Homogenization of nonlinear integrals via the periodic unfolding method

Calculus of Variations

Homogenization of nonlinear integrals via the periodic unfolding method

Doina Cioranescu

, Alain Damlamian

, Riccardo De Arcangelis

aUniversité Pierre et Marie Curie (Paris VI), laboratoire J.-L. Lions CNRS UMR 7598, 175, rue du Chevaleret, 75013 Paris, France bUniversité Paris XII Val de Marne, laboratoire d’analyse et de mathématiques appliquées, CNRS UMR 8050, 94010 Créteil cedex, France

cUniversità di Napoli “Federico II”, Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”, Via Cintia, Complesso Monte S. Angelo, 80126 Napoli, Italy

Received 3 March 2004; accepted 11 March 2004 Available online 25 May 2004 Presented by Philippe G. Ciarlet

Abstract

We consider the periodic homogenization of nonlinear integral energies with polynomial growth. The study is carried out by the periodic unfolding method which reduces the homogenization process to a weak convergence problem in a Lebesgue space.

To cite this article: D. Cioranescu et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).

2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.

Résumé

Homogénéisation non linéaire par éclatement périodique. Cette Note présente un résultat d’homogénéisation périodique pour des énergies intégrales à croissance polynômiale. On utilise la méthode d’éclatement périodique qui réduit la démonstration à de la convergence faible dans un espace de Lebesgue. Pour citer cet article : D. Cioranescu et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).

2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.

Version française abrégée

Soit A0 l’ensemble des ouverts liptschitziens de Rⁿ et Y la cellule de référence ]0,1[ⁿ. On considère des densités d’énergiesf de Carathéodory satisfaisant (H0) et (H1) pourp∈ ]1,+∞[. Le résultat principal de cette Note est le Théorème 1 qui établit la convergence variationnelle pourεh→0, pour des énergies intégrales du type u→

Ωf (_ε^x

h,∇u(x))dx. Si de plus,f vérifie (H₂), la valeur limite est

Ωf_hom^p (∇u(x))dx,oùf_hom^p est définie par (2).

E-mail addresses: [email protected] (D. Cioranescu), [email protected] (A. Damlamian), [email protected] (R. De Arcangelis).

1631-073X/$ – see front matter 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.

doi:10.1016/j.crma.2004.03.028

Ce résultat avait été démontré par une méthode abstraite deΓ-convergence dans [14] et [7]. Nous donnons ici une démonstration par une méthode directe et élémentaire reposant sur la méthode d’éclatement périodique de [10]. Comme dans le cas linéaire, on est conduit à étudier la semi-continuité inférieure faible d’une fonctionnelle intégrale dans l’espaceL^p(Ω;W_per^1,p(Y )). La preuve de (1) est contenue dans les Lemmes 3.1 et 3.2. La preuve de la seconde partie du résultat, utilise le Théorème 4 de Castaing [9], et est donné par le Lemme 3.3.

1. Introduction

The homogenization of periodic structures has been carried out in the last thirty years for various kinds of problems involving differential equations as well as integral energies. Starting from the basic works [13,4,15], several methods have been developed to approach the analytic study of the asymptotic behaviour of such structures, which generated a wide bibliography, cf., for example, [3,5,6,11,12] and references therein. In the 1980s, a two- scale convergence technique was introduced in [16] then applied in [1]. In [2] a “dilation” operation was defined for the homogenization of periodic media with double porosity. These two techniques were used in [8] to study perforated domains and thin structures. The periodic unfolding method, presented in [10] for the homogenization of multi-scale periodic problems, combines the dilation technique with ideas from Finite Elements approximations.

This approach reduces two-scale convergence to a mere weak convergence in an appropriate space. In particular, when applied to linear problems, this method greatly simplifies the homogenization by reducing it to a weak convergence in aL²space.

In the present Note we consider the homogenization problem in the general case of nonlinear integral energies with polynomial growth. Here again, the use of the periodic unfolding method simplifies the homogenization process by reducing it to a weak convergence problem in anL^pspace.

Denote byA0the class of bounded open subsets ofRⁿhaving a Lipschitz boundary, and byYthe unit reference cell]0,1[ⁿ. Consider a Carathéodory energy densityf satisfying

(H₀)

f: (x, z)∈Rⁿ×Rⁿ→f (x, z)∈ [0,+∞[,

f (·, z)Lebesgue measurable, Y-periodic∀z∈Rⁿ; f (x,·)convex for a.e.x∈Rⁿ. Forp∈ [1,+∞[, andM >0, consider the following two growth conditions:

(H1) f (x, z)M(1+ |z|^p)for a.e.x∈Rⁿ,and everyz∈Rⁿ, (H2) |z|^pf (x, z)for a.e.x∈Rⁿ, and everyz∈Rⁿ.

The main result, the proof of which is provided in the next sections, is the following:

Theorem 1.1. Letf satisfy(H₀), and assume that(H₁)holds for somep∈ ]1,+∞[. LetΩ be inA0, and let {εh} ⊆ ]0,+∞[converge to 0. Then,

inf

lim inf

h→+∞

Ω

f x

ε_h,∇uh(x)

dx: {uh} ⊆W^1,p(Ω), uh uinW^1,p(Ω)

=inf

lim sup

h→+∞

Ω

f x

ε_h,∇u_h(x)

dx: {u_h} ⊆W^1,p(Ω), u_h uinW^1,p(Ω)

=inf

Ω×Y

f

y,∇u(x)+ ∇yV (x, y)

dxdy:V ∈L^p

Ω;W_per^1,p(Y )

. (1)

If in addition,(H₂) holds with the same p, this common value equals

Ωf_hom^p (∇u(x))dx,where f_hom^p is the homogenized energy density defined by

f_hom^p :z∈Rⁿ→inf

Y

f

y, z+ ∇v(y)

dy:v∈W_per^1,p(Y )

, (2)

andW_per^1,p(Y )is the Banach space ofY-periodic functions inW_loc^1,p(Rⁿ)with theW^1,p(Y )-norm.

The original proof (which is much more elaborate) of this theorem can be found in [14] and [7], where an abstractΓ-convergence method is used. Our approach here turns out to be elementary.

2. Preliminary results

2.1. The unfolding operator, and its main properties (cf. [10])

For everyx∈Rⁿ,[x]denotes the vector whose coordinates are the integer parts of the corresponding coordinates ofx.

Definition 2.1. LetΩ∈A0. Forε >0, the unfolding operatorTε:L¹(Ω)→L¹(Rⁿ×Y )is defined as Tε(v)(x, y)= ˜v

ε x

ε

+εy

for everyv∈L¹(Ω), and a.e.(x, y)∈Rⁿ×Y, wherev˜is the extension ofvby zero outsideΩ.

DefineΩ_εas the union of all setsε(ξ+ Y ), asξ∈Zⁿ, andε(ξ+ Y )∩ Ω= ∅. Then one has

Ωε×Y

Tε(v)(x, y)dxdy=

Ω

v(x)dx for everyε >0, andv∈L¹(Ω). (3)

In particular,Tε(v)L^p(Ωε×Y )= vL^p(Ω)for everypin[1,+∞], ε >0, andvinL^p(Ω), so that, the unfolding operator restricted toL^p(Ω)is continuous fromL^p(Ω)toL^p(Ω×Y ). Moreover,

Tε(v)→ ˜v inL^p

Rⁿ×Y

asε→0 for everyp∈ [1,+∞[, andv∈L^p(Ω). (4) The main tool in periodic unfolding is

Proposition 2.2. LetΩ∈A0,p∈ ]1,+∞[. Let{v_ε}be a sequence converging weakly inW^1,p(Ω)to somevas ε→0. Then, there exist a subsequence{εk}, andV ∈L^p(Ω;Wper^1,p(Y ))such that,

Tεk(∇v_εk) ∇v+ ∇yV in

L^p(Ω×Y )n

.

2.2. Castaing’s theorem on measurable selections

LetΩ,X be sets, andΓ a multifunction fromΩ toX. A functionσ:Ω →Xwill be said to be a selection ofΓ ifσ (x)∈Γ (x)for everyx∈Ω. The measurable selection result below is proved in [9, Theorem III.6 and Proposition III.11].

Theorem 2.3. LetXbe a separable metric space,(Ω,M)a measurable space, andΓ a multifunction from Ω toX. Assume that for everyx∈Ω,Γ (x)is nonempty and complete inX. Assume moreover, that for every closed subsetF ofX,{x∈Ω: Γ (x)∩F = ∅}belongs toM. ThenΓ admits aM-measurable selection.

3. Proof of Theorem 1.1

The proof of Theorem 1.1 is contained in the following three lemmas:

Lemma 3.1. Assume that f satisfies (H₀). Let Ω ∈A0, p∈ ]1,+∞[, and {ε_h} ⊆ ]0,+∞[converge to 0 as h→ +∞. Then, for everyu∈W^1,p(Ω),

inf

lim inf

h→+∞

Ω

f x

ε_h,∇uh(x)

dx: {uh} ⊆W^1,p(Ω), uh uinW^1,p(Ω)

inf

Ω×Y

f

y,∇u(x)+ ∇yV (x, y)

dxdy:V ∈L^p

Ω;W_per^1,p(Y ) .

Proof. Foru∈W^1,p(Ω), let{u_h}converge weakly touinW^1,p(Ω). Assume for simplicity, that lim_h→+∞

Ω

f (_ε^x

h,∇u_h(x))dx exists and is finite. Then, Proposition 2.2 implies that there exists {ε_hk} ⊆ {ε_h} and U ∈ L^p(Ω;W_per^1,p(Y ))with

Tε_hk(∇u_hk) ∇u+ ∇yU in

L^p(Ω×Y )_n

. (5)

According to (3), we have

Ωf (_ε^x

hk,∇u_hk(x))dx=

Ωε×Yf (y,Tε_hk(∇u_hk)(x, y))dxdy

Ω×Yf (y,Tε_hk(∇u_hk) (x, y))dxdy,where the last functional is sequentially weakly(L^p(Ω))ⁿ-lower semicontinuous (a well-known con- sequence of Fatou’s Lemma under hypothesis(H0)). Therefore, using (5), we get the following inequality, from which the lemma follows:

lim inf

h→+∞

Ω

f x

ε_h,∇u_h(x)

dx

Ω×Y

f

y,∇u(x)+ ∇yU (x, y) dxdy

inf

Ω×Y

f

y,∇u(x)+ ∇yV (x, y)

dxdy:V ∈L^p

Ω;W_per^1,p(Y ) . 2

Lemma 3.2. Assume that f satisfies (H₀), and that (H₁) holds for some p∈ ]1,+∞[. Let Ω ∈A0, and let {ε_h} ⊆ ]0,+∞[converge to 0 ash→ +∞. Then, for everyu∈W^1,p(Ω),

inf

lim sup

h→+∞

Ω

f x

εh

,∇u_h(x)

dx: {u_h} ⊆W^1,p(Ω), u_h uinW^1,p(Ω)

inf

Ω×Y

f

y,∇u(x)+ ∇yV (x, y)

dxdy:V ∈L^p

Ω;Wper^1,p(Y ) .

Proof. Letu∈W^1,p(Ω)andU∈C¹(Rⁿ×Rⁿ)withU (x,·) Y-periodic for everyx∈Ω. For everyh∈Nand x∈Ω, we setu_h(x)=u(x)+ε_hU (x,_ε^x

h). Clearly∇u_h(x)= ∇u(x)+ε_h∇xU (x,_ε^x

h)+ ∇yU (x,_ε^x

h)for everyh∈ N, x∈Ω.

From the periodicity ofU, one has

Ω

f x

ε_h,∇u_h(x)

dx=

Ω_εh×Y

f

y,Tε_h(∇u_h)(x, y) dxdy

=

Ω_εh×Y

f

y,Tε_h(∇u)(x, y)+εh∇xU

εh

x ε_h

+εhy, y

+ ∇yU

εh

x ε_h

+εhy, y

dxdy.

Due to continuity properties of∇xUand to the periodicity of∇yU, we haveε_h∇xU (·,_ε^·

h)→0 uniformly inΩ, and∇yU (·,_ε^·

h)→

Y∇yU (·, y)dy=0 weakly^∗in(L^∞(Ω))ⁿ.This implies thatu_h uinW^1,p(Ω). Moreover, inf

lim sup

h→+∞

Ω

f x

εh

,∇v_h(x)

dx:{v_h} ⊆W^1,p(Ω), v_h uinW^1,p(Ω)

lim sup

h→+∞

Ω_εh×Y

f

y,Tε_h(∇u)(x, y)+εh∇xU

εh

x ε_h

+εhy, y

+ ∇yU

εh

x ε_h

+εhy, y

dxdy.

On the other hand, again by the continuity properties of∇xUand of∇yU, one hasε_h∇xU (ε_h[_ε^·_h] +ε_h·,·)→0 and∇yU (εh[_ε^·_h] +εh·,·)→ ∇yU uniformly in Ω. Note that that by (4), Tε_h(∇u)→∇u in(L^p(Rⁿ×Y ))ⁿ. This, together with (H1), yields limh→+∞

Ω_εh×Yf (y,Tε_h(∇u)(x, y)+εh∇xU (εh[_ε^x_h] +εhy, y)+ ∇yU (εh[_ε^x_h] + εhy, y))dxdy =

Ω×Yf (y,∇u(x)+ ∇yU (x, y))dxdy, since ∂Ω is a null set. Consequently, for every U in C¹(Rⁿ×Rⁿ)withU (x,·) Y-periodic for everyx∈Ω, we have

inf

lim sup

h→+∞

Ω

f x

ε_h,∇v_h(x)

dx:{v_h} ⊆W^1,p(Ω), v_h uinW^1,p(Ω)

Ω×Y

f

y,∇u(x)+ ∇yU (x, y)

dxdy. (6)

The set of functionsUinC¹(Rⁿ×Rⁿ), withU (x,·) Y-periodic for everyx∈Ω, is dense inL^p(Ω;W_per^1,p(Y )).

We conclude the proof by observing that (H₀) and (H₁) imply the continuity onL^p(Ω;W_per^1,p(Y ))of the right-hand side of (6). 2

Lemma 3.3. Assume thatf satisfies(H0), and that(H1)and(H2)hold for somep∈ ]1,+∞[. LetΩ∈A0. Then, for everyu∈W^1,p(Ω), one has

inf

Ω×Y

f

y,∇u(x)+ ∇yV (x, y)

dxdy: V∈L^p

Ω;W_per^1,p(Y )

=

Ω

f_hom^p

∇u(x) dx.

Proof. One inequality is straightforward from definition (2), since foruinW^1,p(Ω)andV inL^p(Ω;W_per^1,p(Y )), the following inequality holds for a.e.x∈Ω,

Yf (y,∇u(x)+ ∇yV (x, y))dyinf{

Yf (y,∇u(x)+ ∇v(y))dy: v∈W_per^1,p(Y )} =f_hom^p (∇u(x)).

The reverse inequality is obvious if

Ωf_hom^p (∇u(x))dx= +∞. To prove it in the case where

Ωf_hom^p (∇u(x))dx

<+∞, we make use of Castaing’s selection theorem. Note first, that due to (H₀) and (H₁),f_hom^p is convex and continuous onRⁿ. Due to (H₂) and the Poincaré–Wirtinger inequality, the infimum defining f_hom^p (z)in (2), is achieved for everyz∈Rⁿ. This and (H₁) imply that forz∈Rⁿ,Γ (z)is nonempty, and strongly closed, whereΓ is the multifunction defined by

Γ:z∈Rⁿ→

v∈W_per^1,p(Y ):

Y

v(y)dy=0,

Y

f

y, z+ ∇v(y)

dy=f_hom^p (z)

. (7)

We now claim that Γ has a B(Rⁿ)-measurable selection, where B(Rⁿ) denotes the Borel σ-algebra of Rⁿ. By Theorem 2.3, it is enough to show that for every strongly closed subset F of W_per^1,p(Y ), Γ⁻(F )= {. ζ ∈ Rⁿ:Γ (ζ )∩F = ∅}belongs toB(Rⁿ).

Assume first that F is a closed ball in W_per^1,p(Y ). We prove that Γ⁻(F ) is closed. To do so, let {z_h} ⊆ Γ⁻(F ), z∈Rⁿ withz_h →z. For everyh∈N, let v_h∈Γ (z_h)∩F. The continuity and the finiteness of f_hom^p provide that lim sup_h→+∞

Y|zh+ ∇vh(y)|^pdy limh→+∞

Yf (y, zh+ ∇vh(y))dy=limh→+∞f_hom^p (zh)= f_hom^p (z) <+∞. Hence there is a subsequence {v_hk} of {v_h} and some v_∞ in F such that v_hk v_∞ in W^1,p(Y ). Moreover,

Yv_∞(y)dy=0. From the continuity off_hom^p and the weakW^1,p(Y )-lower semicontinuity of w→

Yf (y,∇w(y))dy, we getf_hom^p (z)

Yf (y, z+ ∇v_∞(y))dylim infk→+∞

Yf (y, z_hk+ ∇v_hk(y))dy= lim_k→+∞f_hom^p (z_hk)=f_hom^p (z).Thusv_∞belongs toΓ (z)∩F andztoΓ⁻(F ).

Suppose now thatF is a strongly closed subset of W_per^1,p(Y ). SinceW_per^1,p(Y )is separable,F can be written as a countable intersection of countable unions of balls. Consequently,Γ⁻(F )itself a countable intersection of countable unions of closed sets, belongs toB(Rⁿ).

By Theorem 2.3, the multifunctionΓ admits aB(Rⁿ)-measurable selectionσ. Fixu∈W^1,p(Ω). For a.e.x∈Ω, setU (x)=σ (∇u(x)). ThenU isL(Ω)-measurable, with values inW_per^1,p(Y )and, by (7)

f_hom^p

∇u(x)

=

Y

f

y,∇u(x)+ ∇yU (x)(y)

dy for a.e.x∈Ω. (8)

Integrating (8) over Ω yields

Ω×Yf (y,∇u(x)+ ∇yU (x)(y))dydx < +∞, so that by (H₂), ∇yU is in (L^p(Ω×Y ))ⁿ. By the Poincaré–Wirtinger inequality,Ubelongs toL^p(Ω;W_per^1,p(Y )), hence

Ωf_hom^p (∇u(x))dx inf{

Ω×Yf (y,∇u(x)+ ∇yV (x, y))dxdy: V ∈L^p(Ω;W_per^1,p(Y ))}. 2 Acknowledgements

This work was carried out in the framework of HMS 2000 European Network (HPRN-2000-00109), and of the 2003-G.N.A.M.P.A. Project “Metodi Variazionali per Strutture Sottili, Frontiere Oscillanti ed Energie Vincolate”.

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