)→sin(2πy) if computed with the homogenization periodε,i.e., whenp= 1, andu₀: (x, y)→sin(2πy) + sin(πy) if computed with the homogenization period 2ε, i.e., whenp= 2. Furthermore, if we choosep= 1/2, then the two-scale limit is none other than the null function.

Worse, the scale factorpcould be irrational.

The choice of the scale factor p used in the homogenization process is therefore of utmost importance in two-scale convergence. Using a badly chosen scale factorpmay and will often cause a huge loss of information.

At worst, we recover no more information than the one obtained by the standard weak L² limit: if pεis the correct choice of homogenization period, the two-scale limit computed with periodλpεwhereλis an irrational number should, intuitively, carry no information about what happens at scalepε.

Fortunately, there is usually a natural choice of period: the coefficients of the partial differential equation are often chosen locally ε-periodic. The most natural choice is to choose p = 1, i.e., to consider the correct microscopic scale for u_ε is ε itself. If there are two important periods to consider pε and pε, the intuitive solution is to choose a period that is an entire multiple of both. However, this can only be done if the ratiop/p between the two scale factors is a rational number.

When the two-scale limit depends on the fast variable, we may consider an homogenization period of p₂ε instead ofp₁εwherep₂/p₁is a positive integer. The two-scale limit computed with the homogenization periodp₂ε contains more information than the two-scale limit computed with the homogenization period p₁ε. It is then natural to study the behavior of the two-scale limit as the scale factor tends to +∞. Allaire and Conca studied in [3] a similar problem and established, for an elliptic problem, the behavior of the spectra of the equation satisfied by the two-scale limit as the scale factorpgoes to +∞. Ben Arous and Owhadi [4] studied the behavior of the Brownian motion in a periodic potential using multiscale homogenization when the ratio between two successive scales is bounded from above and below.

In this paper, we consider various two-scale limits, each computed with a different homogenization period. In particular, we consider a sequence of periods (p_n)_n∈N such that for all integersn,p_n+1/p_n is a positive integer and we study the two-scale limit of (u_ε)_ε>0computed with the homogenization periodp_nε. This two-scale limit, denoted u_0,pn, is p_n-periodic in each component of its fast variable. Since p_n+1 is always an entire multiple ofp_n, one can always recover the two-scale limitu_0,pn from the two scale limitu_0,pn+1. Ifp_n+1=m_np_n and in dimensiond≥1:

u_0,pn(x,y) = 1 m^d_n

α∈0,mn−1^d

u_0,pn+1(x,y+p_nα).

The sequence of two-scale limits (u_0,pn)_n∈Nyields increasing information on the asymptotic behavior of (u_ε)_ε>0. A natural question is whether the two-scale limits u_0,pn themselves converge whenever n tends to +∞. I.e., does there exist a function that carry the information of all thep_n-two-scale limits? The goal of our paper is to answer this question. The answer is positive. We show in this paper that the sequence of two-scale limits is, after a measure preserving rearrangement, a bounded martingale in L² and therefore converges both strongly in L² and almost everywhere to a function we call the Two-Scale Shuffle limit.

In Section 2, we remind the reader of previously known results: two-scale convergence and the convergence properties of bounded martingales. In Section3, we show how the different two-scale limits are related to each other through martingale-like equalities and explain how to transform these two-scale limits to get a genuine martingale. This leads to our stating of our main theorem: Theorem 3.8 in which we show that in a certain meaning the two-scale limits themselves converge to the Two-Scale Shuffle limit. In addition, we also state in Corollary3.9that all the information present in all the two-scale limits is contained in the Two-Scale Shuffle limit. In Section4, we use this result on the heat equation in multilayers with transmission conditions between adjacent layers and establish, for this particular example, the equation satisfied by the Two-Scale Shuffle limit in Theorem4.1.

2. Notations, prerequisites and known results

Throughout this paper, ifxis inR, we denote by xthe integer part ofx. We also denote by n₁, n₂the set [n₁, n₂]∩N. To make the present paper as self-contained as possible, we recall in this section known results on the two main mathematical tools we use to prove our main theorem: two-scale convergence in Section 2.1, and classical results on the convergence of bounded martingales in Section2.2.

2.1. The classical notion of two-scale convergence

First, as in [1], we introduce some notations. In this paper,palways refer to a scale factor. It remains constant while taking the two-scale limit. However, the goal of this paper is to observe the behavior of the two-scale limits asptends to +∞.

By Ω, we denote a bounded open domain ofR^d whered≥1. By Y_p, we denote the cube [0, p]^d. By L²_#(Y_p), we denote the space of measurable functions defined over R^d, that are p-periodic in each variable and that are square integrable over Y_p. By C_#(Y_p), we denote the set of continuous functions defined on R^d that are p-periodic in each variable.

We reproduce the now classical definition of two-scale convergence found in [1,10]. For convenience, we added the scale factorp.

Definition 2.1 (two-scale convergence).Letpbe a positive real. A sequence (u_ε)_ε>0belonging to L²(Ω) is said top-two-scale converge if there existsu_0,pin L²(Ω×Y_p) such that:

ε→0lim

Ωu_ε(x)ψ x,x

ε

dx= 1 p^d

Ω

Yp

u_0,p(x,y)ψ(x,y) dydx, (2.1) for allψin L²(Ω;C#(Y_p)).

It is a common abuse of notation to also designate by u_0,p the unique extension of u_0,p to Ω×R^d that is p-periodic in the lastdvariables.

Allaire, see [1], and Nguetseng, see [10], proved that any sequence of functions bounded in L²has a subsequence that two-scale converges. Let’s reproduce this precise compactness result.

Theorem 2.2. Let(u_ε)_ε>0be a sequence of functions bounded inL²(Ω). Then, there existu_0,pinL²(Ω×]0, p[^d) and a subsequenceε_k converging to0 such that

k→∞lim

Ωu_εk(x)ψ

x, x ε_k

dx= 1 p^d

Ω

Yp

u_0,p(x,y)ψ(x,y) dydx, (2.2) for allψ inL²(Ω;C#(Y_p)).

Proof. See Allaire [1], Theorem 1.2 and Nguetseng [10], Theorem 2. The presence of the scale factorphas no

impact on the proof.

We also have the classical proposition

Proposition 2.3. Let u_εp-two-scale converges tou_0,p. Then, 1

p^d/2 u0,p L²(Ω×Yp)≤lim inf

ε→0 uε L²(Ω).

Proof. See Allaire [1], Proposition 1.6. The presence of the scale factorphas no impact on the proof.

The next proposition is easy to derive from Theorem2.2.

Proposition 2.4. Let (p_n)_n∈N be an increasing sequence of positive real numbers. Let(u_ε)_ε>0 be a sequence of functions bounded inL²(Ω). Then, there exist a subsequence(ε_k)_k∈Nconverging to0, and a sequence of functions u_0,pn inL²(Ω×]0, p_n[^d)such that, for any non-negative integern, the sequence(u_εk)_k∈Np_n-two-scale converges tou_0,pn. I.e., such that for all integersn:

k→∞lim

Ω

u_εk(x)ψ

x, x ε_k

dx= 1 p^d_n

Ω

Y_pn

u_0,pn(x,y)ψ(x,y) dydx, for allψ inL²(Ω;C_#(Y_pn)).

Proof. Apply Theorem2.2multiple times and proceedviadiagonal extraction.

Our goal in this paper is to study the limit ofu_0,pn as p_n tends to +∞.

2.2. Convergence of bounded martingales

In this section, we recall the notions of probability theory needed to prove our main theorem. In particular, we are interested in using the convergence properties of bounded martingales. For more details, the reader may consult [7]. We assume the reader to be familiar with the notions ofσ-field andσ-additivity in measure theory.

We use the following common notations:

• IfC is a subset of P(X), we denote byσ(C) the smallestσ-field inX that containsC.

• If D is a topological space, we denote by B(D) the set of all Borel sets in D, i.e., the smallest σ-field containing all the open subsets ofD.

Definition 2.5 (measurable space).A pair (X,F) is said to be a measurable space if F is aσ-field inX. Definition 2.6 (measure space). A triplet (X,F, μ) is said to be a measure space if (X,F) is a measurable space and ifμis a positiveσ-additive measure on (X,F).

A measure space (X,F, μ) is said to be finite ifμ(X)<+∞. A measure space (X,F, μ) is said to beσ-finite ifX is the countable union ofF-measurable sets of finite measure. A measure space (X,F,P) is said to be a probability space ifP(X) = 1.

We start by recalling the definition of conditional expectation, see [7] (Chapter 6, Thm. 6.1) for more details.

Usually, the conditional expectation is defined for probability spaces. The definition extends without problem to finite measure spaces and even, to some extent, toσ-finite measure spaces.

Definition 2.7 (conditional expectation). Let (X,F, μ) be a measure space with μ being positive and σ-additive. LetG be a σ-field such that G ⊂ F and (X,G, μ) is alsoσ-finite. Letf :X →Rbe F-measurable and inL¹_loc(X, μ). The conditional expectation of f with respect to theσ-field G is denoted by E(f|G), and is defined as the unique, up to a modification on a set of null measure,G-measurable functiongsuch that

B

g(ω) dω=

B

f(ω) dω, for allB inG.

The existence of the conditional expectation is given by Radon–Nikodym theorem. The measureμneed not be a probability measure. However, to apply Radon-Nykodim theorem, (X,G, μ) needs to be σ-finite, hence the restriction in the definition. A statement and a proof of the Radon-Nikodym theorem can be found in [11], Theorem 6.10.

It is not enough that (X,F, μ) beσ-finite in Definition2.7.

Remark 2.8. WhenG ⊂ F, it does not follow from (X,F, μ) beingσ-finite that (X,G, μ) is also σ-finite. A counter-example is easily obtained by settingG:={∅, X}wheneverμ(X) = +∞.

In our main theorem, we restrict ourselves to the case of finite measures. However, Remark2.8 will explain why the martingale approach doesn’t quite work for the most natural attempt to define a convergence for two-scale limits, see Section3.1.

In order to define martingales, we remind the reader of the definition of filtration. We limit ourselves to filtrations indexed by the setN. See [7], chapter 7, p. 120, for more details.

Definition 2.9 (filtrations). Let (X,F) be a measurable space. A sequence (Fn)_n∈Nof σ-fields,Fn ⊂ F is a filtration if, for all non-negative integersn,Fn is a subset ofF_n+1.

We now recall the definition of martingales.

Definition 2.10 (martingales). Let (X,F, μ) be a σ-finite measure space. Let (Fn)_n∈N be a filtration on (X,F, μ) such that (X,F₀, μ) isσ-finite.

A sequence (f_n)_n∈Nis said to be a (Fn)_n∈N-martingale, if for all non-negative integersnandj, f_n=E(f_n+j|Fn).

I.e., iff_n isFn-measurable and if for allF in Fn:

F

f_n(ω) dω=

F

f_n+j(ω) dω. (2.3)

We now reproduce the convergence results of bounded martingales:

Theorem 2.11 (convergence of bounded martingales). Let (X,F, μ)be a measure space with finite measure.

Let (Fn)_n∈N be a filtration on the measurable space (X,F). Let q be in ]1,+∞[. Let (f_n)_n∈N be a (Fn)_n∈N- martingale such that the sequence (f_n)_n∈N is bounded in L^q(X). Then, the sequence (f_n)_n∈N converges both almost everywhere and strongly in L^q(X,P).

Proof. See [7], Corollary 7.22 for the strong L^q convergence. The almost everywhere convergence is stated in [7], Theorem 7.18 and holds even forq= 1. While these two results are stated for probability measures, the finite measures case is easily deduced from the probability measure case by considering the probability measure

μ(·)/μ(X).

The above theorem extends, at least partially, to σ-finite measures:

Remark 2.12. In Theorem 2.11, if the probability space (X,F,P) is replaced with σ-finite measure space (X,F, μ) such that (X,F0, μ) is alsoσ-finite, then the bounded martingales converge almost everywhere and at least inL^q_loc. It is unknown to the author if the strongL^q convergence can be generalized to theσ-finite case.

3. Two-scale limits and bounded martingales

In this section, we always assume both of the following assumptions are satisfied:

Assumption 3.1 (integer scale ratios).We are given a real sequence(p_n)_n∈N, such that for allninN,p_n >0 and p_n+1 is an entire multiple of p_n. Moreover, we set for n ≥ 1, m_n := p_n/p_n−1 ∈ N, and for n ≥ 0, M_n:=p_n/p₀∈N.

Assumption 3.2. We are given a sequence of functions(u_ε)_ε>0 bounded in L²(Ω) and a decreasing sequence of positive(ε_k)_k∈Nsuch that the sequence (u_εk)_k∈N p_n-two-scale converges for all integersnto a function u_0,pn that belongs to L²(Ω×]0, pn[^d).

This last assumption is justified by Proposition2.4.

Our goal is to study the convergence of the two-scale limitsu_0,pnwhen ngoes to infinity. In this section, we proceed as follows: we begin by establishing a useful equality that looks like a martingale equality in Section3.1, then we propose a rearrangement of the two-scale limits in Section3.2, and finally propose another rearrangement of the two-scale limits in Section3.3, the shuffle, which transform the sequence of two-scale limits into a bounded martingale.

3.1. An almost martingale equality

Proposition 3.3. Suppose both assumptions3.1and3.2 are satisfied. Then, for allj inN, alln inN, almost allx inΩand almost ally inY_p:

u_0,pn(x,y) = p_n

p_n+j

_d

α∈0,pn+j/pn−1^d

u_0,pn+j(x,y+αpn). (3.1)

Proof. Letφbelong toC^∞(Ω×R^d) bep_n-periodic in the lastdvariables. Sincep_n+j/p_n is an integer,φis also p_n+j-periodic in the lastdvariables. We take the limit of

Ωu_ε(x)φ(x,x/ε) dx, asεtends to 0, in the sense of two-scale convergence for both scale factorsp_n+j andp_n:

1 p^d_n

Ω

Y_pn

u_0,pn(x,y)φ(x,y) dydx= 1 p^d_n+j

Ω

Y_pn+j

u_0,pn+j(x,y)φ(x,y) dydx

= 1

p^d_n+j

Ω

Y_pn

α∈(0,pn+j/pn−1^d

u_0,pn+j(x,y+αp)

φ(x,y) dydx.

The most natural approach is to consider theu_0,pnas functions defined over Ω×R^dand to study their convergence in some meaning in Ω×R^d. Such a convergence result would be ideal as the intuitive meaning of the limit would be easy to grasp. Equality (3.1) is similar to the martingale defining equality (2.3). Would it be possible to use the classical convergence properties of martingales, see Theorem2.11, to prove the existence of a limit to the u_0,pn? Unfortunately, the martingale approach doesn’t work in this setting but the attempt is, nevertheless, instructive. First, we try to construct a filtration (Fn)_n∈N for the u_0,pn. For all positive integer n, the u_0,pn are p_n-periodic with respect to the last dvariables. LetFn be the set of all Borel subsets of Ω×R^d that are invariant by translation of ±pn along any of the last ddirections of Ω×R^d. Clearly,u_0,pn is Fn-measurable.

However, the measure space (Ω×R^d,Fn, μ) whereμis the Lebesgue measure is notσ-finite: anyFn measurable subset of Ω×R^d is either of null measure or of infinite measure. Therefore, the concept of (Fn)_n∈N-martingale is ill-defined, see Definition2.10and Remark2.8. Should we attempt to verify whether the martingale defining equality (2.3) hold, we would get either +∞or 0 on both sides of the equation.

However, the martingale defining equality (2.3) is satisfied if one replaces the Lebesgue integral ofR^d by the limit of the mean over a ball as its radius tends to +∞.I.e., we have for allF inFn

R→+∞lim 1

|B(0, R)|

Ω

B(0,R)

½{(x,y)∈F}u_0,pn(x,y) dydx= lim

R→+∞

1

|B(0, R)|

×

Ω

B(0,R)

½{(x,y)∈F}u_0,pn+j(x,y) dydx, whereB(0, R) is the open ball ofR^dcentered on 0 and of radiusRand where|A|is the Lebesgue measure of set A. Unfortunately, we were unable to derive a direct convergence result using this pseudo-martingale equality.

To proceed further, we need to transform the two-scale limitsu_0,pn in order to get genuine martingales.

3.2. Rearrangement of the two-scale limits with integers

In the previous section, we established a “martingale-like” equality for the two-scale limits u_0,pn. To get genuine martingales in the sense of Definition 2.10, we need to rearrange the u_0,pn. While we are unable to prove a convergence for the rearrangement of the two-scale limits presented in this section, the ideas behind this rearrangement provide insight on the next section where we introduce another rearrangement and prove its convergence.

In this section, we rearrange theu_0,pnby introducing a new variableαthat belongs toZ^d. The rearrangement, denoted byv_pn, depends on the slow variablex∈Ω, on a fast variabley∈[0, p₀[^d, and on the new variableα.

To rearrange the u_0,pn into the v_pn, we subdivide Ω×[0, p_n[^d intoM_n^d= (p_n/p₀)^d sets Ω×_d

i=1[α_i, α_i+p₀[.

Each of these sets is the product of Ω with an hypercube indexed byα= (α₁, . . . , α_d) and we definev_pn(·,α,·) as taking in Ω×[0, p₀[^d the same values u_0,pn does in Ω×_d

i=1[α_i, α_i+p₀[. The variable y represents the position of the fast variable inside each hypercube.I.e., we set:

v_pn : Ω×Z^d×Y_p0 →R,

(x,α,y)→u_0,pn(x,y+p₀α).

We have the following proposition

Proposition 3.4. For allninN, for almost allxinΩandyinY_p0, theα-indexed sequence(v_pn(x,α,y))_α∈Zd

isM_n-periodic in each direction ofα. Moreover:

v_pn(x,α,y) = M_n

M_n+j

_d

β∈0,Mn+j/Mn−1^d

v_pn+j(x,α+M_nβ,y), (3.2)

for allα in Z^d.

Proof. This is a direct consequence of Proposition3.3.

This in turn should encourage us to look at the following problem.

Problem 3.5. Let’s call “imbricated (M_n)_n-periodic d-dimensional sequences”, sequences that satisfy the following properties (t_n,α)_n∈N,α∈Zd such that

• for allninN, theα-indexed sequence (tn,α)_α∈Zd isM_n-periodic in each direction ofα,i.e. such that for all nin N, for allα inZ^d, and for allβin Z^d:

t_n,α=t_n,α+Mnβ,

• for allnin N, and for allα inZ^d,

t_n,α= M_n

M_n+j

_d

β∈0,Mn+j/Mn−1^d

t_n+j,α+Mnβ.

Study the convergence of (t_n,α)_n∈N,α∈Zd asntends to +∞. Under which condition does there exist a sequence t_∞,α such that for all non-negative integersn

t_n,α= lim

N→+∞

1 N^d

β∈0,N−1^d

t_∞,α+Mnβ.

or such that

t_n,α= lim

N→+∞

1 2^dN^d

β∈−N,N−1^d

t_∞,α+Mnβ.

or both?

By Proposition 3.4, for almost all x in Ω and y in Y_p0, the α-indexed sequences (vpn(x,α,y))_α∈Z^d are imbricated (M_n)_n-periodic d-dimensional sequences. Solving Problem 3.5 would be the first step in having a very elegant limit to the v_pn as a function defined on Ω×Z^d×Y_p0. Unfortunately, we do not have an answer for Problem3.5. While this sequence is morally a martingale with respect to the filtration made of the σ-fields {α+MnZ^d,α∈0, Mn−1^d}, it technically is not: we have the same problem we had in the previous section. To conclude with bounded martingales on the convergence, we would need a measureμonZ^dsuch thatμ(Z^d) = 1, invariant by translation and such thatμ(mZ^d) = 1/mwhenevermis an integer different from 0. Such a measure cannot be σ-additive. If we remove theσadditivity constraint, thenμexists: just set

μ(A) := lim

N→+∞

#(A∩−N, N^d) (2N+ 1)^d ·

It is unknown to the author if bounded martingales converge when they are defined on a nonσ-additive measure.

To avoid that problem, we introduce, in the next section, a different less natural rearrangement for the u_0,pn, the shuffle, for which we finally prove a convergence result.

3.3. Shuffle rearrangement of two-scale limits

In the previous section, we investigated a rearrangement where the set Ω×[0, p_n[^d was subdivided intoM_n^d subsets indexed byα∈Z^d. In this section, we finally construct a rearrangement, the shuffle, that results in a bounded martingale; thus establishing a convergence result for thep_n-two-scale limits asntends to +∞. To do so, we replace the variableαbelonging toZ^d with the variabley that belongs to [0,1[^d. Like the variableαof the previous section, the variabley indicates which hypercube of edge lengthp₀ we consider. The variable y remains unchanged and continue to represent the location inside the hypercube indexed by y. We set for x in Ω,y in [0, p₀]^d andy in [0,1]^d,

w_n(x,y,y) :=v_pn(x,α(y),y),

where α(y)_i =Mny_ifor all integersiin 1, d. Using Proposition3.4, we derive that for almost allxin Ω, yin Y_p0, (j, n) inN², andαin 0, Mn−1^d

_d

i=1[_Mn^αi,^αi_Mn⁺¹[w_n(x,y,y) dy=

β∈0,^Mn+j_Mn −1^d

_d

i=1[^Mnβi_Mn+j^+αi,^Mnβi_Mn+j^+αi+1[w_n+j(x,y,y) dy. (3.3) To transform thew_n into martingales, we need to shuffle the hypercubes as in Figure 1where, to simplify the drawing, homogenization was only performed on the last component ofR^d, hence the presence of layers instead of hypercubes. In that figure, we show one step of the rearrangement. As seen in the drawing, each step of the rearrangement is measure preserving, therefore the full rearrangement is also measure preserving. We needn−1 such steps to fully rearrangew_n.

To define rigorously this rearrangement, we begin by defining the function that maps the rearranged layer index onto the unrearranged layer index:

R_M,m:0, M m−1→0, M m−1 i→M ·(i modm) +

i m

· (3.4a)

The applicationR_M,m mapskm+j tojM+kwhenk belongs to0, M −1andj belongs to0, m−1. We also haveR_M,m◦R_m,M =R_m,M◦R_M,m= Id.

Then, we set the function that maps the rearranged layer onto the unrearranged one:

h^∗_M,m: [0,1[→[0,1[, y→ R_M,m(M my)

M m +

y−M my) M m

· (3.4b)

Original w

w

after one step

Same mean Same mean

Layer 0 Layer 1

Original w

Layer 0 Layer 2 Layer 4 Layer 1 Layer 3 Layer 5

Layer 0 Layer 1 Layer 2 Layer 3 Layer 4 Layer 5

Figure 1. One step of the measure preserving rearrangementM₁= 2 andM₂= 6.

0 1 2 3 4 5 6 7 8 9 10 11

0 6 1 7 2 8 3 9 10 4 5 11

0 6 2 8 10 4 1 7 3 9 5 11

Second step First step Original

Figure 2.Two steps of the measure preserving rearrangementM₁= 2,M₂= 6 andM₃= 12.

This represents only one step of the rearrangement on one component. For hypercubes, the permutation is the same but is done componentwise: we set

h_M,m: ]0,1[^d→]0,1[^d,

(y₁, . . . , y_n)→(h^∗_M,m(y₁), . . . , h^∗_M,m(y_n)).

And obtain one step of the rearrangement on all dcomponents. For the complete rearrangement on one com- ponent, see Figure2, we set

H_n^∗:=h^∗_Mn−1,mn◦. . .◦h^∗_M1,m2◦h^∗_M0,m1. (3.4c) To get the complete rearrangement on all components we set

H_n: [0,1[^d→[0,1[^d,

(y₁, . . . , y_n)→(H_n^∗(y₁), . . . , H_n^∗(y_n)). (3.4d)

We also have

H_n=h_Mn−1,mn◦. . .◦h_M1,m2◦h_M0,m1. The functionH_n shuffles the hypercubes_d

i=1[β_i/M_n,(β_i+ 1)/M_n[, hence we callH_n the shuffle function.

Finally, we define

w_n(x,y,y) :=w_n(x,y, H_n(y)). (3.5) This measure preserving rearrangement, the shuffle, is purposefully constructed so the w_n form a martingale for the following filtration ofσ-fieldsFn=B(Ω)× B([0, p0]^d)×σ_d

i=1

βi

Mn,^β_Mⁱ⁺¹

n

,β∈0, Mn−1^d . Remark 3.6. The above rearrangement of hypercubes is similar to the one used for computing in place the Discrete Fast Fourier Transform: the bit reversal. In the special case whereM_n= 2ⁿ, the rearrangement simply exchanges layersi,i.e., [i/2ⁿ,(i+1)/2ⁿ[, andi,i.e., [i/2ⁿ,(i+1)/2ⁿ[, wheniandiare bit reversal permutations of each other.I.e wheni=_n−1

j=0b_j2^j andi=_N−1

j=0 b_j2^n−1−j.

Remark 3.7. For generalM_n, the rearrangement of hypercubes is also a bit reversal but for a mixed basis. If M_ny=_n

j=1b_jM_j−1 withb_j in 0, m_j−1, then H_n^∗

⎛

⎝ 1 M_n

n j=1

b_jM_n M_j +

y−Mny M_n

⎞⎠=y.

We now state our main result as a self contained theorem.

Theorem 3.8 (Two-Scale Shuffle convergence).LetΩbe a bounded open domain ofR^d withd≥1. Let(u_ε)_ε>0 be a bounded sequence of functions belonging toL²(Ω). Let(p_n)_n∈Nbe an increasing sequence of positive numbers that satisfy Assumption 3.1. Set for all n ≥0 M_n := p_n/p₀ and for all n ≥ 1 m_n := p_n/p_n−1. Let (ε_k)_k∈N be a decreasing sequence of positive real numbers converging to0 such that the sequence (u_εk)_k∈N p_n-two-scale converges to u_0,pn for all non-negative integer n.

Set

w_n: Ω×[0, p₀]^d×[0,1]^d→R

(x,y,y)→u_0,pn(x, p₀M_nH_n(y)+y). whereH_n is defined by equations (3.4).

Then, the sequence w_n is a bounded martingale inL²(Ω×[0, p₀]^d×[0,1]^d), in the sense of Definition 2.10, for the filtration

Fn=B(Ω)× B([0, p₀]^d)×σ _d

i=1

β_i

M_n,β_i+ 1 M_n

,β∈0, Mn−1^d

. (3.6)

And, the sequencew_n converges both strongly inL²(Ω×[0, p₀]^d×[0,1]^d)and almost everywhere inΩ×[0, p₀]^d× [0,1]^d tow_∞, which we call the Two-Scale Shuffle limit. Moreover,

A

w_n(x,y,y) dydydx=

A

w_∞(x,y,y) dydydx, for all sets Ain Fn. I.e., by Definition2.7,w_n=E(w_∞|Fn).

Proof. Thew_n were constructed specifically so as to be a martingale for the filtration (3.6). To prove they are a martingale for the filtration (Fn)_n∈N, we only need to prove that for all non-negative integern, for almost allxin Ω, almost allyin [0, p₀]^d and for allβ in0, M_n−1^d, we have

_d

i=1[_Mn^βi ,^βi_Mn⁺¹[w_n+1(x,y,y) dy=

_d

i=1[_Mn^βi ,^βi_Mn⁺¹[w_n(x,y,y) dy.

I.e., we need to show that

_d

i=1[_Mn^βi ,^βi_Mn⁺¹[w_n(x,y, H_n(y)) dy=

_d

i=1[_Mn^βi ,^βi_Mn⁺¹[w_n+1(x,y, H_Mn,mn+1◦H_n(y)) dy. ButH_nmaps any hypercube_d

i=1[β_i/M_n,(β_i+1)/M_n[ to another hypercube_d

i=1[β_i/M_n,(β_i+1)/M_n[ andH_n is measure preserving. Therefore, we only need to prove that for almost allxin Ω, almost ally in [0, p₀]^d and for allβin 0, Mn−1^d

_d

i=1[_Mn^βi ,^βi_Mn⁺¹[w_n(x,y,y) dy=

_d

i=1[_Mn^βi ,^βi_Mn⁺¹[w_n+1(x,y, HMn,mn+1(y)) dy. is satisfied. But this equality is equivalent to

1

M_n^dv_pn(x,β,y) = 1 M_n+1^d

β∈0,mn+1−1^d

v_pn+1(x, RMn,mn+1(m_n+1β+β),y)

= 1

M_n+1^d

β∈0,mn+1−1^d

v_pn+1(x,(β+βM_n),y),

which is true by Proposition3.4. Therefore, the sequencew_n is a martingale for the filtration (Fn)_n∈N. By Proposition2.3, this martingale is bounded in L². It converges both strongly in L²(Ω×[0, p₀]^d×[0,1]^d)

and almost everywhere to a functionw_∞, see [7], Corollary 7.22.

Corollary 3.9. It is possible to recoveru_0,pnfrom the Two-Scale Shuffle limitw_∞. First, for allβ in0, M_n− 1^d, ally in_d

i=1[β_i/M_n,(β_i+ 1)/M_n[, and almost all(x,y)inΩ×[0, p₀]^d, we have

w_n(x,y,y) =M_n^d

_d

i=1[_Mn^βi ,^βi_Mn⁺¹[w_∞(x,y,y) dy

because w_n = E(w_∞|Fn). Since the shuffle function H_n is one to one from [0,1[^d to [0,1[^d, see Remark 3.7, we have w_n(x,y,y) = w_n(x,y, H_n⁻¹(y)). Finally, u_0,pn(x,y) is equal to the constant value taken by y → w_n(x,y−p₀y/p₀,y)wheny belongs to the hypercube[y/p₀/M_n,(y/p₀+ 1)/M_n[.

4. Application: heat equation in multilayers

In this section, we consider the multilayer heat equation with three spatial dimensions which we homogenize along the vertical space variable,i.e., along the direction perpendicular to the layers.

In [13], the author established the equations satisfied by the two-scale limits of the heat equation in multi- layers with transmission conditions between adjacent layers. When the magnitude of the interlayer conductivity between adjacent layers is weak, see [13], Section 6.1, the two-scale limit depends on the number of layers present in the homogenization period,i.e., on the scale factorsp_n. For given values of the slow variables (x, t), the two scale limit is piecewise constant in its scalar fast variable y and takes as many values as there are layers in a single homogenization cell. Our goal is to establish the equation satisfied by the limit of two-scale limits,i.e., the Two-Scale Shuffle limit, as defined in Theorem3.8.

To do so, we first recall previously known results in Section 4.1, then derive new results using Two-Scale Shuffle Convergence in Section4.2.

Γ

Γ

Γ

First layer

Second layer

Third layer Fourth layer

Fifth layer Sixth layer

Heat conductive material

Heat isolation

δ

δ

Figure 3.The multilayer geometry, six layers:N = 6.

4.1. The two-scale limit of the multilayer heat equation

We start by recalling some results we obtained in [13]. To avoid unnecessary complications, we consider here a simpler problem than the one considered in [13], System (4.1). Let Ω be B×]0,1[ where B is a convex bounded open subset ofR²with smooth boundary. Letδ, 0< δ <1/2. LetIbe the interval ]δ,1−δ[. For allN, letI_N be_N−1

j=0 ](j+δ)/N,(j+ 1−δ)/N[. Let Ω^N be the domain B×I_N. LetΓ_j^N,+ =B× {(j+δ)/N} and Γ_j^N,−=B× {(j−δ)/N}. LetΓ^N,+=_N−1

j=1 Γ_j^N,+ andΓ^N,−=_N−1

j=1 Γ_j^N,−. Let Γ_e=∂B×]0,1[∪Γ₀^N,+∪Γ_N^N,−. Let γ be the application on ∂Ω^N that maps u in H¹(Ω^N) to its trace on ∂Ω^N. Let γ be the trace operator swapped betweenΓ_j^N,+andΓ_j^N,−:i.e.,γu( ˆx,(j±δ)/N) =γu( ˆx,(j∓δ)/N) for all ˆxinBand alljin1, N−1. See Figure3where we schematized the three dimensional domain Ω^N by projecting it onto the two-dimensional plane.

LetA, K andJ be positive reals:Arepresents the heat conductivity inside Ω^N, andJ is the magnitude of the surfacic interlayer conductivity. For all positive integerN, we consider the multilayer heat equation

∂u_N

∂t −Au_N = 0 in Ω^N×R⁺ (4.1a)

with the boundary conditions A∂u_N

∂ν =

0 onΓ_e×R⁺

−^K_Nγu_N +_N^J(γu_N −γu_N) on (Γ^N,+∪Γ^N,−)×R⁺, (4.1b) and the initial condition

u_N(·,0) =u⁰_N. (4.1c)

We also define the energy

E^N(v) = A 2

Ω^N ∇v(x) ²dx+ K 2N

Γ^N,+∪Γ^N,−|γv|²dσ(x) + J

2N

N−1

j=1

B

v( ˆx,j+δ

N )−v( ˆx,j−δ N )

²d ˆx, for allv inH¹(Ω^N). We suppose

sup

N

E^N(u⁰_N)<+∞,