The "strange term" in the periodic homogenization for multivalued Leray-Lions operators in perforated domains

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The ”strange term” in the periodic homogenization for multivalued Leray-Lions operators in perforated domains

Alain Damlamian, Nicolas Meunier

To cite this version:

Alain Damlamian, Nicolas Meunier. The ”strange term” in the periodic homogenization for multival- ued Leray-Lions operators in perforated domains. Ricerche di matematica, Springer Verlag, 2010, 59 (2), pp.281-312. �hal-00523347�

The “strange term” in the periodic homogenization for multivalued Leray-Lions operators in perforated

domains.

Alain Damlamian^∗and Nicolas Meunier^† 1^eroctobre 2010

R´esum´e

Nous ´etudions par la m´ethode de l’´eclatement priodique de Cio- ranescu, Damlamian et Griso, l’homog´en´eisation des ´equations de la forme

−divdε=f, with ∇uε,δ(x), dε,δ(x)

∈Aε(x)

o`uAε est une fonction dont les valeurs sont des op´erateurs maximaux monotones et le domaine est perfor´e compos´e de trous de taille εδ p´eriodiquement r´epartis dans le domaine. Sous des hypoth`eses de crois- sance et de coercivit´e surAε, si les deux suites d’op´erateurs maximaux monotone ´eclat´es (suivant deux op´erations distinctes) convergent au sens des graphes vers des op´erateurs maximaux monotonesA(x, y) et A⁰(x, z) pour presque tout (x, y, z) ∈ Ω×Y ×R^N, quand ε → 0, alors toute valeur d’adh´erence (u⁰, d⁰) de la suite (uε,δ, dε,δ) pour la topologie faible dans l’espace de Sobolev naturellement associ est une solution du probleme homog´en´eis´e qui s’exprime en terme de la fonc- tion u⁰ seule. Ce rsultat s’applique au cas o Aε(x) est de la forme B(x/ε) oB(y) est priodique et continu eny = 0, et en particulier au cas dup-Laplacien oscillant.

Abstract

Using the periodic unfolding method of Cioranescu, Damlamian and Griso, we study the homogenization for equations of the form

−divdε=f, with ∇uε,δ(x), dε,δ(x)

∈Aε(x)

in a perforated domain with holes of sizeεδperiodically distributed in the domain, whereAεis a function whose values are maximal monotone

∗Laboratoire d’Analyse et de Math´ematiques Appliqu´ees, CNRS UMR 8050, Universit´e Paris 12 Val de Marne 94010 Cr´eteil Cedex, France. ([email protected])

†MAP5, CNRS UMR 8145, Universit´e Paris Descartes, 45 rue des Saints P`eres 75006 Paris, France. ([email protected]).

graphs (on R^N). Two different unfolding operators are involved in such a geometric situation. Under appropriate growth and coercivity assumptions, if the corresponding two sequences of unfolded maximal monotone graphs converge in the graph sense to the maximal monotone graphsA(x, y) and A0(x, z) for almost every (x, y, z)∈Ω×Y ×R^N, asε→0, then every cluster point (u0, d0) of the sequence (uε,δ, dε,δ) for the weak topology in the naturally associated Sobolev space is a solution of the homogenized problem which is expressed in terms ofu⁰ alone. This result applies to the case whereAε(x) is of the formB(x/ε) whereB(y) is periodic and continuous aty= 0, and, in particular, to the oscillatingp-Laplacian.

1 Introduction

This article is devoted to periodic homogenization for nonlinear partial differential equations with oscillating coefficients in domains perforated by small holes of size εδ and periodically distributed with period ε. The ho- mogeneous Dirichlet condition is imposed on the boundaries of these holes.

This type of equation models various physical problems arising in media with holes or heterogeneous materials with various competing length-scales.

Our starting point is the following theorem, where LL(O,R^N, p, α, m) denotes the measurable maps from the open domain O in R^N to the set M(R^N×R^N) of all maximal monotone maps fromR^N into itself such that

α kξk^p

p +kηk^p′ p^′

≤ hη, ξi+m(x). (1.1)

The functionm is assumed to be L¹(O).

Theorem 1.1. Let O be a bounded domain inR^N, p∈(1,∞), m∈L¹(O), and f ∈ L^p′(O). Let (A_n)_n∈N be a sequence in LL(O,R^N, p, α, m) and suppose that for each n∈N, u_n is a solution of







−divd_n=f in D^′(O),

∇un(x), dn(x)

∈An(x), u_n∈W₀^1,p(O).

(1.2) Suppose furthermore that for a.e. x ∈ O, the sequence An(x) converges in the sense of multivalued operators to A₀(x).

Then,A₀ belongs toLL(O,R^N, p, α, m), the sequence(u_n, d_n)_n∈N is bounded in the space W₀^1,p(O)×L^p′(Ω) and every one of its weak limit-points is a solution of







−divd=f in D^′(O),

∇u(x), d(x)

∈A(x), u∈W₀^1,p(O).

(1.3)

Actually, the well-known result of J. Leray and J.-L. Lions ([18]) implies that problem (1.2) has a unique solution in the case where Aε(x) is single- valued and strictly monotone for a.e. x ∈ Ω. Theorem 1.1 itself allows to obtain solutions of the same problem in the general case (by approximating in the sense of graphs a multivalued element of LL(Ω,R^N, p, α, m) by a sequence of such single-valued elements) (see also [9, 17]). Such solutions need not be unique.

In the present, work we consider the homogenization (when the small parameters ε, δ go to 0) of the same problem in a domain Ω^∗_ε,δ with small Dirichlet holes: 





−divd_ε,δ =fε inD^′(Ω^∗_ε,δ),

∇uε,δ(x), d_ε,δ(x)

∈A_ε(x), u_ε,δ ∈W₀^1,p(Ω^∗_ε,δ).

(1.4)

The domain Ω^∗_ε,δ is constructed as follows. Y is a reference period in R^N associated to the set of periodsb(which is a basis ofR^N). ThenB is a (not empty) bounded open subset ofR^N and, forδ small enough,Y_δ^∗ isY \δB.

Finally

Ω^∗_ε,δ =n

x∈Ω, such thatx∈ [

k∈Z^N

ε{Y_δ^∗+k·b}o .

In problem (1.4), it is assumed that 1< p <∞,p⁻¹+p^′−1 = 1,f_ε belongs toW^−1,p′(Ω) andA_εtoLL(Ω,R^N, p, α, m). The solutionsu_ε,δare naturally extended by 0 in the holes and as such belong to W₀^1,p(Ω). One can also extendd_ε,δ measurably in the holes by an element ofA_ε(x,0) so as to belong toL^p′(Ω).

Let (u_ε,δ, d_ε,δ) ∈ W₀^1,p(Ω)×L^p(Ω;R^N) be solutions of problem (1.4).

Suppose furthermore that the two sequences of unfolded graphsTε(A_ε)(x, y) and A_ε,δ(x, z) converge in the sense of maximal monotone graphs, where A_ε,δ(x, z) is given by

(ξ, η)∈A_ε,δ(x, z)⇔(δ^−N/pξ, δ^−N/p′η)∈ Tε,δ(A_ε)(x, z). (1.5) Then we show that, providedεandδconverge to 0 in a controlled way, every weak cluster point (u0, d0) inW₀^1,p(Ω)×L^p(Ω,R^N) of a sequence (u_ε,δ, d_ε,δ) of solutions of problem (1.4) is itself a solution of the homogenized equation:

(−divd−k₁Θ⁻¹(−k1u)∋f,

∇xu, d

∈Ahom(x)u∈W₀^1,p(Ω), (1.6)

where A_hom ∈ LL(Ω,R^N, p, α,m) is defined in terms of the limit of the¯ sequence of unfolded graphs Tε(Aε)(x, y) alone, while Θ is the maximal monotone graph (on R×R) obtained from the limit A₀ of the sequence of second unfolded graphs A_ε,δ(x, z) alone. The constant k₁ is given as

(ε.δ)lim→(0,0)

δ^p−1N

ε , which, without loss of generality, can be assumed to exist.

The outline of the paper is as follows. Section 2 is devoted to the defini- tion of the unfolding operators and their main properties. In subsection 2.1, we recall the definition of the unfolding operatorT_ε for the periodic case in fixed domains (see [11] and [15] as well as [19] and [20]) and in subsection 2.2, we recall the unfolding operatorTε,δ depending of two small parameters εand δcorresponding to the scalesεandεδ (see [12] for the proofs). It was first introduced in a similar form in [6] and [7].

In section 3 we consider the homogenization problem and we state our main result in Theorem 3.4. For simplicity, we assume a homogeneous Dirichlet boundary condition on the outer boundary of the domain, but more general boundary conditions can be assumed there, provided the outer boundary is Lipschitz and the perforations do not intersect it. In each case, we obtain both the unfolded and the classical (standard) form for the limit problem. The operator Tε allows to homogenize the coefficients of the dif- ferential operators, whereas the operatorT_ε,δ generates the “strange term”

Θin the limit in the spirit of [10].

In the Appendix, we recall the definition of maximal monotone operators and of the notion of convergence of maximal monotone graphs (see [16]

for the proofs). We also consider sequences of maximal monotone-valued measurable functions, their canonical extensions and we prove key results about their convergence (including a new generalization of Minty’s method in Proposition 4.10).

Note that there are many papers in the litterature which study the ho- mogenization problem for perforated domains in the non-linear case. We only refer to [13], [6] and the bibliographies therein. The present paper is the first making use of the unfolding method

2 The periodic unfolding operator

In this section we recall, the general properties of the periodic unfolding operator introduced in [11] and of its variants and generalizations introduced in [12].

Let Y be a set containing the origin and having the paving property in R^N (cf. [15]) with respect to a set ofN periods (b₁,· · · , b_N). The classical

example is the unit cube ofR^N centered in the origin,Y =

−1 2,1

2 N

. We consider the subgroup generated by the periods in R^N and all the corre- sponding translates ofY. To eachx∈R^N we can associate its integer part, [x]_Y belonging to the subgroup of periods, such thatx−[x]Y ∈Y, the latter being defined as its fractional part, i.e,{x}Y =x−[x]_Y. Therefore we have x=ε_x

ε Y +ε_x

ε

Y for anyx∈R^N. Without ambiguity, we will drop the subscriptY for the integer part and fractional part functions in the sequel.

Remark 1. This definition is ambiguous, but only on a set of zero measure, which is enough for our purpose.

Let Ω be an open and bounded set inR^N, the setsΩb_εand Λ_εare defined respectively as the largest finite union ofεY cells contained in Ω and as the subset of Ω consisting of the the parts ofεY cells intersecting the boundary

∂Ω. More precisely, we have:

Ωb_ε=n

x∈Ω, εhx

ε i

+εY

⊂Ωo

and Λ_ε = Ω\Ωb_ε. (2.1) 2.1 The case of fixed domains: the operator Tε

We give here the definition of the unfolding operator and its main prop- erties (for details and proofs, the reader is referred to [11] and [15]).

Definition 2.1. The unfolding operatorTε:L^p(Ω)→L^p(Ω×Y) is defined as follows for φ∈L^p(Ω):

T_ε(φ)(x, y) = ( φ

εhx ε

i+εy

if (x, y)∈Ωb_ε×Y,

0 if (x, y)∈ Λ_ε×Y.

Theorem 2.2. (Properties of the operator Tε) 1. For any v, w∈L^p(Ω), Tε(vw) =Tε(v)Tε(w).

2. The “exact integration” formula. For everyw∈L^p(Ω), 1

|Y| Z

Ω×Y

Tε(w)(x, y) dx dy = Z

Ω

w(x) dx− Z

Λε

w(x) dx= Z

Ωbε

w(x) dx.

3. TheL¹ control. For every u∈L¹(Ω), 1

|Y| Z

Ω×Y

|Tε(u)| dx dy≤ Z

Ω

|u| dx.

4. For everyu∈L¹(Ω),

Z

Ω

u dx− 1

|Y| Z

Ω×Y

Tε(u) dx dy ≤

Z

Λε

|u|dx. (2.2)

5. Let {wε} be a sequence which converges strongly in L^p(Ω) to some w, (for 1≤p <+∞). Then,

Tε(wε)→w strongly in L^p(Ω×Y).

6. Let {w_ε} be a sequence which converges weakly in W^1,p(Ω) to some w (1≤p <+∞). Then, there exists a subsequence and wb∈L^p

Ω;W_per^1,p(Y) such that

Tε(∇wε)⇀∇xw+∇ywb weakly in L^p(Ω×Y).

Remark 2. Ifu:R^N →S and f :S→S^′, then Tε(f◦u) =f ◦ Tε(u).

In particular ifu:R^N →Sandv:R^N →T, the preceding property applied to the projectionsP : (u, v)7→u and Q: (u, v)7→v yields

T_ε (u, v)

= T_ε(u),T_ε(v) . Therefore, ifF :S×T →R,

Tε F(u, v)

=F Tε(u),Tε(v)

. (2.3)

Useful particular cases are when S = R, T = R and F : (s, t) → st and whenS =R^N,T =R^N and F is the dot product.

The previous formulas apply also for functions defined on an open set Ω, with the obvious modifications on the set Λ_ε.

Remark 3. Property 4 shows that every integral of a function w on Ω, is

“almost equivalent” to the integral of its unfolded on Ω×Y, the ”integra- tion defect” arises only from the cells intersecting the boundary ∂Ω and is controlled by the right hand side integral in (2.2). Hence we deduce the following result.

Proposition 2.3. If {wε} is a sequence inL¹(Ω)satisfying Z

Λε

|w_ε| dx→0,

then Z

Ω

w_ε dx− 1

|Y| Z

Ω×Y

Tε(w_ε) dx dy→0.

We end this subsection by recalling the notion of local average of a function.

Definition 2.4. The local averageM_Y^ε :L^p(Ω)7→L^p(Ω), is defined for any φinL^p(Ω),1≤p <∞,by

M_Y^ε(φ)(x) = 1

|Y| Z

Y

Tε(φ)(x, y) dy.

Remark 4. The function M_Y^ε(φ) is indeed a local average, since

MY^ε(φ)(x) = 1

|Y| Z

Y

Tε(φ)(x, y) dy=





 1 ε^N|Y|

Z εx

ε

+εY φ(ζ) dζ, ifx∈Ωbε,

0, ifx∈Λε.

Remark 5. Note thatTε(M_Y^ε(φ)) =M_Y^ε(φ) on the set Ω×Y.

The next proposition, which is also frequently used, is classical (and also follows from the previous theorem):

Proposition 2.5. Let {wε} be a sequence such that w_ε → w strongly in L^p(Ω)where 1≤p <∞. Then we have

M_Y^ε(w_ε)→w strongly in L^p(Ω).

2.2 Unfolding in domains with volume-distributed “small”

holes: the operator Tε,δ

We consider domains with holes of size εδ ( δ will go to 0 with ε) and εY-periodically distributed. More precisely, for a given a given bounded open set B and for δ small enough (so that δB ⊂⊂ Y), we denote Y_δ^∗ the setY \δB and define the perforated domain Ω^∗_ε,δ as

Ω^∗_ε,δ =n

x∈Ω, such thatnx ε

o∈Y_δ^∗o

. (2.4)

The sets B andY_δ^∗ and the corresponding Ω^∗_ε,δ

This geometry with ”small” holes leads to the definition of a new un- folding operator T_ε,δ (see [12]), depending on both parameters ε and δ. In the next sections, we will consider functions v_ε,δ that vanish on the whole boundary of the perforated domain Ω^∗_ε,δ, namely belonging to the space W₀^1,p(Ω^∗_ε,δ). They are naturally extended by zero to the whole of Ω and these extensions still denoted v_ε,δ are functions in W₀^1,p(Ω). This justifies the introduction of T_ε,δ on the fix domain Ω (but keeping in mind that our aim will be to apply it to the extensions).

Definition 2.6. For φ ∈ L^p(Ω), (1 ≤ p < ∞), the unfolding operator Tε,δ:L^p(Ω)→L^p(Ω×R^N) is defined as

T_ε,δ(φ)(x, z) = (

Tε(x, δz) if (x, z)∈Ωbε× 1 δY,

0 otherwise.

The next results follow directly from Theorem 2.2 by using the change of variablez= (1/δ)y.

Theorem 2.7. (Properties of the operator Tε,δ) 1. For everyv, w∈L^p(Ω), T_ε,δ(vw) =T_ε,δ(v)T_ε,δ(w).

2. Foru∈L¹(Ω), δ^N

|Y| Z

Ω×RN

|Tε,δ(u)| dx dz≤ Z

Ω

|u| dx.

3. For everyu∈L^p(Ω), 1≤p <+∞,

kTε,δ(u)k^p_Lp(Ω×R^N) ≤ |Y|

δ^Nkuk^p_Lp(Ω). 4. For everyu∈L¹(Ω),

Z

Ω

u dx− δ^N

|Y| Z

Ω×R^N

Tε,δ(u) dx dz ≤

Z

Λε

|u| dx.

5. Let u∈W^1,p(Ω). Then, T_ε,δ(∇xu) = 1

εδ∇z T_ε,δ(u)

in Ω×1 δY.

Suppose N > p ≥ 1, set p^∗ = pN/(N −p) and denote the Sobolev- Poincar´e-Wirtinger constant for W^1,p(Y) by C.

6. Let ω be open and bounded inR^N. Then, the following estimates hold:

k∇z Tε,δ(u)

k^p_Lp(Ω×¹δY)≤ ε^p|Y|

δ^N−pk∇uk^p_Lp(Ω), (2.5)

kTε,δ u−M_Y^ε(u)

k^p_Lp(Ω;L^p∗(RN))≤ Cε^p|Y|

δ^N−p k∇uk^p_Lp(Ω), (2.6) and

kTε,δ u)k^p_Lp(Ω×ω) ≤ 2Cε^p|Y|

δ^N−p k∇uk^p_Lp(Ω)+ 2|ω| kuk^p_Lp(Ω).

7. Let{vε,δ} be a sequence of functions in W^1,p(Ω) which converges weakly to some v0 when both ε and δ go to zero. Suppose furthermore that ε^p

δ^N−p remains bounded. Then, up to a subsequence, there exist a function U in L^p(Ω;L^p_loc(R^N))and W inL^p(Ω;L^p∗(R^N))with∇zW inL^p(Ω×R^N)such that

M_Y^ε(v_ε,δ)1¹

δY − T_ε,δ(v_ε,δ)

⇀ W weakly inL^p(Ω;L^p∗(R^N)),

∇z Tε,δ(v_ε,δ) 1¹

δY ⇀−∇zW weakly inL^p(Ω×R^N), Tε,δ(v_ε,δ)⇀ U weakly inL^p(Ω;L^p_loc(R^N)) and

U(x, z) +W(x, z) =v₀(x) for a.e. x andz in Ω×Rⁿ.

Remark 2 above applies in an analogous way to the definition of the unfolded of a dot product .

Concerning the integral formulas, we have the following results, similar to those of the previous subsection which follow from Property 4 in Theorem 2.7:

Proposition 2.8. If {wε} is a sequence inL¹(Ω)satisfying Z

Λε

|wε| dx→0,

then Z

Ω

w_ε dx− δ^N

|Y| Z

Ω×R^N

T_ε,δ(w_ε) dx dy→0.

3 Homogenization in domains with small holes which are periodically distributed in volume

In this section, we state our main homogenization result. First, we give the definitions of some functional spaces, then we give a convergence result and its proof. Finally, we study the form of the limit problem. Our proofs use the maximal monotone graph theory. For the convenience of the reader we only give the definition of the graphs we will consider while their properties are given in the appendix.

3.1 The functional setting We begin with some definitions.

Let X be a locally uniformly convex reflexive separable Banach space and let X^′ be its dual. The duality product in X^′×X is denoted byh·,·i. The set of maximal monotone operators fromX toX^′is denoted byM(X×X^′).

Let Ω be an open set ofR^N.

Definition 3.1. For a measurable functionmfrom Ω toR⁺, forα >0 and 1< p <+∞,LL(Ω, X, p, α, m) is the set ofA: Ω→M(X×X^′) such thatA is measurable and such that for almost everyt∈Ω, for every (ξ, η)∈A(t),

αkξk^p

p +kηk^p′ p^′

≤ hη, ξi+m(t), (3.1)

wherep⁻¹+p^′−1 = 1.

Remark 6. The definition of the measurability of A is understood in the sense of [8]. Let (Ω,T, µ) will be a σ-finite µ-complete measure-space. A functionA: Ω→M(X×X^′) ismeasurable if and only if for every open set U ⊂X×X^′ (resp closed set, Borel set, open ball, closed ball),

t∈Ω :A(t)∩U 6=∅ is measurable in Ω.

Remark 7. This condition is equivalent to the fact that the maximal mono- tone graphs are coercive with respect tok·k^p and that they satisfy a growth condition of the type kηk^p′ ≤ Ckξk^p+m₁(t). It also has the advantage of being symmetrical with respect to ξ and η.

Remark 8. If α > 1, then LL(Ω, X, p, α, m) is empty. Indeed, by Young’s inequality, the graph A(t) should be bounded in X×X^′ for almost every t∈Ω, in contradiction with the maximality of almost everyA(t).

Remark 9. From now on, “ ֌ ” will denote the convergence of maximal monotone graphs (this is explained in the appendix).

When Ω(R^N,A: Ω→M(X×X^′) is unfolded as follows:

Definition 3.2. Let Ω(R^N andA: Ω→M(X×X^′). FirstAis extended toR^N byαkξk^p−2F(ξ), whereF is the duality mapping ofX. This extension is still denoted byA. The unfolded graphT_ε(A) is now defined onR^N×Y, but only its restriction to Ω×Y will be used.

The perforated domain Ω^∗_ε,δis defined by (2.4). Assume that the operator A_ε belongs toLL(Ω, X, p, α, m). Forf_ε∈W^−1,p′(Ω), consider the following

problem denotedPε,δ:













Find u_ε,δ ∈W₀^1,p(Ω^∗_ε,δ) such that R

Ωhdε,δ,∇ϕi dx=hfε, ϕi_W−1,p′

(Ω),W0^1,p(Ω)

∇uε,δ(x), d_ε,δ(x)

∈A_ε(x),

∀ϕ∈W₀^1,p′(Ω^∗_ε,δ).

(3.2)

Remark 10. The previous variational problem can be expressed as follows:







−divd_ε,δ =fε inD^′(Ω^∗_ε,δ),

∇uε,δ(x), d_ε,δ(x)

∈A_ε(x), u_ε,δ ∈W₀^1,p(Ω^∗_ε,δ).

(3.3)

The following result is a direct application of Theorem 1.1 above.

Theorem 3.3. There exists at least one solution (u, d) in W₀^1,p(Ω^∗_ε,δ) × L^p′(Ω;R^N) for problem Pε,δ. Moreover, if A_ε satisfies some strict mono- tonicity condition for a.e. x∈Ω^∗_ε,δ, the solution is unique.

Remark 11. We still denote u_ε,δ and d_ε,δ the extension by zero of u_ε,δ and d_ε,δ to the whole Ω.

In the sequel, we will suppose thatN > pand we will study the asymp- totic behavior of problemPε,δ, asεand δ =δ(ε) are such that there exists a positive constantk1 satisfying

k1 = lim

ε→0

δ^Np−1

ε , with 0≤k1 ≤ ∞. (3.4)

Remark 12. Sinceεandδ satisfy (3.4), for simplicity we will say thatεgoes to zero instead of (ε, δ) goes to (0,0).

Remark 13. The cases k₁ = 0 and k₁ = ∞ are simpler and will be briefly discussed later.

We introduce the following functional spaces which play a crucial role in the statement of our main result:

KB = {Φ∈L^p∗(R^N) ; ∇Φ∈L^p(R^N), Φ|B is constant } (3.5) K_B⁰ = {Φ∈L^p∗(R^N) ; ∇Φ∈L^p(R^N), Φ|B= 0}. (3.6) We denote the constant value which an element Φ of K_B takes on B by Φ(B). It is known thatK_B is a Banach space when endowed with the norm

kΦkKB

=.

|Φ(B)|^p+k∇Φk^p_Lp(R^N)

¹_p , and thatK_B⁰ is a closed hyperplane ofK_B.

We also denote by Ke_B: Ke_B .

={Φ−Φ(B);Φ∈K_B}={Ψ∈L^p_loc^∗(R^N),Ψ = 0 on B,∇Ψ∈L^p(R^N)}.

For Ψ = Φ−Φ(B), let ℓ(Ψ) denote the real number −Φ(B). Then, Ke_B is also a Banach space for the norm

kΨk_Ke

B

=.

|ℓ(Ψ)|^p+k∇Ψk^p_Lp(R^N)

¹_p ,

Note thatℓis a continuous linear form on Ke_B. The numberℓ(Ψ) represents the limit at infinity for Ψ (in the sense that Ψ−ℓ(Ψ)∈L^p∗(R^N)). Its kernel is exactly K_B⁰.

3.2 The unfolded homogenization result

We derive here the unfolded formulation of the limit problem for P_ε,δ. In the limit we will observe the contribution of the periodic oscillations as well as the contribution of the perforations.

Here is the main result of this section.

Theorem 3.4. Assume 1< p <∞,p⁻¹+p^′−1 = 1,m_ε is in L¹(Ω), α >0 and letΩ^∗_ε,δ be given by (2.4). LetA_ε ∈LL(Ω,R^N, p, α, m_ε)for everyε >0.

Assume that there exists measurable mapsm₁ ∈L¹(Ω×Y),m₂ ∈L¹(Ω×

R^N), A∈LL(Ω×Y,R^N, p, α, m1) andA0∈LL(Ω×R^N,R^Np, α, m2)such that asε goes to0, the following convergences hold

Tε(mε) → m1 strongly inL¹(Ω×Y) (3.7) δ^NTε,δ(m_ε) → m₂ strongly inL¹(Ω×R^N) (3.8)

Tε(A_ε)(x, y) ֌ A(x, y), (3.9)

A_ε,δ(x, z) ֌ A₀(x, z), (3.10)

with

(ξ, η)∈A_ε,δ(x, z)⇔(δ^−N/pξ, δ^−N/p′η)∈ T_ε,δ(Aε)(x, z). (3.11) Assume f_ε→f strongly inW^−1,p′(Ω)as ε→0.

Then, the set(u_ε,δ, d_ε,δ)ε>0 of solutions of all the problemsP_ε,δ is weakly compact in W₀^1,p(Ω)×L^p′(Ω;R^N).

For every weak limit point (u0, d0) ∈ W₀^1,p(Ω)×L^p′(Ω;R^N), there exists (ˆu,d)ˆ ∈Wper^1,p(Y)×L^p′(Y;R^N) and (U, ξ₀)∈L^p(Ω;Ke_B)×L^p′(Ω;R^N) such that there is a subsequence(ε_n, δ_n)_n≥1 withε_n→0, δ_n→0 satisfying (3.4)

and

































u_εn,δn⇀ u₀ weakly in W₀^1,p(Ω),

Tεn(∇u_εn,δn)⇀∇u0+∇yuˆ weakly in L^p(Ω×Y), d_εn,δn ⇀ d₀ weakly inL^p′(Ω;R^N),

Tεn(d_εn,δn)⇀ d₀+ ˆdweakly in L^p′(Ω×Y), T_εn,δn(u_εn,δn)⇀ U weakly inL^p(Ω;L^p_loc(R^N)), ℓ(U) =u₀ for a.e. x∈Ω,

∇z T_εn,δn(u_εn,δn) 1¹

δY ⇀∇zU weakly in L^p(Ω×R^N), δ^N/p_n ^′Tεn,δn(d_εn,δn)⇀ ξ₀ weakly inL^p′(Ω×R^N),

(3.12)

as n→ ∞, with Z

Y

dˆdy= 0 and Z

Y

hd,ˆ∇yϕi dy= 0 ∀ϕ∈W_per^1,p(Y), (3.13) Z

Ω×(R^N\B)

hξ0, ϕ∇zvi dz= 0 ∀(ϕ, v)∈W₀^1,p(Ω×R^N)×K_B⁰. (3.14)

Moreover, Z

Ω

hd0,∇xψi dx− k1

|Y| Z

Ω×∂B

(ξ0 · ν_B)ψ dx dσz = Z

Ω

f ψ dx, (3.15) for allψ∈W₀^1,p(Ω), where ν_B is the inward normal vector on∂B and dσ_z is its surface measure.

Furthermore,

∇u0+∇yu, dˆ 0+ ˆd

∈A(x, y) for a.e. (x, y)∈Ω×Y, (k₁∇zU, ξ₀)∈A₀(x, z) for a.e. (x, z)∈Ω×(R^N \B).

Remark 14. For the linear case, the proof of theorem 3.4 was given in [12].

Remark 15. The case of the periodic homogenization for similar problems involving pseudo-monotone operators was treated in [6] using a result of [2] which immediately implies that the sequence (∇uε,δ) converges almost everywhere. In the case of Theorem 3.4, this is not true, and the crux of the proof is to overcome this difficulty, using a non-standard form of Minty’s method (see Lemma 4.2).

Remark 16. Note thatA_ε,δ belongs to LL( ˆΩ_ε×¹_δY,R^N, p, α, δ^NTε,δ(m)).

Indeed, first recalling the definition (3.11) ofA_ε,δ we have that for almost ev- ery (x, z)∈Ωˆε×¹_δY, (ξ, η)∈A_ε,δ(x, z) iff (δ^−N/pξ, δ^−N/p′η)∈ T_ε,δ(Aε)(x, z).

Next, from relation (3.1) in the definition of LL(Ω,R^N, p, α, m) together with the fact thatTε,δ(A_ε) belongs toLL( ˆΩ_ε×¹_δY,R^N, p, α, δ^NTε,δ(m_ε)), we

deduce that for almost every (t, u)∈ Ωˆ_ε×¹_δY, for every (ξ, η)∈A_ε,δ(t, u), we have

αkδ^−N/pξk^p

p +kδ^−N/p′ηk^p′ p^′

≤ hδ^−N/p′η, δ^−N/pξi+Tε,δ(m_ε)(t, u),

hence the result by multiplying this inequality by δ^N. A similar (albeit simpler) result holds forTε(A_ε).

Remark 17. Note that ifm_εconverges strongly tominL¹then convergences (3.7) and (3.8) hold.

Remark 18. For simplicity, we will not relabel the subsequences and denote them all as (ε, δ).

Example 1. The case of periodic A_ε. Suppose that A_ε is of the form Aε(x) =B(^x_ε), withBinLL(Y,R^N, p, α, m0) and extended byY-periodicity.

Then, it easily follows that condition (3.9) is satisfied withA(x, y) .

=B(y).

If furthermore,B(y) is continuous at y= 0 (in the sense of maximal mono- tone graphs), then condition (3.10) is satisfied,A₀ being the following graph associated withB(0):

(ξ, η)∈A₀ ⇐⇒

there is a sequence (δ^−Np ξ_δ, η_δ)∈B(0) such that lim

δ→0(ξ_δ, δ^Np′η_δ) = (ξ, η).

(3.16) This is the normalized asymptote graph of B(0) because the factors δ^−Np and δ^Np′ take into account the growth condition satisfied by B.

Example 2. The case of the oscillating p-Laplacian corresponds to the single-valued B(y) given by η = b(y)|ξ|^p−2ξ, where b is measurable, Y- periodic, bounded below byα and above byα^1−p′. Then, one easily checks that A(x, y)(ξ) .

= b(y)|ξ|^p−2ξ. If b is continuous at y = 0, the maximal monotone graph A0 is also single-valued and given by A0(ξ) .

=b(0)|ξ|^p−2ξ.

We begin the proof of theorem 3.4 with two elementary lemmas.

Lemma 3.5. Let δ0>0. Then, forN > p, the set [

0<δ<δ⁰

{φ∈W_per^1,p(Y);φ= 0 on δB}

is dense in Wper^1,p(Y).

Proof. Let ψ∈C_per^∞(Y) be fixed. Consider φ∈Wper^1,p(Y) smooth such that supp(φ) ⊂B(O,2), φ= 1 on B(O,1). Let φ_δ be such that φ_δ(x) = φ(^x_δ), then|∇φδ| ≤ ^C_δ. Define Φ_δ= (1−φ_δ)ψ, we see that

||Φ_δ−ψ||_L^p_(Y)+||∇Φ_δ− ∇ψ||_L^p_(Y)≤ Z

2δB(O,1)

|ψ|^p dy+ Z

2δB(O,1)

|∇ψ|^p dy

+ Z

2δB(O,1)

|∇φδ|^p|ψ|^p dy.

Using the definition ofφ_δ, one gets Z

2δB(O,1)

|∇φδ|^p|ψ|^p dy≤Cδ^N−pkψk^p_L∞(Y), hence, we deduce that

Φ_δ →ψ strongly in W_per^1,p(Y) as δ→0.

Since Wper^1,p(Y) is the closure of C_per^∞(Y) in the W^1,p-norm, a density argu- ment completes the proof.

Lemma 3.6. Assume (3.4), and let v be in K_B. For δ small enough, set w_ε,δ(x) =v(B)−v1

δ nx

ε o

Y

for x∈R^N. Then, for 0< k₁ <∞,

w_ε,δ ⇀ v(B) weakly inW_loc^1,p(R^N) (hence weakly inW^1,p(Ω)). (3.17) Fork₁ = 0,w_ε,δ→v(B) strongly inW_loc^1,p(R^N)(hence strongly inW^1,p(Ω)).

Furthermore, w_ε,δ vanishes outside Ω^∗_ε,δ and Tε,δ(∇wε,δ) =−1

εδ∇zv in Ωb_ε× 1

δY. (3.18)

The proof of this lemma is similar to the proof of Lemma 4.2 from [12].

The proof of Theorem 3.4 uses several intermediate results. Throughout the proof, to simplify the notations, we only writeε, δ instead ofε_n, δ_n.

The first step is

Proposition 3.7. Under assumptions of Theorem 3.4, the set of solutions (u_ε,δ, d_ε,δ)ε>0of the problemP_ε,δ is weakly compact inW₀^1,p(Ω)×L^p′(Ω;R^N).

For every weak limit point (u₀, d₀) ∈ W₀^1,p(Ω)×L^p′(Ω;R^N), there exists (ˆu,d)ˆ ∈ Wper^1,p(Y)×L^p′(Y;R^N) and (U, ξ₀) ∈L^p(Ω;Ke_B)×L^p′(Ω;R^N) and a subsequence (ε_n, δ_n)_n≥1 going to (0,0) and satisfying (3.4) such that, as n→ ∞, (3.12), (3.13), (3.14)and (3.15)are satisfied.

Proof. The proof (given here for the case k₁ > 0) is broken down in four steps.

–Step one: weak compactness of the unfolded sequences Tε(∇uε,δ),Tε(d_ε,δ) .

From inequality (3.1) and takingu_ε,δas a test function in equation (3.2), we deduce that the sequence (u_ε,δ)εis bounded inW^1,p(Ω) and the sequence (d_ε,δ)_ε is bounded in L^p′(Ω;R^N). Hence they are weakly compact.

Assume nowu_ε,δ⇀ u₀weakly inW₀^1,p(Ω) andd_εn,δ⇀ d₀ inL^p′(Ω;R^N).

By Theorem 2.2 (6), up to a subsequence, there is some ˆu∈L^p Ω;W_per^1,p(Y) such that

Tε(∇uε,δ)⇀∇xu₀+∇yuˆ

weakly inL^p(R^N×Y;R^N). Consequently, the sequenceTε(d_ε,δ) is bounded in the space L^p′(R^N ×Y). Using Theorem 2.2 (3), again up to a sub- sequence, there exists some η ∈ L^p′(R^N ×Y), so that Tε(d_ε,δ) ⇀ η and d₀ = ¹

|Y|

R

Y η(y) dy. So, setting ˆd=η−d₀, one has Z

Y

dˆdy= 0 and

Tε(d_ε,δ)⇀ d₀+ ˆdweakly inL^p′(R^N×Y).

–Step two: the equation satisfied by the corresponding weak limits.

Letϕ∈ D(Ω) andψ∈C^∞(R^N), such thatψisY-periodic and vanishes on a neighborhood of the origin. The function ϕ_ε(x) = εϕ(x)ψ(x/ε) is a valid test function for (3.2). As ε → 0, by Theorem 2.2, ϕ_ε ⇀ 0 weakly inW^1,p(Ω) andTε(∇ϕε)→ϕ(x)∇ψ(y) strongly inL^p(Ω;Wper^1,p(Y)), so that going to the limit in

1

|Y| Z

RN×Y

hTε(d_ε,δ),Tε(∇ϕε)i dx dy= Z

Ω

f_ε,δϕ_ε dx.

gives

1

|Y| Z

Ω×Y

hd0(x) + ˆd(x, y), ϕ(x)∇ψ(y)i dx dy= 0.

Since ϕ is arbitrary and N > p and making use of Lemma 3.5, we deduce that for almost everyx∈Ω, for everyψ∈W_per^1,p(Y),

1

|Y| Z

Y

hd(x, y),ˆ ∇yψ(y)i dy= 0,

i.e. −div ˆd(x,·) = 0 in (C_per^∞)^′(Y). This is (3.13), the first equation of the unfolded formulation for the limit problem.

– Step three: weak convergence (up to a subsequence) of the unfolded sequences (T_ε,δ(u_ε,δ)),(∇zT_ε,δ(u_ε,δ)) and (δ^N/p′T_ε,δ(d_ε,δ)).

Using property 7 of Theorem 2.7, we know that there exists a U in L^p(Ω;L^p_loc(R^N)) such that (again, up to a subsequence),

Tε,δ(u_ε,δ)⇀ U weakly in L^p(Ω;L^p_loc(R^N)). (3.19)

Furthermore, Proposition 2.5 gives M_Y^ε(u_ε,δ)1¹

δY →u₀ strongly inL^p(Ω;L^p_loc(R^N)). (3.20) On the other hand, using property 7) of Theorem 2.7 we deduce that there exists a W inL^p(Ω;L^p∗(R^N)) with∇zW inL^p(Ω×R^N) with

M_Y^ε(u_ε,δ)1¹

δY − Tε,δ(u_ε,δ)⇀ W (3.21) weakly inL^p(Ω;L^p∗(R^N)). From (3.19), (3.20) and (3.21), it follows that

U+W =u₀ and ∇zU =−∇zW.

Finally, by Theorem 2.7 (7) again, the following weak convergence inL^p(Ω× R^N) holds:

∇z Tε,δ(u_ε,δ) 1¹

δY =εδTε,δ(∇uε,δ)⇀∇zU. (3.22) Recalling Definition 2.6, we know that T_ε,δ(u_ε,δ) = 0 in Ω×B, so that using (3.19), we obtain that

U = 0 on Ω×B.

Therefore,W =u₀−U belongs to L^p(Ω;K_B) and this implies that W(·, B) =ℓ(U) =u₀ inL^p(Ω).

Now set

ξ_ε,δ =δ^N/p′Tε,δ(d_ε,δ). (3.23) By definition,

ξ_ε,δ ∈A_ε,δ(δ^NpTε,δ(∇uε,δ)). (3.24) Since the sequence (d_ε,δ) is bounded inL^p′(Ω×R^N), Theorem 2.7 (3) implies that there existsξ₀ such that (up to a subsequence),

ξ_ε,δ ⇀ ξ₀ weakly inL^p′(Ω×R^N). (3.25) Furthermore, using (3.24) together with Theorem 2.7 (5), it follows that

ξ_ε,δ ∈A_ε,δδ^N/p−1

ε ∇zT_ε,δ(u_ε,δ)

. (3.26)

–Step four: the equations satisfied by the corresponding weak limits.

In order to describe the contribution of the perforations, we use the functions w_ε,δ introduced in Lemma 3.6 associated withv∈K_B∩ D(R^N). Recall that these functions vanish on the holes. For ψ in D(Ω), using w_ε,δψ as a test function in (3.2), one has

Z

Ω^∗_ε,δ

hd_ε,δ, ψ∇w_ε,δ+w_ε,δ∇ψi dx= Z

Ω^∗_ε,δ

fεw_ε,δψ dx.

For simplicity, let us denote byE_ε¹ andE_ε² the terms:

E_ε¹ = Z

Ω^∗_ε,δ

hd_ε,δ,∇w_ε,δiψ dx, E_ε² =

Z

Ω^∗_ε,δ

hd_ε,δ,∇ψiw_ε,δ dx.

First, recalling convergence (3.17) together with the strong convergence of the sequence (f_ε), we know that

εlim→0

E_ε¹+E²_ε

=v(B) Z

Ω

f ψ dx. (3.27)

We study separately each term of the left-hand side of the previous expres- sion. Let us start with the second one. From the strong convergence of (w_ε,δ) in L^p_loc to v(B) (which follows from (3.17)), together with the weak convergence of (d_ε,δ) in L^p′, we deduce that

εlim→0E_ε² =v(B) Z

Ω

hd0,∇ψi dx. (3.28)

We now apply the unfolding operator Tε,δ to the the first term (equality follows from the fact thatψ has a compact support in Ω):

E_ε¹ = δ^N

|Y| Z

Ω×¹δY

hT_ε,δ(d_ε,δ),T_ε,δ(∇w_ε,δ)i T_ε,δ(ψ) dx dz

= 1

|Y| Z

Ω×¹δY

hδ^N/p′T_ε,δ(d_ε,δ), δ^N/pT_ε,δ(∇w_ε,δ)i T_ε,δ(ψ) dx dz.

With our notations, E_ε¹ can be rewritten as E_ε¹ = 1

|Y| Z

Ω×¹δY

hξ_ε,δ,δ^N/p−1

ε (−∇zv)i T_ε,δ(ψ) dxdz.

On the other hand, using the inequality

||Tε,δ(ψ)−ψ||_L∞(Ωbε×¹δY)≤C ε||∇ψ||_L^∞_(Ω), we deduce that the following convergence holds uniformly

Tε,δ(ψ)→ψ. (3.29)

Putting now (3.4), (3.25) and (3.29) together, we obtain that

εlim→0E_ε¹ = k₁

|Y| Z

Ω×RN

hξ0,(−∇zv)iψ dx dz. (3.30)

From (3.27), (3.28) and (3.30), the limit equation becomes:

v(B) Z

Ω×Y

hd₀,∇ψi dx− k₁

|Y| Z

Ω×R^N\B

hξ₀, ψ∇_zvi dx dz

=v(B) Z

Ω

f ψ dx, (3.31)

which, by density, holds true for all ψ ∈ W₀^1,p(Ω) and v ∈ K_B. Choosing v(B) = 0 in (3.31) gives (3.14). This implies that

−divzξ₀= 0 inD^′(Ω×(R^N \B)). (3.32) Now, Stoke’s formula associated with (3.32), gives for every v inK_B

Z

R^N\B

hξ0,∇zvidz=v(B) Z

∂B

(ξ0 · νB) dσz. (3.33) Multiplying (3.33) by ψ, integrating over Ω and combining with (3.31) gives (3.15).

We introduce a new functional spaceK^p(Ω;B) to which the functionW obtained in (3.21) already belongs:

K^p(Ω;B) = {V ∈L^p(Ω;K_B) such thatV(x, B)∈W₀^1,p(Ω)},(3.34) with the norm:

kVk^p_Kp(Ω;B)=kVk^p_Lp(Ω×RN)+k∇zVk^p

L^p(Ω;L^p∗(R^N\B))+k∇xV(., B)k^p_Lp(Ω). (3.35) Proposition 3.8. For all V ∈K^p(Ω, B),

1

|Y| Z

Ω×Y

hd0,∇xV(x, B)i dx dy− k₁

|Y| Z

Ω×RN\B

hξ0,∇zV(x, B)i dx dz

= Z

Ω

f V(x, B) dx. (3.36)

Proof. For almost everyx∈Ω, V(x,·) belongs toKB, so by (3.33), one gets Z

R^N\B

hξ0,∇zV(x, z)i dz=V(x, B) Z

∂B

(ξ₀ · ν_B) dσ_z. (3.37) Denote byψthe function V(x, B) and integrate (3.37) over Ω. Combin- ing the result with (3.15) gives the required conclusion (3.36).