Keywords: Degenerating problem, multi-structure, anisotropic, reduction of dimension, nonlocal problem. AMS Classification: 35B25, 35B27, 35B40

members of the "Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA)" of the "Istituto Nazionale di Alta Matematica (INdAM)" , A. Sili

Introduction

We investigate a degenerating elliptic problem in a multi-structure Ω ε of R 3 , in the framework of the thermal stationary conduction with highly contrasting diffusivity.

Precisely, for a small parameter ε in ]0, 1], we consider a multi-structure Ω ε in R 3 consisting of a fixed base Ω -surmounted by a thin cylinder Ω + ε with height 1 and crosssection with diameter of order ε (see Fig. 1).

We assume that Ω ε is filled with two different materials. Specifically, Ω + ε contains a cylindrical core F ε , always with height 1 and cross-section with diameter of order ε, Figure 1: The domain Ω ε filled with a material with conductivity of order 1, surrounded by an insulation material in M ε = int(Ω + ε \ Fε ) with conductivity of order ε 2 . Also Ω -is filled with a material with conductivity of order ε 2 . Namely, the diffusivity matrix is given by

A ε (X ′ , x 3 ) = χ Fε + ε 2 χ Mε (X ′ , x 3 )A X ′ ε , x 3 + ε 2 χ Ω -(X ′ , x 3 )A(X ′ , x 3 ), a.e. in Ω ε ,
where A is a uniformly elliptic matrix with coefficients in L ∞ (Ω + ), X ′ denotes a generic element (X 1 , X 2 ) of R 2 , and Ω + = Ω 1 (i.e., Ω + ε with ε = 1). We assume that the source term is of order 1 in Ω + ε and of order ε 2 in Ω -. Namely, the source term is given by

f ε (X ′ , x 3 ) = f X ′ ε , x 3 χ Ω + ε (X ′ , x 3 ) + ε 2 f χ Ω -(X ′ , x 3 ), a.e. in Ω ε , with f ∈ L 2 (Ω + ∪ Ω -).
By assuming that the temperature is zero on the top and on the bottom Γ D ε of the boundary of Ω ε , while the flux is zero on the remaining part Γ N ε of the boundary, we are interested in the following problem

           -div(A ε ∇ū ε ) = f ε in Ω ε , A ε ∇ū ε • ν = 0 on Γ ε N ,
ūε = 0 on Γ D ε , where ν denotes the exterior unit normal to ∂Ω ε .

A numerical analysis of this problem could be not available since the required discretization step could become too small for the machine precision. So an asymptotic analysis of the problem, as ε vanishes, is needed.

First note that a similar problem was studied in [?], in the case where Ω -is filled with a material with conductivity of order 1. In this case the authors proved that the diffusion through the base Ω -is not seen at the limit, and only the diffusion in thin cylinder Ω + ε is taken into account. Moreover, they proved that the diffusion in Ω + ε occurs mainly along the vertical axis of its conductive part and it is described especially by a function v depending only on the vertical variable x 3 . More precisely, they proved that the limit temperature involves also other two functions which are related to the diffusion in the insulation part. Hence the limit problem appears to be nonlocal. About nonlocal limit problems arising from contrasting diffusivity in thin cylinders see also [?] and [?].

The novelty of our paper is the presence of material with conductivity of order ε 2 also in Ω -. In this case we prove that also a diffusion through the base Ω -appears at the limit, but it is completely independent of the limit diffusion along thin cylinder Ω + ε . The limit behavior in the thin cylinder is almost the same as in [?]. Then, two uncoupled problems appear at the limit, but some boundary conditions in the two limit problem retain memory of the initial junction. We refer to Section ?? for the formulation of the problem and the statement of the main results (see Theorem ??, Theorem ??, and Remark ??).

By following [?], the problem is rescaled from Ω + ε to Ω + . Precisely, setting

u + ε (x ′ , x 3 ) = ūε (εx ′ , x 3 ) a.e. in Ω + , u - ε (x ′ , x 3 ) = ūε (x ′ , x 3 ) a.e. in Ω -,
the following interface condition

u + ε (x ′ , 0) = u - ε (εx ′ , 0) (1)
appears on the surface separating Ω + and Ω -. Section ?? is devoted to obtaining some sharp a priori norm-estimates for (u + ε , u - ε ). These estimate are sharp in the sense that they depend on the zone where the material has a low conductivity, or a high conductivity. Then, a first convergence result is obtained.

From a mathematical point of view, one of the main difficulties of this paper is to build suitable test functions that, roughly speaking, at the same time take into account the interface condition in (??), but they also allow to split the problem in two uncoupled problems at the limit. These test functions are build in Section ?? and they are based on a density result proved in Section ??. Section ?? is devoted to the proof of the main results (i.e., Theorem ?? and Theorem ??).

We refer to [?] for problems in thin multi-structures. We refer to [?], [?], and [?] for other problems in the same multi-structure of this paper.

Numerous studies have been carried out on the homogenization of high-contrast problems. We just recall the founding work of T. Arbogast, J. Douglas, and U. Hornung For spectral stiff problems in domains surrounded by thin layers, we refer to [?], [?], [?] and references therein.

Setting of the problem and main results

In what follows, X ′ (resp. x ′ ) denotes a generic element (X 1 , X 2 ) (resp.

(x 1 x 2 )) of R 2 , (X ′ , x 3 ) (resp. (x ′ , x 3 )) denotes a generic element X = (X 1 , X 2 , X 3 ) (resp. (x 1 , x 2 , x 3 )) of R 3 , ∇ ′ v denotes ∂v ∂X 1 , ∂v ∂X 2
or ∂v ∂x 1 , ∂v ∂x 2 , as appropriate, v ′ denotes ∂v ∂x 3 if v depends only on x 3 , and χ A denotes the characteristic function of a set A. Moreover, ε denotes a parameter taking values in a vanishing sequence in ]0, 1[.

For each ε, we consider the multi-structure in R 3 (see Fig. 1)

Ω ε = Ω -∪ (εω × {0}) ∪ Ω + ε ,
consisting of the fixed base

Ω -=] -1, 1[ 2 ×] -1, 0[, surmounted by the thin cylinder Ω + ε = εω × I, where ω = - 1 2 , 1 2
2 , I =]0, 1[.
The set εω × {0} is the surface separating Ω -from Ω + ε . We assume that Ω ε is filled with two different materials. Precisely, Ω + ε contains a cylindrical core

F ε = εD × I
filled with a material with conductivity of order 1, where D is the circle in R 2 with centre the origin and radius r ∈]0, 1 2 [, surrounded by an insulation material in

M ε = int(Ω + ε \ Fε )
with conductivity of order ε 2 . Also Ω -is filled with a material with conductivity of order ε [START_REF] Bellieud | Homogenization of elliptic problems in a fiber reinforced structure. Nonlocal effects[END_REF] .

For what follows, we set

Ω + = ω × I and Ω = Ω -∪ (ω × {0}) ∪ Ω + .
The aim of this paper is to study the asymptotic behaviour, as ε tends to zero, of the weak solution to the following problem

           -div(A ε ∇ū ε ) = f ε in Ω ε , A ε ∇ū ε • ν = 0 on Γ ε N , ūε = 0 on Γ D ε , (2)
where ν denotes the exterior unit normal to ∂Ω ε , Γ ε D denotes the top and the bottom of Ω ε , while Γ N ε denotes the remaining part of the boundary of Ω ε , i.e.

Γ D ε = Γ -∪ Γ + ε , Γ -=] -1, 1[ 2 ×{-1}, Γ + ε = - ε 2 , ε 2 2 × {1}, and Γ N ε = ∂Ω ε \ Γ D ε ;
A ε is the diffusivity matrix defined by

A ε (X ′ , x 3 ) = χ Fε + ε 2 χ Mε (X ′ , x 3 )A X ′ ε , x 3 +ε 2 χ Ω -(X ′ , x 3 )A(X ′ , x 3 ), a.e. in Ω ε , (3) with A a 3 × 3 matrix-valued function on Ω such that    A ∈ (L ∞ (Ω)) 3×3 , ∃α ∈]0, +∞[ : A (x) λλ ≥ α |λ| 2 , a.e. x ∈ Ω, ∀λ ∈ R 3 ; (4)
and f ε is the source term defined by

f ε (X ′ , x 3 ) = f X ′ ε , x 3 χ Ω + ε (X ′ , x 3 ) + ε 2 f (X ′ , x 3 )χ Ω -(X ′ , x 3 ), a.e. in Ω ε , with f ∈ L 2 (Ω) . ( 5
)
The weak formulation of problem (??) is given by

                                       ūε ∈ H 1 (Ω ε ), ūε = 0 on Γ D ε , Fε A X ′ ε , x 3 ∇ū ε ∇vdX ′ dx 3 + ε 2 Mε A X ′ ε , x 3 ∇ū ε ∇vdX ′ dx 3 +ε 2 Ω - A (X ′ , x 3 ) ∇ū ε ∇vdX ′ dx 3 = Ω + ε f X ′ ε , x 3 vdX ′ dx 3 + Ω - f (X ′ , x 3 ) vdX ′ dx 3 , ∀v ∈ H 1 (Ω ε ) : v = 0 on Γ D ε . ( 6
)
The Lax-Milgram Theorem ensures the existence and the uniqueness of the solution to problem (??).

As it is usual (see [?], [?]), problem (??) will be reformulated on the fixed domain Ω through the following map

T ε : (x ′ , x 3 ) ∈ Ω → (εx ′ , x 3 )χ Ω + ∪(ω×{0}) + (x ′ , x 3 )χ Ω -∈ Ω ε .
Precisely, setting

u ε (x ′ , x 3 ) =    u + ε (x ′ , x 3 ) = ūε (εx ′ , x 3 ) a.e. in Ω + ∪ (ω × {0}) , u - ε (x ′ , x 3 ) = ūε (x ′ , x 3 ) a.e. in Ω -, (7)
the following interface condition

u + ε (x ′ , 0) = u - ε (εx ′ , 0) a.e. in ω,
holds true. Consequently, introducing the space

H 1 D,ε (Ω) = ϕ ε = (ϕ + ε , ϕ - ε ) : ϕ + ε ∈ H 1 (Ω + ), ϕ - ε ∈ H 1 (Ω -), ϕ + ε = 0 on Γ + , ϕ - ε = 0 on Γ -, ϕ + ε (x ′ , 0) = ϕ - ε (εx ′ , 0) a.e. x ′ ∈ ω ,
where

Γ + = - 1 2 , 1 2 2 × {1}, problem (??) is rescaled in the following one                        u ε ∈ H 1 D,ε (Ω), F A    1 ε ∇ ′ u + ε ∂u + ε ∂x 3       1 ε ∇ ′ ϕ + ε ∂ϕ + ε ∂x 3    dx + M A   ∇ ′ u + ε ε ∂u + ε ∂x 3     ∇ ′ ϕ + ε ε ∂ϕ + ε ∂x 3   dx + Ω - A∇u - ε • ∇ϕ - ε dx = Ω + f ϕ + ε dx + Ω - f ϕ - ε dx, ∀ϕ ε ∈ H 1 D,ε (Ω), (8)
where

F = D × I and M = Ω + \ F .
To stating the main result of this paper, the following spaces are needed.

U = {ϕ ∈ L 2 (I; H 1 (ω)) : ϕ| F = 0}, (9)
H 1 d (I) = {ϕ ∈ H 1 (I) : ϕ(1) = 0}, (10)
W = ϕ ∈ L 2 (I; H 1 (D)) : D ϕ(x ′ , x 3 )dx ′ = 0 a.e. x 3 ∈ I , (11)
H 1 d (Ω -) = ϕ ∈ H 1 (Ω -) : ϕ(x ′ , -1) = 0 a.e. x ′ ∈ (-1, 1) 2 , ( 12
)
In what follows, the subscript t will refer to test functions.

The main result of this paper is the following one.

Theorem 2.1. For every ε, let u ε be the unique weak solution to problem (??), under assumptions (??) and (??). Then, there exist a quadruple (u

+ , v, w, u -) in U × H 1 d (I) × W × H 1 d (Ω -) such that u + ε → u + + v strongly in L 2 (M ), (13)
∇ ′ u + ε → ∇ ′ u + strongly in L 2 (M ) 2 , ( 14
)
ε ∂u + ε ∂x 3 → 0 strongly in L 2 (M ), (15)
u + ε → v strongly in H 1 (F ), (16)
1 ε ∇ ′ u + ε → ∇ ′ w strongly in L 2 (F ) 2 , ( 17
)
u - ε → u -strongly in H 1 (Ω -), (18)
as ε tends to zero, where (u + , v, w) is the unique solution to the following problem

                   (u + , v, w) ∈ U × H 1 d (I) × W, F A ∇ ′ w v ′ ∇ ′ w t v ′ t dx + M A ∇ ′ u + 0 ∇ ′ u + t 0 dx = Ω + f (u + t (x) + v t (x 3 )) dx, ∀(u + t , v t , w t ) ∈ U × H 1 d (I) × W, ( 19
)
while u -is the unique solution to the following problem

       u -∈ H 1 d (Ω -), Ω - A∇u -∇u - t dx = Ω - f u - t dx, ∀u - t ∈ H 1 d (Ω -). (20)
In order to highlight the nonlocal character of the diffusion in the upper part of the domain, now another representation of problem (??) will be given. Moreover, Theorem ?? will be translated in terms of the asymptotic behavior of the solution ūε of problem (??). Theorem 2.2. For every ε, let u ε be the unique solution to (??), under assumptions (??) and (??). Then

1 ε 2 |ω| εω u ε (X ′ , •) dX ′ → v(•) + 1 |ω| 1 |ω \ D| ω\D f (x ′ , •) dx ′ ω\D û(x ′ , •) dx ′ + ω\D ũ(x ′ , •) dx ′ strongly in L 2 (I), as ε → 0, ( 21
)
where ũ is the unique solution to the following problem

       ũ ∈ U , M A ∇ ′ ũ 0 ∇ ′ ũt 0 dx = M f - 1 |ω \ D| ω\D f dx ′ ũt dx, ∀ũ t ∈ U , ( 22
)
for a.e. x 3 in I, û(•, x 3 ) is the unique solution to the following problem

               û(•, x 3 ) ∈ H 1 (ω), û(x ′ , x 3 ) = 0 a.e. x ′ ∈ D, ω\D A ∇ ′ û 0 ∇ ′ ût 0 dx ′ = ω\D ût dx ′ , ∀û t ∈ H 1 (ω), ût (x ′ , x 3 ) = 0 a.e. x ′ ∈ D, ( 23
)
and v is the unique solution to the following problem

       v ∈ H 1 d (I), I a 0 (x 3 )v ′ (x 3 )v ′ t (x 3 )dx 3 = I ω f (x ′ , x 3 ) dx ′ v t (x 3 )dx 3 , ∀v t ∈ H 1 d (I). (24)
In (??) the coefficient a 0 is defined by

a 0 (x 3 ) = D 2 j=1 a 3j (x ′ , x 3 ) ∂ ŵ ∂x j (x ′ , x 3 ) + a 33 (x ′ , x 3 ) dx ′ , a.e. x 3 ∈ I,
where, for a.e. x 3 in I, ŵ(•, x 3 ) is the unique solution to the elementary equation

           ŵ(•, x 3 ) ∈ H 1 (D), D ŵ(x ′ , x 3 )dx ′ = 0, D A ∇ ′ ŵ 1 ∇ ′ ŵt 0 dx ′ = 0, ∀ ŵt ∈ H 1 (D). ( 25
)
Moreover, ŵ belongs to W ∩ L ∞ (I; H 1 (D)) and û belongs to U ∩ L ∞ (I; H 1 (ω)).
Remark 2.1. From (??) it appears clearly that the diffusion in the fixed base Ω -of the multi-domain is completely independent of the diffusion along the thin cylinder Ω + ε . The right-hand side of (??) represents the limit cross-section average of the temperature in the upper part of the multi-domain. The term v appearing in (??) means that the diffusion in the thin cylinder occurs mainly along the vertical axis of its conductive part. On the other side, v is not sufficient to determine the average of the limit temperature since such average requires to compute û and ũ which are related to the diffusion in the insulation part M of Ω + ; hence the limit problem is not local. Moreover, the function ŵ which depends only on the diffusivity matrix is due to the anisotropy of the conductive part of the thin cylinder, and it is taken into account in the effective diffusivity coefficient a 0 (x 3 ). When the conductive part is made up of an isotropic material for which the entries of the matrix A are such that A 13 = A 23 = 0, then w = 0 (for instance, [?], [?] and [?]).

Note also that the interface condition disappears at the limit giving rise to Neumann boundary conditions on v and u -on the interface x ′ = 0.

Finally, we point out that in the terminology of correctors, the strong convergences stated in the Theorem 1.1 provide a corrector for u ε which means that u + ε behaves as u + + v + εw while u - ε behaves as u -.

A priori norm-estimates and convergence results

This section is devoted to proving some a priori norm-estimates for the solutions to problem (??). First, we recall a well-known result, the proof of which is just stated for the sake of clarity.

L p (A) ≤ C∥∇u∥ L p (A) , ∀u ∈ W 1,p (A). ( 26
)
Proof. Set

u A = 1 |A| A u dx, u B = 1 |B| B u dx. Assume p in [1, +∞[. The Poncaré-Wirtingen inequality provides the existence of C 1 in ]0, +∞[ such that ∥u -u B ∥ L p (A) ≤ ∥u -u A ∥ L p (A) + ∥u A -u B ∥ L p (A) ≤ C 1 ∥∇u∥ L p (A) + |u A -u B ||A| 1 p , ∀u ∈ W 1,p (A). (27)
In the other side, the Hölder inequality and again the Poncaré-Wirtingen inequality give

|u A -u B | ≤ 1 |B| B |u -u A | dx ≤ 1 |B| A |u -u A | dx ≤ |A| 1-1 p |B| ∥u -u A ∥ L p (A) ≤ |A| 1-1 p |B| C 1 ∥∇u∥ L p (A) , ∀u ∈ W 1,p (A). (28)
Eventually, (??) follows from (??) and (??). The proof in the case p = +∞ is similar.

Proposition 3.1. For every ε, let u ε be the solution to problem (??). Then there exists a positive constant C, independent of ε, such that

∥u + ε ∥ L 2 (Ω + ) ≤ C, ( 29
)
∂u + ε ∂x 3 L 2 (F ) ≤ C, ( 30
)
1 ε ∥∇ ′ u + ε ∥ (L 2 (F )) 2 ≤ C, ( 31
)
ε ∂u + ε ∂x 3 L 2 (M ) ≤ C, ( 32
)
∥∇ ′ u + ε ∥ (L 2 (M )) 2 ≤ C, ( 33
) ∥u - ε ∥ L 2 (Ω -) ≤ C, ( 34
)
∥∇ ′ u - ε ∥ (L 2 (Ω -)) 2 ≤ C, ( 35
)
for every ε.

Proof. Obviously,

∥u + ε ∥ 2 L 2 (Ω + ) ≤ 2 Ω + u + ε (x ′ , x 3 ) - 1 |D| D u + ε (y ′ , x 3 )dy ′ 2 dx ′ dx 3 + 2 Ω + 1 |D| D u + ε (y ′ , x 3 ) dy ′ 2 dx ′ dx 3 , ∀ε. ( 36
)
As far as the first integral in (??) is concerned, Lemma ?? ensures the existence of

C in ]0, +∞[ such that ω u + ε (x ′ , x 3 ) - 1 |D| D u + ε (y ′ , x 3 )dy ′ 2 dx ′ ≤ C ω ∇ ′ u + ε (x ′ , x 3 ) 2 dx ′ , ∀ε, a.e. x 3 ∈ I,
which implies

Ω + u + ε (x ′ , x 3 ) - 1 |D| D u + ε (y ′ , x 3 )dy ′ 2 dx ′ dx 3 ≤ C Ω + ∇ ′ u + ε (x ′ , x 3 ) 2 dx ′ dx 3 . ∀ε, ( 37
)
by an integration on x 3 in I.

As far as the second integral in (??) is concerned, the Cauchy-Schwarz inequality and the Poincaré inequality provide that

Ω + 1 |D| D u + ε (y ′ , x 3 ) dy ′ 2 dx ′ dx 3 ≤ |ω| F |u + ε | 2 dx ≤ |ω| F ∂u + ε ∂x 3 2 dx, ∀ε. (38)
Combining (??), (??), and (??) gives the existence of C in ]0, +∞[ such that

∥u + ε ∥ 2 L 2 (Ω + ) ≤ C ∥∇ ′ u + ε ∥ 2 (L 2 (Ω + )) 3 + ∂u + ε ∂x 3 2 L 2 (F ) , ∀ε. ( 39
)
Choosing u ε as test function in (??), thanks to (??) and the Cauchy-Schwarz inequality, get to

α ε 2 ∥∇ ′ u + ε ∥ 2 (L 2 (F )) 2 + α ∂u + ε ∂x 3 2 L 2 (F ) + α∥∇ ′ u + ε ∥ 2 (L 2 (M )) 2 + αε 2 ∂u + ε ∂x 3 2 L 2 (M ) +α∥∇u - ε ∥ 2 (L 2 (Ω -)) 3 ≤ ∥f ∥ L 2 (Ω) (∥u + ε ∥ L 2 (Ω + ) + ∥u - ε ∥ L 2 (Ω -) ), ∀ε. (40)
Combining (??), (??) and Poincaré inequality on Ω -, leads to existence of C in ]0, +∞[ such that

1 ε ∥∇ ′ u + ε ∥ (L 2 (F )) 2 + ∂u + ε ∂x 3 L 2 (F ) + ∥∇ ′ u + ε ∥ (L 2 (M )) 2 + ε ∂u + ε ∂x 3 L 2 (M ) +∥∇u - ε ∥ (L 2 (Ω -)) 3 2 ≤ C 1 ε ∥∇ ′ u + ε ∥ (L 2 (F )) 2 + ∂u + ε ∂x 3 L 2 (F ) + ∥∇ ′ u + ε ∥ (L 2 (M )) 2 + ε ∂u + ε ∂x 3 L 2 (M ) +∥∇u - ε ∥ (L 2 (Ω -)) 3 , ∀ε, (41)
Eventually, estimates (??), (??),(??), (??), and (??) follow from (??). Estimate (??) follows from (??), (??), (??), and (??). Estimates (??) follows from (??) and the Poincaré inequality in Ω -.

Proposition ?? allows us to obtain the following convergence result.

Proposition 3.2. For every ε, let u ε be the solution to problem (??). Let U, H 1 d (I), W, and H 1 d (Ω -) be tha spaces defined in (??), (??), (??), and (??). Then, there exist a subsequence of {ε} , still denoted by {ε}, a quadruple (u

+ , v, w, u -) in U × H 1 d (I) × W × H 1 d (Ω -)
, depending possibly on the selected subsequence, such that

u + ε ⇀ u + + v weakly in L 2 (M ), (42)
∇ ′ u + ε ⇀ ∇ ′ u + weakly in L 2 (M ) 2 , ( 43
)
ε ∂u + ε ∂x 3 ⇀ 0 weakly in L 2 (M ), (44)
u + ε ⇀ v weakly in H 1 (F ), (45)
1 ε ∇ ′ u + ε ⇀ ∇ ′ w weakly in L 2 (F ) 2 , ( 46
)
u - ε ⇀ u -weakly in H 1 (Ω -), (47)
as ε → 0.

Proof. Estimate (??), (??) and (??) provide the existence of a subsequence of {ε} , still denoted by {ε} , and

u 0 ∈ L 2 (I; H 1 (ω)), (48)
depending possibly on the selected subsequence, such that

u + ε ⇀ u 0 weakly in L 2 (Ω + ), as ε → 0, (49)
∇ ′ u + ε ⇀ ∇ ′ u 0 weakly in (L 2 (Ω + )) 2 , as ε → 0, (50)
and

∇ ′ u 0 = 0, a.e. in F. ( 51
)
Equation (??) implies that in F u 0 depends only on x 3 , i.e. there exists a function

v ∈ L 2 (I) (52) such that u 0 (x) = v(x 3 ), a.e. in F. ( 53
)
Consequently, setting

u + : x = (x ′ , x 3 ) ∈ Ω + → u + (x) = u 0 (x) -v(x 3 ), (54)
thanks to (??), (??) and (??) one has that

u + ∈ U (55)
and convergences (??) and (??) can be rewritten as

u + ε ⇀ u + + v weakly in L 2 (Ω + ), as ε → 0, ( 56
)
∇ ′ u + ε ⇀ ∇ ′ u + weakly in (L 2 (Ω + )) 2 , as ε → 0. ( 57
)
Now, let us prove that v ∈ H 1 (I).

Indeed, (??) provides the existence of a subsequence of {ε} , still denoted by {ε} , and of a function

u 1 ∈ L 2 (Ω + ), (59)
depending possibly on the selected subsequence, such that

∂u + ε ∂x 3 χ F ⇀ u 1 χ F weakly in L 2 (Ω + ), as ε → 0. ( 60
)
Moreover, since u belongs to U, (??) gives that

u + ε χ F ⇀ vχ F weakly in L 2 (Ω + ), as ε → 0, (61)
which implies

∂u + ε ∂x 3 χ F = ∂ ∂x 3 (u + ε χ F ) → ∂ ∂x 3 (vχ F ) = v ′ χ F in D ′ (Ω + ), as ε → 0, (62)
since χ F is independent of x 3 . From (??) and (??) it follows that

u 1 (x) = v ′ (x 3 ), a.e. in F, ( 63
)
which combinined with (??) and (??) proves (??). Now, let us prove that v(1) = 0.

(64)

Indeed, thanks to (??), (??), and (??)

                   Ω + u + ε χ F φ dx → Ω + vχ F φ dx, Ω + ∂u + ε ∂x 3 χ F φ dx → Ω + v ′ χ F φ dx, as ε → 0, ∀φ ∈ L 2 (I), (65)
that is

                   I D u + ε dx ′ φ dx 3 → |D| I vφ dx 3 , I D u + ε dx ′ ′ φ dx 3 = I D ∂u + ε ∂x 3 dx ′ φ dx 3 → |D| I v ′ φ dx 3 , as ε → 0, ∀φ ∈ L 2 (I). ( 66
)
This means that

D u + ε dx ′ ⇀ |D|v weakly in H 1 (I), (67)
which implies (??), since

D u + ε (x ′ , 1) dx ′ = 0, ∀ε. ( 68
)
At this stage, we have proved (??), (??), (??), (??), and (??). Indeed, (??) and (??) follow from (??), (??), (??), (??) and (??). Convergence (??) follows from (??) and (??). Convergence (??) follows from (??), (??), and (??). Eventually, (??) is an immediate consequence of estimates (??) and (??).

Let as prove (??). For every ε, let us set

w ε : x = (x ′ , x 3 ) ∈ F → u ε (x) ε - 1 |D| D u ε (x ′ , x 3 ) ε dx ′ . ( 69
)
Thanks to (??),

∥∇ ′ w ε ∥ (L 2 (F )) 2 ≤ C, ∀ε. ( 70
)
On the other side, thanks to the Poincaré-Wirtinger inequality there exists c in ]0, +∞[ such that

D |w ε (x ′ , x 3 )| 2 dx ′ ≤ c D |∇ ′ w ε (x ′ , x 3 )| 2 dx ′ , a.e. x 3 ∈ I, ∀ε, ( 71
) since D w ε (x ′ , x 3 ) dx ′ = 0, a.e. x 3 ∈ I, ∀ε. ( 72
)
Integrating (??) on x 3 over I and using (??) give

∥w ε ∥ L 2 (F ) ≤ √ c C, ∀ε. ( 73
)
Hence (??), (??), and (??) provide the existence of w in L 2 (I; W) such that

w ε ⇀ w weakly in L 2 (I; W), as ε → 0, (74)
which implies (??).

Density results

This section is devoted to prove a density result, which proof is obtained by adapting the proof of Proposition 3.1. in [?].

Let us set

V = (v + , v -) ∈ C 1 ([0, 1]) × W 1,∞ (Ω -) : ∃I v -⊂ R 3 neighborhood of (0 ′ , 0) depending on v -: v -= v + (0), in I v -∩ Ω -, v + (1) = 0, v -(x ′ , -1) = 0, a.e. x ′ ∈] -1, 1[ 2 .
(75) Proposition 4.1. Let V be defined in (??) respectively. Then V is dense in

H 1 d (]0, 1[) × H 1 d (Ω -) in H 1 -norm. Proof. Let us set V = (v + , v -) ∈C 1 ([0, 1]) × C 1 (Ω -) : v + (1) = 0, v -(x ′ , -1) = 0, a.e. x ′ ∈] -1, 1[ 2 .
(76)

To obtain the thesis it is enough to prove that V is dense in V in H 1 -norm.

Let (v + , v -) ∈ V . Let us prove that there exists a sequence

{(v + n , v - n )} n∈N in V such that v + n → v + in H 1 (]0, 1[), (77)
v - n → v -in H 1 (Ω -). (78)
To this aim, for every n in N, let us set

B 1 n = x = (x ′ , x 3 ) : |x| < 1 n , x 3 < 0 , B 2 n = x = (x ′ , x 3 ) : |x| < 2 n , x 3 < 0 ,
and

φ n : x ∈ B 2 n \ B 1 n → φ n (x) = n dist x, ∂B 2 n = 2 -n|x|. Obviously,                              φ n ∈ C 1 (B 2 n \ B 1 n ), ∀n, 0 ≤ φ n ≤ 1, in B 2 n \ B 1 n , ∀n, |∇φ n | = n, in B 2 n \ B 1 n , ∀n, φ n = 1, on ∂B( 1 n ), ∀n, φ n = 0, on ∂B( 2 n ), ∀n. (79)
Now, for every n in N, let us set

v + n = v + , in Ω + , ( 80
) v - n =              v + (0), in B 1 n , φ n v + (0) + (1 -φ n )v -, in B 2 n \ B 1 n , v -, in Ω -\ B( 1 n , (81)
Obviously, the sequence {(v + n , v - n )} n∈N belongs to V and convergence (??) is satisfied. To complete the proof it remains to prove that also converge (??) is satisfied. Indeed, (??) provides that

Ω - |v - n -v -| 2 dx = B( 1 n ) |v + (0) -v -| 2 dx + B( 2 n )\B( 1 n ) |φ| 2 |v + (0) -v -| 2 dx ≤ ∥v + (0) -v -∥ 2 L ∞ (Ω -) 16π 3n 3 -→ 0, as n → +∞,
and

Ω - |∇v - n -∇v -| 2 dx = B( 1 n ) |∇v -| 2 dx + B( 2 n )\B( 1 n ) | -φ n ∇v -+ (v + (0) -v -)∇φ n | 2 dx ≤ 2∥∇v -∥ 2 (L ∞ (Ω -)) 3 16π 3n 3 + 2∥v + (0) -v -∥ 2 L ∞ (Ω -) B( 2 n )\B( 1 n ) |∇φ n | 2 dx ≤ 2∥∇v -∥ 2 (L ∞ (Ω -)) 3 16π 3n 3 + 2∥v + (0) -v -∥ 2 L ∞ (Ω -) 16π 3n -→ 0, as n → +∞,
which prove (??).

5 Proof of Theorem ?? and Theorem ??

Proof of Theorem ??

Proposition ?? ensures the existence of a subsequence of {ε} , still denoted by {ε}, and of a quadruple

(u + , v, w, u -) in U × H 1 d (I) × W × H 1 d (Ω -)
, depending possibly on the selected subsequence, satisfying (??)÷(??). Let V be defined in (??). Moreover, let us set

U reg = {ϕ ∈ C 1 (Ω + ) : ϕ = 0 in F, ϕ(•, 1) = 0 in ω} (82)
and

C 1 d (Ω + ) = ϕ ∈ C 1 (Ω + ) : ϕ(•, 1) = 0 in ω . ( 83
)
Now, in (??) choosing

ϕ ε (x ′ , x 3 ) =                εw t (x) + u + t (x) + v t (x 3 ), in ω × (η ϵ , 1), εw t (x ′ , η ε ) + u + t (x ′ , η ε ) + v t (η ε ) x 3 η ε + u - t (εx ′ , 0) η ε -x 3 η ε , in ω × (0, η ε ), u - t (x),
in Ω -, (84) as test function, with

u + t ∈ U reg , (v t , u - t ) ∈ V , w t ∈ C 1 d (Ω + ), (85)
and {η ε } ε a decreasing sequence of positive numbers included in ]0, 1[, such that

lim ε→0 η ε = 0, lim ε→0 ε √ η ϵ = 0, (86)
gives

I 1,ε + I 2,ε + I 3,ε + I 4,ε + I 5,ε = I 6,ε + I 7,ε + I 8 ∀ε, (87)
with

I 1,ε = 1 ηε ω χ F A    1 ε ∇ ′ u + ε ∂u + ε ∂x 3      ∇ ′ w t ε ∂w t ∂x 3 + v ′ t   dx ′ dx 3 , I 2,ε = ηε 0 ω χ F A    1 ε ∇ ′ u + ε ∂u + ε ∂x 3       ∇ ′ w t (x ′ , η ε ) x 3 η ε + (∇ ′ u - t )(εx ′ , 0) η ϵ -x 3 η ϵ 1 η ε εw t (x ′ , η ε ) + v t (η ε ) -u - t (εx ′ , 0)    dx ′ dx 3 , I 3,ε = 1 ηε ω χ M A   ∇ ′ u + ε ε ∂u + ε ∂x 3     ε∇ ′ w t + ∇ ′ u + t ε 2 ∂w t ∂x 3 + ε ∂u + t ∂x 3 + εv ′ t   dx ′ dx 3 , I 4,ε = ηε 0 ω χ M A   ∇ ′ u + ε ε ∂u + ε ∂x 3      [ε∇ ′ w t (x ′ ,η ε )+∇ ′ u + t (x ′ , η ε )] x 3 η ε + ε∇ ′ u - t (εx ′ , 0) η ϵ -x 3 η ϵ ε η ε εw t (x ′ , η ε ) + u + t (x ′ , η ε ) + v t (η ε ) -u - t (εx ′ , 0)    dx ′ dx 3 , I 5,ε = Ω - A∇u - ε ∇u - t dx, I 6,ε = 1 ηε ω f εw t + u + t + v t dx ′ dx 3 , I 7,ε = ηε 0 ω f εw t (x ′ , η ε ) + u + t (x ′ , η ε ) + v t (η ε ) x 3 η ε + u - t (εx ′ , 0) η ε -x 3 η ε dx ′ dx 3 , I 8 = Ω - f u - t dx.
Now, let us pass to the limit, as ε tends to zero, in each term of (??). Convergences (??) and (??) provide that

lim ε→0 I 1,ε = F A ∇ ′ w v ′ ∇ ′ w t v ′ t dx. ( 88
)
Since (v t , u - t ) belong to V , one has that

1 η ε |v t (η ε ) -u - t (εx ′ , 0)| = 1 η ε |v t (η ε ) -v t (0)| ≤ ∥v ′ t ∥ L ∞ (0,1) , ∀x ′ ∈ ω,
∇ ′ u - t (εx ′ , 0) = 0, ∀x ′ ∈ ω, for ε small enough. Consequently, thanks also to (??), (??), and (??), one obtains

lim ε→0 I 2,ε = 0. (89)
Convergences (??) and (??) give

lim ε→0 I 3,ε = M A ∇ ′ u + 0 ∇ ′ u + t 0 dx. ( 90
)
Using (??), (??), and (??) provides that

lim ε→0 I 4,ε = 0. (91)
Convergence (??) provides that

lim ε→0 I 5,ε = Ω - A∇u -∇u - t dx. ( 92
)
Moreover, it is obviously that

lim ε→0 I 6,ε = Ω + f (u + t + v t ) dx, ( 93
)
lim ε→0 I 7,ε = 0. (94)
Eventually, passing to the limit, as ε tends to 0, in (??), and using (??) ÷ (??), get

                           (u + , v, w, u -) ∈ U × H 1 d (I) × W × H 1 d (Ω -), F A ∇ ′ w v ′ ∇ ′ w t v ′ t dx + M A ∇ ′ u + 0 ∇ ′ u + t 0 dx + Ω - A∇u -∇u - t dx = Ω + f (u + t + v t ) dx + Ω - f u - t dx, ∀u + t ∈ U reg , ∀(v t , u - t ) ∈ V , ∀w t ∈ C 1 d (Ω + ).
Consequently, Proposition ?? and classical density results ensure that (u + , v, w) solves problem (??) and u -solves problem ??. Since these problems have a unique solution, convergences (??)÷(??) hold true for the whole sequence.

It remains to prove that weak convergences (??)÷(??) are in fact strong convergences, that is convergences (??)÷(??) hold true.

To this aim, first let us prove the convergence of energies. Indeed, passing to the limit in (??), as ε tends to zero, with test function ϕ ε = u ε , using convergences (??), (??), and (??), and choosing (u + t , v t , w t ) = (u + , v, w) as test function in (??) and u - t = u -as test function in (??) give the convergence of energies

lim ε→0 M A ∇ ′ u + ε ε ∂u + ε ∂x 3 ∇ ′ u + ε ε ∂u + ε ∂x 3 dx + F A 1 ε ∇ ′ u + ε ∂u + ε ∂x 3 1 ε ∇ ′ u + ε ∂u + ε ∂x 3 dx + Ω - A∇u - ε ∇u - ε dx = F A ∇ ′ w v ′ ∇ ′ w v ′ dx + M A ∇ ′ u + 0 ∇ ′ u + 0 dx + Ω - A∇u -∇u -dx.
Now, for every ε set

J ε = M A ∇ ′ u + ε -∇ ′ u + ε ∂u + ε ∂x 3 ∇ ′ u + ε -∇ ′ u + ε ∂u + ε ∂x 3 dx + F A 1 ε ∇ ′ u + ε -∇ ′ w ∂u + ε ∂x 3 -v ′ 1 ε ∇ ′ u + ε -∇ ′ w ∂u + ε ∂x 3 -v ′ dx + Ω - A(∇u - ε -∇u -)(∇u - ε -∇u -) dx.
The convergence of the energies and convergences (??)÷(??) provide that lim ε→0 J ε = 0, which, thanks to (??), implies all strong convergences stated in Theorem ??, except the first one (??). The proof of (??) will follow from (??), (??), (??), and the following Lemma ?? applied with z = u + ε -(u + + v). Altough Lemma ?? is proved in [?], we reproduce its proof for the convenience of the reader.

Lemma 5.1. There exists

K in ]0, +∞[ such that    ∀ z ∈ L 2 (I; H 1 (ω)) ∩ L 2 (D; H 1 d (I)), ∥ z ∥ L 2 (Ω + ) ≤ K ∥ ∇ ′ z ∥ (L 2 (Ω + )) 2 + ∥ ∂z ∂x 3 ∥ L 2 (F ) . ( 95
)
Proof. We argue by contradiction. Assuming (??) false implies the existence of a sequence

{z n } n∈N in L 2 (0, L; H 1 (ω)) ∩ L 2 (D; H 1 d (I)) such that        ∥ z n ∥ L 2 (Ω + ) = 1 ∀ n, ∥ ∇ ′ z n ∥ L 2 (Ω + ) + ∂z n ∂x 3 L 2 (F ) -→ 0, as n → +∞. (96)
Lemma ?? ensures existence of a contant C in ]0, +∞[ such that

ω z n (x ′ , x 3 ) - 1 |D| D z n (y ′ , x 3 ) dy ′ 2 dx ′ ≤ C 2 ω |∇ ′ z n (x ′ , x 3 )| 2 dx ′ , a.e. x 3 , ∀n ∈ N,
from which, integrating over x 3 in I, it follows that

Ω + z n (x ′ , x 3 ) - 1 |D| D z n (y ′ , x 3 ) dy ′ 2 dx ′ dx 3 1 2 ≤ C ∥ ∇ ′ z n ∥ L 2 (Ω + ) , ∀n ∈ N. ( 97
)
On the other side, first applying Hölder inequality and then the Poincaré inequality ensure the existence of C in ]0, +∞[ such that

Ω + 1 |D| D z n (y ′ , x 3 ) dy ′ 2 dx ′ dx 3 1 2 ≤ |ω| 1 2 |D| -1 2 ∥z n ∥ L 2 (F ) ≤ |ω| 1 2 |D| -1 2 C ∂zn ∂x 3 L 2 (F )
, ∀n ∈ N.

(98)

Combining (??), (??), and (??) leads to the contradiction:

1 = ∥z n ∥ L 2 (Ω + ) ≤ C ∥ ∇ ′ z n ∥ L 2 (Ω + ) +|ω| 1 2 |D| -1 2 C ∂z n ∂x 3 L 2 (F )
-→ 0, as n → +∞.

Proof of Theorem ??

Taking (u + t , v t ) = (0, 0) in (??) provides that w is the unique solution to following problem

       w ∈ W, I D A ∇ ′ w v ′ ∇ ′ w t 0 dx ′ dx 3 = 0, ∀ w t ∈ W. ( 99
)
On the other hand, multiplying equation in (??) by ϕ(x 3 )v ′ (x 3 ), with ϕ in L 2 (I), integrating over I, and using a density argument provide that ŵv ′ solves the equation in (??). Moreover, one can prove that ŵ belongs to L ∞ (I, H 1 (D)) (for instance, compare the proof of (2.28) in [?]). Consequently, ŵv ′ belongs to L 2 (I; H 1 (D)). Then, the uniqueness of the solution to problem (??) ensure that w(x ′ , x 3 ) = ŵ(x ′ , x 3 )v ′ (x 3 ), a.e. (x ′ , x 3 ) ∈ F.

Replacing w by ŵv ′ and choosing (u + t , w t ) = (0, 0) in the equation in (??) privide that v is the unique solution to problem (??).

Choosing (u t , 0, 0), with u t in U, as test function in (??) implies that u + is the unique solution to the following problem where û is the unique solution to (??). Now, for each ε let ūε be the unique solution to (??). Then, (??), (??), and a change of variable ensure that

       u + ∈ U , M A ∇ ′ u + 0 ∇ ′ u t 0 dx = M f u t dx, ∀u t ∈ U .
1 ε 2 εω ūε (X ′ , •) dX ′ → ω\D u + (x ′ , •)dx ′ + |ω|v(•) strongly in L 2 (I), as ε → 0. ( 104
)
Eventually, (??) follows from (??), (??), and (??).

  [?] (see also[?], [?], [?], and [?]). Here we just focalize the reader's attention on the homogenization of high-contrast problems in fibered structures: [?], [?], [?], [?], [?], [?], [?], [?], [?], [?], [?], [?], [?] and [?].

  f in the following wayf (x ′ , x 3 ) = 1 |ω \ D| ω\D f (x ′ , x 3 ) dx ′ +f (x ′ , x 3 )-1 |ω \ D| ω\D f (x ′ , x 3 ) dx ′ , a.e. (x ′ , x 3 ) ∈ M,and taking advantage of the linearity ensure thatu + (x) = u + 0 (x) + ũ(x), a.e. x ∈ M, ω\D f (x ′ , x 3 ) dx ′ u t dx, ∀u t ∈ U , (102)and ũ is the unique solution to (??). Arguing exactly as we have done for w and bearing in mind that M = (ω \ D) × I infer thatu + 0 (x ′ , x 3 ) = û(x ′ , x 3 ) 1 |ω \ D| ω\D f (x ′ , x 3 ) dx ′ , a.e. (x ′ , x 3 ) ∈ M (103)

  Lemma 3.1. Let A be a bounded connected open set of R n with Lipschitz boundary and B be a measurable subset of A with positive Lebesgue-measure. Let p in [1, +∞]. Then, there exists C in ]0, +∞[ such that

	u -	1 |B| B	u dx

The first two authors are also


Ahmed Boughammoura
email: ahmed.boughammoura@gmail.com

Leila Rahmani
email: rahmani-lei@yahoo.fr

Ali Sili
email: ali.sili@univ-amu.fr

A 2D model for a highly heterogeneous plate

Keywords: plate, thin structure, corrector, nonlocal. AMS Classification:: 35B25, 35B27, 35B40, 76M50, 74K10

In this paper we investigate the 2d-model for a thin plate Ω ε := ω × εI of R 3 having two components: a circular stiff fiber F ε and its complement the soft matrix M ε with 1 ε 2 as a ratio between their respective elasticity coefficients. We prove that the limit model is associated to a nonlocal system involving Kirchoff-Love displacements in the fiber and we exhibit a corrector for the displacements in the initial cylindrical structure of R 3 .

Introduction, notations and setting of the problem

The aim of this work is the study of the asymptotic behavior of the solutions of the linearized system of elasticity posed in a cylindrical domain Ω ε := ω ×εI of R 3 which is the configuration domain of a composite material. The material is made up of two components with high contrast: the first one F ε representing the stiff part of the material has an elasticity tensor with coefficients of order 1. The second component M ε (the soft material) surrounds the first one and the coefficients of its elasticity tensor are of order ε 2 .

Under an appropriate assumption on the volumes forces, we aim to approximate the behavior of the displacements and that of the associated tensors as the small parameter ε tends to zero. Hence the present work may be viewed as the 3d -2d version of the study addressed in [START_REF] Paroni | Non-local effects by homogenization or 3D-1D dimension reduction in elastic materials reinforced by stiff fibers[END_REF] where the 3d -1d reduction of dimension problem was considered as well as the homogenization of a ε-periodic fibered medium inducing a local 3d -1d reduction of dimension. It was proved in [START_REF] Paroni | Non-local effects by homogenization or 3D-1D dimension reduction in elastic materials reinforced by stiff fibers[END_REF] that the homogenized problem in a such setting is a copy of the one-dimensional problem obtained in the 3d -1d study. Although one can also consider here the ε-periodic homogenization problem of a medium containing 1 ε cells which are the translates of Ω ε in such a way that the homogenization process leads to a local 3d -2d reduction of dimension, for the sake of brevity we restrict ourselves to the 3d -2d reduction of dimension problem arising in the single composite structure Ω ε . We consider the critical case where the ratio between the elasticity coefficients of the two components is equal to 1 ε 2 but other scalings may be considered as pointed out in [START_REF] Paroni | Non-local effects by homogenization or 3D-1D dimension reduction in elastic materials reinforced by stiff fibers[END_REF], see also [START_REF] Gaudiello | Homogenization of highly oscillating boundaries with strongly contrasting diffusivity[END_REF][START_REF] Sili | Diffusion through a composite structure with a high contrasting diffusivity[END_REF].

On the other hand, we deal with general elasticity tensors including anisotropic materials, see also [START_REF] Boughammoura | Homogenization and correctors for composite media with coated and highly anisotropic fibers[END_REF][START_REF] Cherednichenko | Nonlocal limits for composite media with highly anisotropic periodic fibers[END_REF][START_REF] Murat | Problèmes monotones dans des cylindres de faible diamètre formés de matériaux hétérogènes[END_REF][START_REF] Sili | Homogénéisation dans des cylindres minces[END_REF]. Several studies on composite materials with hight contrast between their components have been performed during the last years, see for instance [START_REF] Bellieud | Homogenization of a soft elastic material reinforced by fibers[END_REF][START_REF] Braides | Homogenization of non-uniformly bounded periodic diffusion energies in dimension two[END_REF][START_REF] Braides | A variational approach to double-porosity problems[END_REF][START_REF] Brillard | Asymptotic behaviour of a cylindrical elastic structure periodically reinforced along identical fibers[END_REF][START_REF] Gaudiello | Homogenization of highly oscillating boundaries with strongly contrasting diffusivity[END_REF][START_REF] Mabrouk | Homogénéisation d'un milieu élastique fortement hétérogène[END_REF][START_REF] Murat | Problèmes monotones dans des cylindres de faible diamètre formés de matériaux hétérogènes[END_REF].

It is known that for this kind of materials, the limit problem has in general a different structure than the starting problem. In particular, nonlocal phenomena can appear at the limit. We show here that the limit problem obtained after reduction of dimension 3d -2d is indeed a nonlocal problem. Theorem 3.2 below, which gives the limit problem, shows that the displacements in the circular fiber F are essentially of Kirchov-Love type and the associated equation may be obtained from system (29) by choosing z = 0; but to determine the limit displacements in the structure, another equation related to the matrix M (the outside of the fiber) is necessary. That equation is obtained by choosing ū = v = 0 in (29). This nonlocal phenomenon is emphasized in Theorem 3.4 which gives the limit of the average of the displacements ûε in the three-dimensional structure Ω ε . In particular convergences (33) and (34) show clearly that the limit displacements are determined after solving the two equations posed in F and in M respectively.

In terms of correctors, our result shows that the transversal displacements u ε α behave as u α + εv α + z α while the horizontal displacements u ε 3 behave as

u 3 + ε 2 v 3 + εz 3 .
We now make more precise the notations we will use throughout the paper.

A vector x in R 3 is denoted by x = (x , x 3 ) where x 3 denotes the vertical coordinate. Latin indices will usually range from 1 to 3, and Greek ones take values in {1, 2}; the summation convention applies whenever indices are repeated. We write ∂ i := ∂ ∂x i and ∂ ij := ∂ 2 ∂x i ∂x j . The gradient with respect to x and x are denoted by ∇ and ∇ respectively. Given any φ ∈ (D (Ω)) 3 , we denote the strain of φ by Eφ := sym∇φ = 1 2 ∇φ + ∇φ T . We shall also use the following matrices notations

∇φ =   ∂ α φ β ∂ 3 φ β ∂ α φ 3 ∂ 3 φ 3   , Eφ =   (Eφ) αβ (Eφ) α3 (Eφ) α3 (Eφ) 33   .
Let ω ⊂ R 2 be an open, bounded, simply connected set with Lipschitz boundary ∂ω. Let I := (-1 2 , 1 2 ), J := (-r 2 , r 2 ), 0 < r < 1 2 , Ω := ω × I, F := ω × J, M := ω × I \ J , where J denotes the closure of J. According to Remark 1.1 below, physically we can think of Ω as the reference configuration of a rescaled thin plate Ω ε reinforced by the fiber F ε which is surrounded by the matrix M ε occupied by a soft material. For every ε > 0 we denote the diagonal matrix whose entries are 1, 1 and ε by R ε := diag(1, 1, ε), then the scaled gradient ∇ ε φ and the scaled strain E ε φ are defined respectively by

∇ ε φ := (R ε ) -1 ∇φ(R ε ) -1 =   ∂ α φ β 1 ε ∂ 3 φ β 1 ε ∂ α φ 3 1 ε 2 ∂ 3 φ 3   , (1)
and

E ε φ := sym∇ ε φ =   (Eφ) αβ 1 ε (Eφ) α3 1 ε (Eφ) α3 1 ε 2 (Eφ) 33   . (2)
We are now in position to state the problem. Let A ∈ L ∞ (Ω) be a symmetric fourth-order tensor field. We assume that A fulfills the following assumptions:

A ijkl = A jikl = A klij , a.e. in Ω, ∃ C > 0, A ijkl ξ kl ξ ij ≥ Cξ ij ξ ij , ∀ξ ∈ R 9 s.t. ξ T = ξ. (3)
We shall assume that the plate is clamped at the lateral boundary of Ω and subjected to body forces f ∈ L 2 (Ω; R 3 ), we thus set

H 1 L (Ω) := {u ∈ H 1 (Ω; R 3 ) : u = 0 on ∂ω × I}.
Consider the displacement field u ε solution of the following system:

     u ε ∈ H 1 L (Ω), ∀φ ∈ H 1 L (Ω), Ω χ F + ε 2 χ M AE ε (u ε ).E ε (φ)dx = Ω f φdx. (4)
By virtue of the assumptions on the tensor A and the body forces F , for every ε > 0, the problem (4) admits a unique solution by the Lax-Milgram Theorem.

Remark 1.1 As in [START_REF] Ciarlet | A justification of the twodimensional linear plate model[END_REF], the homothety along the vertical axis defined by r ε (x) := (x , εx 3 ) transforms Ω, F and M respectively into Ω ε := ω × εI, F ε := ω × εJ and M ε := ω × ε I \ J . Then, the problem (4) is the variational version in the fixed reference configuration Ω of the elasticity problem (6) below posed in the variable thin domain Ω ε . Indeed, for any

v : Ω -→ R 3 we define v : Ω ε -→ R 3 by v(x , εx 3 ) := (R ε ) -1 v • (r ε ) -1 (x , εx 3 ) = v α (x), 1 ε v 3 (x) . (5)
In addition

∇v = H ε v • (r ε ) -1 , Ev = E ε v • (r ε ) -1 .
With these new unknowns, the problem (4) may be rewritten as

     ûε ∈ H 1 L (Ω ε ), ∀φ ∈ H 1 L (Ω ε ), Ωε χ Fε + ε 2 χ Mε A ε E(û ε ).E(φ)dx = Ωε f ε φdx (6)
where

H 1 L (Ω ε ) := {u ∈ H 1 (Ω ε ; R 3 ) : u = 0 on ∂ω × εI}, A ε := A • (r ε ) -1 , f ε = R ε f • (r ε ) -1 .
Thus, the components of the loads are

f ε = f α • (r ε ) -1 , εf 3 • (r ε ) -1 ) . (7)
We study the behavior as ε → 0 of the sequence {u ε }, solution of (4), through the forthcoming steps. That behavior will be described through Theorem 3.2 and then we will deduce the behavior of the sequence ûε solution of (6) in Theorem 3.4.

A priori estimates

First, we shall recall a rescaled Korn's inequality proved in [1]

Ev 2 L 2 (Fε) ≥ Cε 2 v 2 L 2 (Fε) + ∇v 2 L 2 (Fε) , ∀v ∈ H 1 L (F ), (8)
where

H 1 L (F ) := {u ∈ H 1 (F ; R 3 ) : u = 0 on ∂ω × J} and v = (R ε ) -1 v • (r ε ) -1 . As a consequence, we deduce E ε v 2 L 2 (F ) ≥ Cε 2 (R ε ) -1 v 2 L 2 (F ) + ∇ ε v 2 L 2 (F ) , ∀v ∈ H 1 L (F ). ( 9
)
On the other hand, repeating the same arguments used in [12, Lemma 4.4], the following inequality holds true

∇ ε v 2 L 2 (F ) + E ε v 2 L 2 (M ) ≥ C ∇ ε v 2 L 2 (M ) , ∀v ∈ H 1 L (Ω). ( 10
) For v ∈ H 1 L (Ω) we set v := (v 1 , v 2 ), ∇ v := ∂ β v α α,β=1,2 and F ε (v) := Ω χ F + ε 2 χ M AE ε v.E ε vdx. (11)
By using the ellipticity assumption (3) we obtain, for all v ∈ H 1 L (Ω)

F ε (v) ≥ C E ε v 2 L 2 (F ) + ε 2 E ε v 2 L 2 (M ) ≥ C Ev 2 L 2 (F ) ≥ C v 2 H 1 (F ) , (12)
where the last inequality follows from the Korn's inequality. Furthermore, by virtue of ( 9) and ( 10) one has

F ε (v) ≥ Cε 2 ∇ ε v 2 L 2 (F ) + E ε v 2 L 2 (M ) ≥ Cε 2 ∇ ε v 2 L 2 (M ) . (13)
From ( 13), one has for ε small enough,

F ε (v) ≥ Cε 2 1 ε 2 ∇ v 3 2 L 2 (M ) + 1 ε 4 ∂ 3 v 3 2 L 2 (M ) ≥ C ∇v 3 2 L 2 (M ) . ( 14
)
By means of the Poincaré's inequality, one has

F ε (v) ≥ C v 3 2 H 1 (M ) . (15)
Likewise, it holds that

F ε (v) ≥ C ε 2 ∇ v 2 L 2 (M ) + ∂ 3 v 2 L 2 (M ) ≥ C ∂ 3 v 2 L 2 (M ) . (16)
From the following Poincaré's type inequality

v 2 L 2 (M ) ≤ C v 2 L 2 (F ) + ∂ 3 v 2 L 2 (M ) , (16)
and ( 12), we get the following inequality

F ε (v) ≥ C v 2 L 2 (M ) . (17)
It follows from ( 15) and (17) that

F ε (v) ≥ C v 3 2 H 1 (M ) + v 2 L 2 (ω;H 1 (I\J)) . (18)
As an immediate consequence of the previous inequalities, the following apriori estimates hold true.

Theorem 2.1 Let u ε be the solution of problem (4). Then

sup ε u ε H 1 (F ) + ε∇ ε u ε L 2 (F ) + E ε u ε L 2 (F ) ≤ C, (19)
sup ε u ε 3 H 1 (M ) + u ε α L 2 (ω;H 1 (I\J)) + ε∇ ε u ε L 2 (M ) ≤ C, (20)
in particular

sup ε u ε 3 H 1 (Ω) + u ε α L 2 (ω;H 1 (I)) + ε∇ ε u ε L 2 (Ω) ≤ C, α = 1, 2. (21)
Proof.

Taking φ = u ε in the problem (4), we derive

F ε (u ε ) ≤ f L 2 (Ω) u ε L 2 (Ω) ,
while from ( 12), ( 15) and ( 17) we find

F ε (u ε ) ≥ C u ε L 2 (Ω) .
Thus

sup ε F ε (u ε ) ≤ C.
Therefore, making use of ( 12)-( 13) and then ( 13)-(18) we deduce ( 19) and (20) respectively.

Convergence results

We first define the following functional spaces:

H KL (Ω) := u ∈ H 1 L (Ω) : (Eu) i3 = 0, i = 1, 2, 3
which is the space of Kirchhoff-Love displacements on Ω; it can be characterized also as

H KL (Ω) = u ∈H 1 L (Ω), ∃(g α , g 3 ) ∈H 1 0 (ω)×H 2 0 (ω) : u α = g α -x 3 ∂ α g 3 , u 3 = g 3 ,
for brevity we set U := H KL (Ω),

V := v ∈ L 2 ω; H 1 (J) 3 : - J v i (x , t)dt = 0, a.e. x ∈ ω, i = 1, 2, 3 , Z := z ∈ L 2 ω; H 1 m (I) 3 : z i = 0, a.e. in F, i = 1, 2, 3 ,
where

H 1 m (I) := ψ ∈ H 1 (I) : I ψ(t)dt = 0 .
The following lemma is concerned with some convergence results.

Lemma 3.1 There exists (u, v, z) ∈ U × V × Z, such that, up to a subsequence, we have

u ε α u α + z α in L 2 (ω; H 1 (I)), u ε 3 u 3 in H 1 (Ω), E ε u ε χ F   (Eu) αβ ∂ 3 v α ∂ 3 v α ∂ 3 v 3   χ F in L 2 (Ω) 3×3 , εE ε u ε χ M   0 1 2 ∂ 3 z α 1 2 ∂ 3 z α ∂ 3 z 3   χ M in L 2 (Ω) 3×3 .
Proof.

Step 1: convergence of (u ε ) ε .

From estimate (19) which implies that u ε α is bounded in H 1 (F ) and from estimate (20) which implies that u ε α is bounded in L 2 (ω; H 1 (I \ J)) for α = 1, 2, we conclude that there exist u α ∈ H 1 (Ω) such that for a subsequence

u ε α u α in H 1 (F ), (E ε u ε ) αβ χ F (Eu) αβ χ F in L 2 (F ). (22)
On the other hand, from (21) we get that up to a subsequence,

     u ε α ũα in L 2 (ω; H 1 (I)), u ε 3 u 3 in H 1 (Ω). (23)
Moreover, due to the estimate

1 ε ∂ 3 u ε 3 L 2 (Ω) ≤ C which is a consequence of (21), we have ∂ 3 u ε 3 → 0 in L 2 (Ω) in such a way that ∂ 3 u 3 = 0.
Thus, there exists g 3 ∈ H 1 0 (ω) such that

u 3 (x) = g 3 (x ) for a.e. x ∈ Ω. ( 24
)
Comparing the first convergences in ( 22) and (23), we derive the equality

ũα = u α in F. (25)
On the other hand, using (19) we infer

(Eu ε ) α3 (Eũ) α3 in L 2 (F ). Since (Eu ε ) i3 → 0, in L 2 (F )
we are led to

(ũ α , u 3 ) ∈ H KL (F ).
By definition of the space H KL (F ), the component u 3 belongs to H 2 0 (ω), so that equality (24) implies that g 3 ∈ H 2 0 (ω). Hence, there exist

g α ∈ H 1 0 (ω) for α = 1, 2, such that ũα (x) = g α (x ) -x 3 ∂ α g 3 (x ), u 3 (x) = g 3 (x ) for a. e. x ∈ F.
For a. e. x ∈ Ω, we set

u α (x) := g α (x ) -x 3 ∂ α g 3 (x ) ∈ H 1 L (Ω), z α (x) := ũα (x) -u α (x) ∈ L 2 (ω; H 1 (I)),
so that z α = 0, a. e. in F as seen from ( 25) and u := (u α , u 3 ) ∈ U and

u ε α u α + z α , in L 2 (ω; H 1 (I)).
Step 2: convergence of (E ε u ε ) ε .

To identify the limit of the sequence (E ε u ε ) i3 χ F , we introduce the following sequences :

ûε α := x 3 -1 2 1 ε ∂ α u ε 3 (x , t)dt, (26)
v ε α := 1 ε u ε α + ûε α -- J 1 ε u ε α + ûε α dx 3 , v ε 3 := 1 ε 2 u ε 3 -- J 1 ε 2 u ε 3 dx 3 . ( 27
)
Obviously, on has

∂ 3 v ε α = 2 ε (Eu ε ) α3 , ∂ 3 v ε 3 = 1 ε 2 (Eu ε ) 33 .
Since 2 ε (Eu ε ) α3 and 1 ε 2 (Eu ε ) 33 are bounded in L 2 (F ), due to (19), and v ε := (v ε α , v ε 3 ) has mean-value zero with respect to x 3 , the sequence v ε is bounded in L 2 (ω; H 1 m (J)) 3 , where

H 1 m (J) := ψ ∈ H 1 (J) : - J ψ(t)dt = 0 .
Therefore there exists some v ∈ L 2 (ω;

H 1 m (J)) 3 such that v ε v in L 2 (ω; H 1 m (J)) 3 .
In particular, one has v ∈ V,

(E ε u ε ) α3 χ F 1 2 ∂ 3 v α χ F in L 2 (Ω),
and

(E ε u ε ) 33 χ F ∂ 3 v 3 χ F in L 2 (Ω).
As a consequence, one obtains the following convergence

E ε u ε χ F   (Eu) αβ 1 2 ∂ 3 v α 1 2 ∂ 3 v α ∂ 3 v 3   χ F in L 2 (Ω) 3×3 ,
where (u, v) ∈ U × V. We now seek for the limit of the sequence εE ε u ε χ M . To that aim, we consider the following sequence

z ε α := u ε α + ûε α - I (u ε α + ûε α ) dx 3 , z ε 3 := 1 ε u ε 3 - I 1 ε u ε 3 dx 3 . ( 28
)
where ûε is defined by (26). Since

∂ 3 z ε α = 2(Eu ε ) α3 , ( 29
) ∂ 3 z ε 3 = 1 ε (Eu ε ) 33 . ( 30
)
Due to (21), 2(Eu ε ) α3 and 1 ε (Eu ε ) 33 are bounded in L 2 (Ω) with mean-value zero with respect to x 3 ; hence the sequence z ε is bounded in L 2 (ω; H 1 m (I)) 3 . Therefore there exists some ẑ ∈ L 2 (ω; H 1 m (I)) 3 , such that

z ε ẑ in L 2 (ω; H 1 m (I)) 3 .
We claim that ẑα = z α . Indeed,

∂ 3 u ε α ∂ 3 u α + ∂ 3 z α in L 2 (Ω) and ∂ 3 z ε α ∂ 3 ẑα in L 2 (Ω). But ∂ 3 z ε α = ∂ 3 u ε α + ∂ α u ε 3 ∂ 3 u α + ∂ 3 z α + ∂ α u 3 in L 2 (Ω). Thus ∂ 3 ẑα = ∂ 3 u α + ∂ 3 z α + ∂ α u 3 .
Since u ∈ U, we have

∂ 3 u α + ∂ α u 3 = 0,
and therefore

∂ 3 ẑα = ∂ 3 z α i.e. ẑα (x) = z α (x) + c(x ), a.e. x ∈ Ω.
In particular ẑα (x) = c(x ), a.e. x ∈ F since z α (x) = 0, a.e. x ∈ F.

On the other hand one has by virtue of (20),

z ε → 0 in L 2 (F ).
Hence c(x ) = 0, a.e. x ∈ F and therefore ẑα (x) = z α (x), a.e. x ∈ Ω.

This proves our claim. Finally, from (29) and (30) we deduce that

ε(E ε u ε ) α3 χ M 1 2 ∂ 3 z α χ M , ε(E ε u ε ) 33 χ M (Ez) 33 χ M in L 2 (Ω),
where

z 3 := ẑ3 . Moreover, since εu ε α -→ 0 in L 2 (M ) then ε(E ε u ε ) αβ χ M 0 in L 2 (Ω). We conclude that εE ε u ε χ M   0 1 2 ∂ 3 z α 1 2 ∂ 3 z α ∂ 3 z 3   χ M in L 2 (Ω) 3×3 .
This ends the proof of Lemma 3.1.

In the following theorem, we state our main result.

Theorem 3.2 There exists (u, v, z) ∈ U × V × Z, such that the sequence of solutionu ε of the problem (4) fulfills the following strong convergences

u ε α -→ u α + z α strongly in L 2 (ω; H 1 (I)), u ε 3 -→ u 3 strongly in H 1 (Ω), E ε u ε χ F -→   (Eu) αβ 1 2 ∂ 3 v α 1 2 ∂ 3 v α ∂ 3 v 3   χ F strongly in L 2 (Ω) 3×3 , εE ε u ε χ M -→   0 1 2 ∂ 3 z α 1 2 ∂ 3 z α ∂ 3 z 3   χ M strongly in L 2 (Ω) 3×3 ,
where the limit (u, v, z) is the unique solution of the problem

                             (u, v, z) ∈ U × V × Z, ∀(ū, v, z) ∈ U × V × Z, Ω A(x)   (Eu) αβ 1 2 ∂ 3 v α 1 2 ∂ 3 v α ∂ 3 v 3   .   (Eū) αβ 1 2 ∂ 3 vα 1 2 ∂ 3 vα ∂ 3 v3   χ F dx+ Ω A(x)   0 1 2 ∂ 3 z α 1 2 ∂ 3 z α ∂ 3 z 3   .   0 1 2 ∂ 3 zα 1 2 ∂ 3 zα ∂ 3 z3   χ M dx = Ω f α (ū α + zα ) + f 3 ū3 dx. (31)
Proof.

We first prove the convergence of the sequence of energies to the energy associated to the limit problem, assuming that the limit problem is the system (31). To that aim we will use the weak convergences proved in Lemma 3.1.

We follow the argument already used in [12, Proof of Theorem 3.1].

Choosing (ū, v, z) = (u, v, z) in (31) and passing to the limit thanks to Lemma 3.1, we get

lim ε→0 Ω χ F + ε 2 χ M AE ε (u ε ).E ε (u ε )dx = lim ε→0 Ω f u ε dx = Ω f α (u α + z α ) + f 3 u 3 dx = Ω (AE F .E F χ F + AE M .E M χ M ) dx,
where

E F :=   (Eu) αβ 1 2 ∂ 3 v α 1 2 ∂ 3 v α ∂ 3 v 3   , E M :=   0 1 2 ∂ 3 z α 1 2 ∂ 3 z α ∂ 3 z 3   .
Using the equivalence of the norms

L 2 (Ω) E -→ Ω AE.Edx, L 2 (Ω) E -→ Ω E.Edx we deduce that lim ε→0 (χ F + εχ M ) E ε (u ε ) L 2 (Ω) = χ F E F + χ M E M L 2 (Ω) .
Hence the strong convergence of the tensor sequences is proved. Now, by using the last strong convergence together with the following Korn's inequality

E ε (u ε ) L 2 (F ) ≥ E(u ε ) L 2 (F ) ≥ C u ε L 2 (F ) ,
we deduce that u ε is a Cauchy sequence in H 1 (F ).

On the other hand, using [START_REF] Gaudiello | Homogenization of highly oscillating boundaries with strongly contrasting diffusivity[END_REF] and [START_REF] Mabrouk | Homogénéisation d'un milieu élastique fortement hétérogène[END_REF] one has

E ε u ε 2 L 2 (F ) ≥ Cε 2 (R ε ) -1 u ε 2 L 2 (F ) + ∇ ε u ε 2 L 2 (F ) and ∇ ε u ε 2 L 2 (F ) + E ε u ε 2 L 2 (M ) ≥ C ∇ ε u ε 2 L 2 (M )
. Hence, ε∇ ε u ε is also a Cauchy sequence in L 2 (Ω). It remains to prove that u ε α and u ε 3 are two Cauchy sequences in L 2 (ω; H 1 (I)) and H 1 (M ) respectively. To that aim, it suffices to use (18) to obtain

(χ F + εχ M ) E ε (u ε ) 2 L 2 (Ω) = F ε (u ε ) ≥ C u ε 3 2 H 1 (M ) + u ε α 2 L 2 (ω;H 1 (I\J))
then we can argue as previously to complete the proof. We now prove that the limit problem is nothing but the system (31). We choose a test function φ ε in (4) in the following form

φ ε α = ūα + εv α + zα φ ε 3 = ū3 + ε 2 v3 + εz 3 (32)
where

(ū, v, z) ∈ U × D ω; H 1 m (J) 3 × D ω; H 1 m (I) 3 : z = 0, a.e. x ∈ F .
Since, from the definition of U, one has (E ū) i3 = 0, for all i = 1, 2, 3, an elementary calculation shows that

     (E ε φ ε ) αβ = (Eū) αβ + ε(Ev) αβ + (Ez) αβ , 2(E ε φ ε ) α3 = ∂ 3 vα + ε∂ α v3 + 1 ε ∂ 3 zα + ∂ α z3 , (E ε φ ε ) 33 = (Ev) 33 + 1 ε (Ez) 33 . Using the following strong convergence E ε φ ε χ F -→   (Eū) αβ 1 2 ∂ 3 vα 1 2 ∂ 3 vα ∂ 3 v3   χ F in L 2 (Ω) 3×3 , εE ε φ ε χ M -→   0 1 2 ∂ 3 zα 1 2 ∂ 3 zα ∂ 3 z3   χ M in L 2 (Ω) 3×3 ,
we can pass to the limit in (4) as ε goes to zero to get (31) thanks to the density of D ω; H 1 m (J) 3 and z ∈ D ω; H 1 m (I) 3 : z i = 0, a.e. x ∈ F, i = 1, 2, 3 in V and Z respectively.

Remark 3.3

The space U × V × Z is a Hilbert space for the following norm

(u, v, z) 2 := α,β=1,2 (Eu) αβ 2 L 2 (Ω) + 3 i=1 ∂ 3 v i 2 L 2 (Ω) + ∂ 3 z i 2 L 2 (Ω) .
Therefore, using the Lax-Milgram Theorem we obtain the well-posedness of the limit problem (31) since the tensor A is coercive by virtue of (3) and f ∈ L 2 (Ω; R 3 ).

We now turn back to the sequence of solutions of problem [START_REF] Brillard | Asymptotic behaviour of a cylindrical elastic structure periodically reinforced along identical fibers[END_REF] to derive the following theorem from Theorem 3.2. Then the sequence of solutions ûε of the problem (6) fulfills the following convergences

                     - εJ Eû ε dx 3 -→ -J E F dx 3 , in L 2 (ω; R 3×3 ), - ε(I\J) Eû ε dx 3 -→ -I\J E M dx 3 in L 2 (ω; R 3×3 ),
-εI ûε α dx 3 -→ U α (x ) := -I (u α + z α ) dx 3 , in L 2 (ω),

- εI εû ε 3 dx 3 -→ u 3 (x ) in L 2 (ω), (33)
Moreover, U α may be written as

U α (x ) = - I u α dx 3 + m 0 α (x )- I\J f α (x , x 3 )dx 3 + m 00 α (x ), (34)
where m 0 α and m 00 α are given by m 0 α (x ) :=

Theorem 3 . 4 1 2 ∂ 3 z α 1 2 ∂ 3 z α ∂ 3 z 3

 34113 Let (u, v, z) ∈ U × V × Z be the solution of the problem (31). Let E

1 2 ∂ 3 zα 1 2

 11 , x 3 )dx 3 , m 00 α (x ) := I\J z 00 α (x , x 3 )dx 3 , and z 0 , z 00 are respectively the solutions of the following problems L ∞ ω, Z 0 , Z 0 := z ∈ (H 1 (I \ J) 3 : z = 0 on ∂J I\J ∂ 3 zα ∂ 3 z3   dx 3 = I\J zα (x 3 )dx 3 , ∀z ∈ Z 0 , a.e. in ω, x) --I\J f α (x)dx 3 zα (x)dx, ∀z ∈ Z.

Proof.

The convergences (33) follow from the corresponding convergences stated in Theorem 3.2 by the change of variables [START_REF] Braides | A variational approach to double-porosity problems[END_REF].

Let us prove for instance the third convergence arising in (33) by the use of the first convergence arising in Theorem 3.2. Setting y 3 = εx 3 for x 3 ∈ I and bearing in mind that ûε is defined according to (5) by u ε α (x , x 3 ) = ûε (x , εx 3 ) for (x , x 3 ) ∈ ω × I and that the length of the interval I is equal to 1, we get thanks to the Cauchy-Schwarz inequality (with respect to

To prove (34), we first notice that one can derive the following equation satisfied by z α by choosing ū = v = 0 in (31),

Taking advantage from the linearity of (38), one can check easily by a uniqueness argument that z α may be identified as z α = z 1 α + z 2 α where z 1 α and z 2 α are solutions of (38) but with right hand sides defined respectively by

By linearity and using once again a uniqueness argument, we conclude that

where z 0 and z 00 are the solutions of ( 35) and (36) respectively. Hence, U α defined in the third convergence (33) may be written as (34). This completes the proof of Theorem 3.4.


A Gaudiello
email: gaudiell@unina.it

A Sili
email: ali.sili@univ-amu.fr

Limit models for thin heterogeneous structures with high contrast *

Keywords: Thin structure, contrasting diffusivity, degenerating equation, correctors. 2020 AMS subject classifications: 35J25, 35J70

Limit models for thin heterogeneous structures with high contrast

A. Gaudiello, Ali

Introduction

In this paper we investigate two linear diffusion problems, with strongly contrasting diffusivity, in a thin heterogeneous cylinder.

In what follows, x (resp. y ) denotes a generic element (x 1 , x 2 ) (resp. (y 1 , y 2 )) of R 2 , moreover (x , x 3 ) (resp. (y , x 3 ) ) denotes a generic element x = (x 1 , x 2 , x 3 ) (resp. (y 1 , y 2 , x 3 )) of R 3 .

Figure 1: Thin cylinder Mn

For every n in N we consider a thin cylinder Mn of R 3 with height l and with a small crosssection h n M . In Mn we distinguish two thin cylinders with the same height l: an inner core Ĉn with small cross-section r n C and its complementary annulus În = h n M \ r n C ×]0, l[ (see Figure 1), where M and C are two open connected sets of R 2 with Lipschitz boundaries such that 0 ∈ C ⊂⊂ M , and h n and r n are two small parameters taking values in two vanishing sequences of positive numbers such that r n << h n .

In Mn we consider the following boundary value problem:    -div (g n (y )A (x 3 ) D u n (y , x 3 )) + b (x 3 ) u n (y , x 3 ) = f n (y , x 3 ), in Mn , (g n (y )A (x 3 ) D u n (y , x 3 )) • ν = 0, on ∂ Mn , (1.1) where A is a coercive matrix with coefficients in L ∞ (]0, l[), b is a coefficient in L ∞ (]0, l[) which is bounded from below away from zero, f n is a function in L 2 ( Mn ), and ν denotes the exterior unit normal to ∂ Mn . For the sake of simplicity in this introduction we assume that the matrix A and the coefficient b depend only on the vertical variable x 3 , but in the paper we treat a more general case with A and b depending on (y , x 3 ). As far as function g n is concerned, we shall consider two cases:

I) g n = δ 2 n χ Ĉn + χ În , II) g n = χ Ĉn + δ 2 n χ În ,
where also δ n is a small parameter taking values in a vanishing sequence of positive numbers, and χ Ĉn and χ În denote the characteristic function of Ĉn and În , respectively. In the first case, Mn has a very small insulation inner cylindrical core Ĉn with low conductivity of order δ 2 n enveloped by the annulus În with conductivity of order 1. In the second case, we have the opposite situation, i.e., Mn has a very small inner cylindrical core Ĉn with conductivity of order 1 enveloped by the insulation annulus În with low conductivity of order δ 2 n . In this paper we study the asymptotic behavior of problem (1.1) as h n → 0, r n → 0, r n h n → 0, δ n → 0, and under suitable assumptions on f n . Precisely, we assume that the rescaling of f n on M ×]0, l[, i.e., f n (x , x 3 ) = f n (h n x , x 3 ), is L 2 -strongly converging to a function f (x , x 3 ). We point out that the peculiarity of this paper is the assumption r n h n → 0 (i.e., r n << h n ) combined with the lost of coercivity of the problems when h n and r n vanish, due to the vanishing coefficient δ 2 n in front of the divergence term. • In the first case (i.e., when g n = δ 2 n χ Ĉn + χ În ), using the Cauchy-Schwarz inequality, Theorem 3.2 provides that

- hnM u n (x , •)dx -→ u strongly in L 2 (]0, l[), (1.2)
and u ∈ H 1 (]0, l[) is the unique weak solution to

           - d dx 3 a 0 du dx 3 + |M |bu = f 0 , in ]0, l[, a 0 du dx 3 (0) = 0 = a 0 du dx 3 (l),
where for a.e.

x 3 in ]0, l[ a 0 (x 3 ) = M A(x 3 ) D x z 1 D x z 1 dx , f 0 (x 3 ) = M f (x , x 3 )dx , and z(•, x 3 ) ∈ H 1 (M ) is the unique weak solution to                    M z(x , x 3 )dx = 0, div x (A (x 3 )D x z) = 0, in M , A (x 3 )D x z + A 13 (x 3 ) A 23 (x 3 ) • ν = 0, on ∂M ,
with A = (A αβ ) α,β=1,2 and ν denoting the unit exterior normal to ∂M . Note that a 0 (x 3 ) = |M |A 33 (x 3 ) if A 13 = A 23 = 0 (since in this case z = 0). In particular,

a 0 = |M | if A = I.
• In the second case (i.e., when g n = χ Ĉn + δ 2 n χ În ), the limit behavior of problem (1.1) depends on the

lim n δ n h n = µ ∈ [0, +∞].
Precisely, using the Cauchy-Schwarz inequality, Theorem 4.1 provides that -if µ ∈]0, +∞[, then

- hnM u n (x , •)dx -→ - M u(x , •)dx strongly in L 2 (]0, l[),
where, for a.e.

x 3 in ]0, l[, u(•, x 3 ) ∈ H 1 (M ) is the unique weak solution to    -µ 2 div x (A (x 3 )D x u(•, x 3 )) + b(x 3 )u(•, x 3 ) = f (•, x 3 ), in M , (A (x 3 )D x u(•, x 3 )) • ν = 0, on ∂M . (1.3) -If µ = 0 or µ = +∞, then - hnM u n (x , •)dx - 1 b - hnM fn (x , •)dx -→ 0 strongly in L 2 (]0, l[).
or equivalently,

- hnM u n (x , •)dx -→ 1 b - M f (x , •)dx strongly in L 2 (]0, l[).
More generally, in Theorem 3.2 and in Theorem 4.1 we consider the case where the matrix A and the coefficient b depend also on y and we prove L 2 -strong convergence results for u n and for its gradient. It is not difficult to extend these results to monotone operators.

As far as the first case is concerned (i.e., g n = δ 2 n χ Ĉn + χ În ), let us point out that convergence (1.2) is the same result which one obtains studying the limit behavior of the following problem (for instance, compare [START_REF] Casado-Díaz | Homogenization and correctors for monotone problems in cylinders of small diameter[END_REF] , [START_REF] Gaudiello | Asymptotic analysis for monotone quasilinear problems in thin multidomains[END_REF], [START_REF] Gaudiello | Asymptotic analysis of the eigenvalues of an elliptic problem in an anisotropic thin multidomain[END_REF], and [START_REF] Murat | Problèmes monotones dans des cylindres de faible diamètre formés de matériaux hétérogènes[END_REF])

   -div (A (x 3 ) D u n (y , x 3 )) + b (x 3 ) u n (y , x 3 ) = f n (y , x 3 ), in Mn , (A (x 3 ) D u n (y , x 3 )) • ν = 0, on ∂ Mn ,
i.e., roughly speaking, in a first approximation (in L 2 (]0, l[)-norm) of the average of the solution on the cross-section of the thin cylinder, the presence of a very small insulation cylindrical core Ĉn with a small conductivity does not give any contribution with respect to considering a conductivity of order 1 everywhere in the original problem. Moreover, in Theorem 3.2 we shall show that u n (y , x 3 ) and its gradient behave as u(x 3 ) + h n du dx 3 (x 3 ) z y hn , x 3 in I n which is a corrector result.

As far as the second case is concerned (i.e., g n = χ Ĉn + δ 2 n χ În ), h n is a critical size for δ n . Precisely, if δ n h n (i.e., µ ∈]0, +∞[), then the L 2 (]0, l[)-limit of the average of u n on the cross-section of the thin cylinder is -M u(x , •)dx , with u the solution to problem (1.3). Moreover in Theorem 4.1 we shall show that u n (y , x 3 ) and its gradient with respect to y behave as u y hn , x 3 in M n . If δ n << h n or δ n >> h n (i.e., µ = 0 or µ = +∞), the L 2 (]0, l[)-limi is given by 1 b -M f (x , •)dx (formally, this result can be obtained choosing µ = 0 or µ = +∞ in problem (1.3)). Indeed, this last result depends on the fact that in this introduction we are assuming b independent of y . In Theorem (4.1) we shall prove that, more generally (i.e., A and b depend also on y ), the case µ = 0 is different from the case µ = +∞.

For the sake of completeness and for better understand our result in the second case, we recall what happens when r n h n δ n in the case g n = χ Ĉn +δ 2 n χ În . Actually, if for instance one assumes r n = rh n , r > 0, then the limit problem is a nonlocal one in the sense that the limit diffusion is described by a nonlocal system of two equations in which a two-dimensional problem like (1.3), posed in that case on M \ rC , is associated with a one-dimensional problem involving the diffusion along the vertical axis x 3 of the core. This result was proved in [START_REF] Murat | A remark about the periodic homogenization of certain composite fibered media[END_REF] for the Laplacian equation with the homogeneous Dirichlet boundary condition on the extremities of the cylinder and the homogeneous Neumann boundary condition elsewhere (also compare [START_REF] Sili | Homogenization of a nonlinear monotone problem in an anisotropic medium[END_REF] for monotone operators), and in [START_REF] Paroni | Non-local effects by homogenization or 3D-1D dimension reduction in elastic materials reinforced by stiff fibers[END_REF] for the linearized elasticity system in a thin beam clamped at the extremities. It is easily seen that the nonlocal effect still appears when the homogeneous Neumann boundary condition is assumed everywhere (just a zero order term in the equation is needed in order to obtain a priori estimates). On the other hand, the results are similar in the opposite situation, i.e., g n = δ 2 n χ Ĉn + χ În and r n h n δ n .

Our present result states that if the radius r n of the core is negligible compared to h n and if g n = χ Ĉn + δ 2 n χ În with δ n h n , then the nonlocal effect disappears since the limit diffusion inside the core is not perceived and only the diffusion outside the core (which is described by (1.3)) gives a contribution in the limit process.

Eventually, we point that when r n h n , but δ n has a different behavior, it is an easy task to prove that the limit problem is local. We also refer to [START_REF] Briane | Homogenization of convex functionals which are weakly coercive and not equi-bounded from above[END_REF] for nonlocal limits in problems with high contrast in a thin cylinder.

In the homogenization setting, there is a vast literature on problems with highly contrasting coefficients since the founding work of T. Arbogast, J. Douglas, and U. Hornung [START_REF] Arbogast | Derivation of the double porosity model of single phase flow via homogenization theory[END_REF] (see also [START_REF] Braides | A variational approach to doubleporosity problems[END_REF] and [START_REF] Braides | Homogenization of discrete highcontrast energies[END_REF]). Here, we just recall those papers devoted to the homogenization of fibered structures with high contrast: [START_REF] Bellieud | Homogenization of elliptic problems in a fiber reinforced structure. Nonlocal effects[END_REF], [START_REF] Bellieud | Homogenization of a soft elastic material reinforced by fibers[END_REF], [START_REF] Briane | Fibered microstructures for some nonlocal Dirichlet forms[END_REF], [START_REF] Caillerie | A perturbation problem with two small parameters in the framework of the heat conduction of a fiber reinforced body[END_REF], [START_REF] Cherednichenko | Non-local homogenized limits for composite media with highly anisotropic periodic fibres[END_REF], [START_REF] Gaudiello | Homogenization of highly oscillating boundaries with strongly contrasting diffusivity[END_REF], [START_REF] Kamotski | Bandgaps in two-dimensional high-contrast periodic elastic beam lattice materials[END_REF], [START_REF] Murat | A remark about the periodic homogenization of certain composite fibered media[END_REF], [START_REF] Paroni | Non-local effects by homogenization or 3D-1D dimension reduction in elastic materials reinforced by stiff fibers[END_REF], [START_REF] Sili | Homogenization of a nonlinear monotone problem in an anisotropic medium[END_REF]. Let us point out that as far as the periodic homogenization of fibered media is concerned, the homogenized problem has the same form as the limit problem obtained by the corresponding reduction of dimension occurring in the reference cell, as remarked in [START_REF] Murat | A remark about the periodic homogenization of certain composite fibered media[END_REF].

Our paper is structured as follows: in Section 2 we introduce some notations and we fix the assumptions on A and b depending on (y , x 3 ). In Section 3 we study the asymptotic behavior of the problem with a low conductivity in the core of the structure. After rescaling the solution on

M = M ×]0, l[ by u n (x) = u(h n x , x 3 ), we prove that δ n 1 hn D x u n , D x 3 u n χ Cn is bounded in L 2 (M )
and, up to a subsequence, u n converges to u weakly in L 2 (M ) and

1 hn D x u n , D x 3 u n χ In converges to ξ = (ξ , ξ 3 ) weakly in (L 2 (M ))
3 , where C n and I n denote the scaling of Ĉn and În , respectively. It should be noted that C n and I n still depend on n since Ĉn and În are dependent also on r n (and not only on h n ). It is not difficult to show that u is independent of x and ξ 3 = D x 3 u, but it is quite complicated to write ξ in terms of u. To do that we need to adapt an uniform extension operator introduced in [START_REF] Conca | Nonhomogeneous Neumann problems in domains with small holes[END_REF]. Eventually, using the convergence of the energies allows us to obtain the strong convergence in L 2 (M ) for u n and for its rescaled gradient (in particular we prove that δ n

1 hn D x u n , D x 3 u n χ Cn converges to zero strongly in L 2 (M )).
In Section 4 we study the asymptotic behavior of the problem with a low conductivity in the annulus of the structure. At first we rewrite the problem so that it can be splited in a part on M n and a part on C n . Then we introduce two different rescalings

u n (x) = u(h n x , x 3 ) in M = M ×]0, l[ and u n (x) = u n (r n x , x 3 ) in C = C ×]0, l[,
and we write the problem satisfied by the couple (u n , u n ). A priori estimates suggest us that the limit problem depends on the ratio µ = lim n δn hn and three different regimes appear according to µ = 0, 0 < µ < +∞, or µ = +∞. The limit problems are obtained passing to the limit in the equation with suitable test functions which, roughly speaking, take into account the relation between u n and u n , and using suitable density results. Eventually, thanks to the convergence of the energies we obtain the strong convergences.

Definitions and notation

Let {h n } n∈N and {r n } n∈N be two sequences in ]0, 1] such that

lim n r n = 0, lim n h n = 0, lim n r n h n = 0. (2.1) Let 0 ∈ C ⊂⊂ M ⊆ - 1 2 , 1 2
2
be two open connected sets with Lipschitz boundaries and let l ∈]0, +∞[. For every n ∈ N set

Mn = h n M ×]0, l[, Ĉn = r n C ×]0, l[, În = h n M \ r n C ×]0, l[. Set M = M ×]0, l[, (2.2)
and let A and b be such that

     A ∈ (L ∞ (M )) 3×3 , ∃λ ∈]0, +∞[ : A(x) ξ ξ ≥ λ |ξ| 2 , a.e. x ∈ M, ∀ ξ ∈ R 3 .
(2.3)

∃γ ∈]0, +∞[ : b ∈ L ∞ (M ), b(x) ≥ γ, a.e. x ∈ M. (2.4) Let {δ n } n∈N ⊂]0, 1] be such that lim n δ n = 0, (2.5)
and for every n ∈ N let

f n ∈ L 2 ( Mn ). (2.6)
In what follows, α, β belong to {1, 2} and repeated indices imply the summation.

3 Problem I: a low conductivity in the core of the structure

For every n ∈ N consider the following problem

       -div δ 2 n χ Ĉn + χ În A y hn , x 3 D u n + b y hn , x 3 u n = f n , in Mn , δ 2 n χ Ĉn + χ În A y hn , x 3 D u n • ν = 0, on ∂ Mn , (3.1)
where ν denotes the exterior unit normal to ∂ Mn .

The weak formulation of (3.1) is

                     u n ∈ H 1 Mn , δ 2 n Ĉn A y h n , x 3 D u n D ϕdy dx 3 + În A y h n , x 3 D u n D ϕdy dx 3 + Mn b y h n , x 3 u n ϕdy dx 3 = Mn f n ϕdy dx 3 , ∀ ϕ ∈ H 1 Mn . (3.2)
which admits a unique solution by virtue of the Lax-Milgram Theorem. The goal of this section is to study the asymptotic behavior of this problem, as n diverges. To this aim set

f n : (x , x 3 ) ∈ M → f n (h n x , x 3 ), (3.3)
and assume that

f n → f strongly in L 2 (M ). (3.4) Let H 1 m (M ) = v ∈ H 1 (M ) :
M vdx = 0 . The Lax-Milgram Theorem ensures that for a.e. x 3 in ]0, l[ the following problem

                   z(•, x 3 ) ∈ H 1 m (M ), M A D x z 0 D x ψ 0 dx = - M A 0 1 D x ψ 0 dx , ∀ψ ∈ H 1 (M ), (3.5)
admits a unique solution. Moreover, the measurability of A ensures (compare [START_REF] Gaudiello | Asymptotic analysis of the eigenvalues of an elliptic problem in an anisotropic thin multidomain[END_REF], pages 749-50) that the function

x 3 ∈]0, l[→ z(•, x 3 ) ∈ H 1 m (M ) is measurable. Furthermore, choosing ψ = z(•, x 3 ) as test function in (3.5) provides sup x 3 ∈]0,l[ D x z(•, x 3 ) (L 2 (M )) 2 < +∞, i.e. z ∈ L ∞ (0, l, H 1 m (M )). (3.6) Set a 0 : x 3 ∈]0, l[→ M A D x z 1 D x z 1 dx . (3.7) b 0 : x 3 ∈]0, l[→ M bdx , (3.8)
and f 0 :

x 3 ∈]0, l[→ M f dx . (3.9)
Then (2.3), (2.4), (3.4), and (3.6) provide that

a 0 ∈ L ∞ (]0, l[), b 0 ∈ L ∞ (]0, l[), f 0 ∈ L 2 (]0, l[),
and

a 0 (x 3 ) ≥ λ|M |, b 0 (x 3 ) ≥ γ|M |, a.e. x 3 ∈]0, l[.
Consequently, the Lax-Milgram Theorem ensures that the following problem

       u ∈ H 1 (]0, l[), l 0 a 0 du dx 3 dv dx 3 dx 3 + l 0 b 0 uvdx 3 = l 0 f 0 vdx 3 , ∀v ∈ H 1 (]0, l[), (3.10)
admits a unique solution which is the weak solution to the following one

           - d dx 3 a 0 du dx 3 + b 0 u = f 0 , in ]0, l[, a 0 du dx 3 (0) = 0 = a 0 du dx 3 (l).
(3.11)

Remark 3.1. If A is independent of x 3 , then also z is independent of x 3 . Consequently, a 0 is a positive constant.

If the matrix A is such that A 13 = A 23 = 0, then z = 0. Consequently, a 0 = M A 33 dx . In particular, a 0 = |M | if A = I. Theorem 3.2. Assume (2.1), (2.3), (2.4), (2.5), (2.6), (3.3), (3.4), For every n ∈ N let u n be the unique weak solution to problem (3.1). Let z ∈ L ∞ (0, l, H 1 m (M )) be the unique solution to (3.5). Let a 0 , b 0 , and f 0 be defined by (3.7), (3.8), and (3.9), respectively. Let u be the unique weak solution to (3.11). Then Proof. The proof will be divided into nine steps.

                       lim n 1 h n u n -u L 2 ( Mn) = 0, lim n 1 h n δ n D u n (L 2 ( Cn)) 3 = 0, lim n 1 h n D y u n - du dx 3 (D x z) y h n , x 3 (L 2 ( In)) 2 + D x 3 u n - du dx 3 L 2 ( In) = 0. ( 3
Step 1. The first step is devoted to reformulating Problem (3.2). Problem (3.2) can be reformulated through the map

(x , x 3 ) ∈ M → (h n x , x 3 ) ∈ Mn .
Precisely, for every n ∈ N set

C n = r n h n C ×]0, l[, I n = M \ r n h n C ×]0, l[, and
D hn : ϕ ∈ H 1 (M ) →      1 h n D x 1 ϕ 1 h n D x 2 ϕ D x 3 ϕ      ∈ (L 2 (M )) 3 . (3.13)
Then u n defined by

u n : (x , x 3 ) ∈ M → u n (h n x , x 3 ) (3.14)
is the unique solution to

                   u n ∈ H 1 (M ) , δ 2 n Cn AD hn u n D hn ϕdx + In AD hn u n D hn ϕdx + M bu n ϕdx = M f n ϕdx, ∀ϕ ∈ H 1 (M ) .
(3.15)

Step 2. This step is devoted to proving the existence of a subsequence of N, still denoted by

N, u ∈ L 2 (M ), and ξ = (ξ , ξ 3 ) ∈ (L 2 (M )) 2 × L 2 (M ) (which can depend on the subsequence) such that u n u weakly in L 2 (M ) (3.16) and χ In D hn u n ξ = (ξ , ξ 3 ) weakly in L 2 (M ) 3 .
(3.17)

Choosing ϕ = u n as test function in (3.15) and using (2.3), (2.4), (3.4), and the Young inequality provide the existence of a positive constant c such that

u n L 2 (M ) ≤ c, ∀n ∈ N, (3.18)
χ Cn δ n D hn u n (L 2 (M )) 3 ≤ c, ∀n ∈ N, (3.19)
χ In D hn u n (L 2 (M )) 3 ≤ c, ∀n ∈ N.
(3.20)

Eventually, (3.16) and (3.17)follow from (3.18) and (3.20).

Step 3. This step is devoted to proving that Step 4. This step is devoted to proving the existence of z in L 2 (0, l, H 1 m (M )) such that

ξ 3 = D x 3 u,
D x u n (L 2 (A k )) 2 ≤ D x u n (L 2 (In)) 2 ≤ ch n , for n large enough, which implies D x u (L 2 (A k )) 2 =
ξ = D x z, a.e. in M. (3.27) For every n ∈ N let Q n ∈ L H 1 h n M \ r n C , H 1 (h n M ) be such that Q n ϕ = ϕ, a.e. in h n M \ r n C , ∀ϕ ∈ H 1 h n M \ r n C , (3.28)
and ∃c ∈]0, +∞[ :

hnM |D y (Q n ϕ)| 2 dy ≤ c hnM \rnC |D y ϕ| 2 dy , ∀ϕ ∈ H 1 h n M \ r n C , ∀n ∈ N.
(3.29)

The existence of such a family is proved in the proof of Lemma A1 in [START_REF] Conca | Nonhomogeneous Neumann problems in domains with small holes[END_REF] when

M = -1 2 , 1 2 2 .
Now, for every n ∈ N and for a.e. x 3 in ]0, l[ set

z n (x , x 3 ) = Q n u n (•, x 3 ) | hnM \rnC (h n x ), a.e. x ∈ M , (3.30)
where u n is the unique solution to (3.2). At first remark that

z n ∈ L 2 (0, l, H 1 (M )), ∀n ∈ N. (3.31) Indeed, u n ∈ H 1 Mn ⊂ H 1 ( În ). Consequently u n ∈ L 2 0, l, H 1 h n M \ r n C . Then, since Q n ∈ L H 1 h n M \ r n C , H 1 (h n M ) , one deduces that (see Proposition 1.2.2 in [13]) Q n u n (•, x 3 ) | hnM \rnC ∈ L 2 (0, l, H 1 (h n M )),
M |D x z n (x , x 3 )| 2 dx = M D x Q n u n (•, x 3 ) | hnM \rnC (h n x ) 2 dx = hnM D y Q n u n (•, x 3 ) | hnM \rnC (y ) 2 dy ≤ c hnM \rnC |D y u n (y , x 3 )| 2 dy = c M \ rn hn C |D x ( u n (h n x , x 3 )) | 2 dx = c M \ rn hn C |D x (u n (x , x 3 )) | 2 dx ,
which implies, by an integration on

x 3 ∈]0, l[, that M |D x z n | 2 dx ≤ c In |D x u n | 2 dx, ∀n ∈ N, (3.33)
where c is the positive constant independent of n ∈ N given in (3.29). Then, combining (3.33) with (3.20) provides the existence of a positive constant c such that

1 h n D x z n (L 2 (M )) 2 ≤ c, ∀n ∈ N,
which implies the existence of a subsequence of N, still denoted by n, and z in L 2 (0, l, H 1 m (M )) (which can depend on the subsequence) such that

1 h n D x z n D x z weakly in L 2 (M ) 2 . ( 3
]0, l[ z n (x , x 3 ) = Q n u n (•, x 3 ) | hnM \rnC (h n x ) = u n (h n x , x 3 ) = u n (x , x 3 ), a.e. x ∈ M \ rn hn C . (3.35)
Eventually, (3.27) follows from (3.17), (3.32) and from the fact that

χ In 1 h n D x z n = χ In 1 h n D x u n , a.e. in M, ∀n ∈ N,
due to (3.35).

Step 5. This step is devoted to proving that

M A   D x z du dx 3     D x w dv dx 3   dx + M buvdx = M f vdx, ∀v ∈ H 1 (]0, l[), ∀w ∈ H 1 (M ) . (3.36)
To this aim, let us rewrite the equation in (3.15) in the following way

δ 2 n Cn AD hn u n D hn ϕdx + In A αβ 1 h n D x β u n + A α3 D x 3 u n 1 h n D xα ϕ + A 3α 1 h n D xα u n + A 33 D x 3 u n D x 3 ϕ dx + M bu n ϕdx = M f n ϕdx, ∀ϕ ∈ H 1 (M ) .
(3.37) Choosing ϕ = v + h n w, with v, w ∈ H 1 (M ) and v independent of x , as test function in (3.37) gives

δ 2 n Cn AD hn u n   D x w dv dx 3 + h n D x 3 w   dx + In A αβ 1 h n D x β u n + A α3 D x 3 u n D xα w + A 3α 1 h n D xα u n + A 33 D x 3 u n dv dx 3 + h n D x 3 w dx + M bu n (v + h n w)dx = M f n (v + h n w)dx, ∀v ∈ H 1 (]0, l[), ∀w ∈ H 1 (M ) .
(3.38) Passing to the limit, as n diverges, in (3.38) and using (2.5), (3.4)

A αβ D x β z + A α3 du dx 3 D xα w + A 3α D xα z + A 33 du dx 3 dv dx 3 dx + M buvdx = M f vdx, ∀v ∈ H 1 (]0, l[), ∀w ∈ H 1 (M ) ,
i.e. (3.36).

Step 6. This step is devoted to identifying z in terms of u, i.e. to proving that

z = du dx 3 z, in L 2 (0, l, H 1 m (M )), (3.39)
where, for a.e. x 3 in ]0, l[, z(•, x 3 ) is the unique solution to (3.5) Choosing v = 0 and w = ψφ, with ψ ∈ H 1 (M ) and φ ∈ C ∞ 0 (]0, l[), as test functions in (3.36) gives

l 0   M A   D x z du dx 3   D x ψ 0 dx   φdx 3 = 0, ∀ψ ∈ H 1 (M ), ∀φ ∈ C ∞ 0 (]0, l[).
Consequently, since H 1 (M ) is separable, for a.e. x 3 in ]0, l[ one has Step 7. This step is devoted to identifying u, i.e. to proving that u is the unique solution to (3.10).

M A D x z 0 D x ψ 0 dx = - M A   0 du dx 3   D x ψ 0 dx , ∀ψ ∈ H 1 (M ). ( 3
Choosing w = 0 as test function in (3.36) and using (3.39) give

M A    du dx 3 D x z du dx 3      0 dv dx 3   dx + M buvdx = M f vdx, ∀v ∈ H 1 (]0, l[), i.e. l 0 M A D x z 1 0 1 dx du dx 3 dv dx 3 dx 3 + l 0 M bdx uvdx 3 = l 0 M f dx vdx 3 , ∀v ∈ H 1 (]0, l[),
which can be rewritten as

l 0 a du dx 3 dv dx 3 dx 3 + l 0 b 0 uvdx 3 = l 0 f 0 vdx 3 , ∀v ∈ H 1 (]0, l[), (3.41)
where

a = M A D x z 1 0 1 dx , a.e. in ]0, l[, (3.42)
and b 0 is defined in (3.8).

Remark that a = a 0 , a.e. in ]0, l[,

with a 0 defined in (3.7). Indeed

a 0 = M A D x z 1 D x z 1 dx = M A D x z 1 0 1 + D x z 0 dx = a + M A D x z 1 D x z 0 dx .
(3.44) On the other side, choosing ψ = z(•, x 3 ) as test function in (3.5) provides that the last integral in (3.44) is zero. So (3.43) holds true. Then, u is the unique solution to (3.10). Consequently, all previous convergences hold true for the whole sequence.

Step 8. This step is devoted to proving that

lim n     Cn |δ n D hn u n | 2 dx + In D hn u n -    du dx 3 D x z du dx 3    2 dx + M |u n -u| 2 dx     = 0. ( 3
    λ Cn |δ n D hn u n | 2 dx + λ In D hn u n -    du dx 3 D x z du dx 3    2 dx + γ M |u n -u| 2 dx     ≤ lim sup n δ 2 n Cn AD hn u n D hn u n dx + M A    χ In D hn u n -χ In    du dx 3 D x z du dx 3          χ In D hn u n -χ In    du dx 3 D x z du dx 3       dx + M b |u n -u| 2 dx = lim sup n δ 2 n Cn AD hn u n D hn u n dx + In AD hn u n D hn u n dx + M b|u n | 2 dx - M Aχ In D hn u n    du dx 3 D x z du dx 3    dx - M A    du dx 3 D x z du dx 3    χ In D hn u n dx + M χ In A    du dx 3 D x z du dx 3       du dx 3 D x z du dx 3    dx -2 M bu n udx + M b |u| 2 dx = 0,
A    du dx 3 D x z du dx 3       du dx 3 D x z du dx 3    ∈ L 1 (M ).
Step 9. Eventually, (3.12) follows from (3.45) by a changing of variable.

Remark 3.4. If assumption (3.4) is replaced with

f n f weakly in L 2 (M ).
One can prove all the previous result until the convergence of the energies (3.46) excluded.

To prove the convergence of the energies, strong convergence (3.4) is needed. Then, to obtain convergences in (3.12), assumption (3.4) is needed.

4 Problem II: a low conductivity in the annulus of the structure

For every n ∈ N consider the following problem

       -div χ Ĉn + δ 2 n χ În A y hn , x 3 D u n + b y hn , x 3 u n = f n in Mn , χ Ĉn + δ 2 n χ În A y hn , x 3 D u n • ν = 0 on ∂ Mn , (4.1)
where Mn , Ĉn , În , δ n , A, b, f n , and ν are defined in the previous section. The weak formulation of (4.1) is

                     u n ∈ H 1 Mn , Ĉn A y h n , x 3 D u n D ϕdy dx 3 + δ 2 n În A y h n , x 3 D u n D ϕdy dx 3 + Mn b y h n , x 3 u n ϕdy dx 3 = Mn f n ϕdy dx 3 , ∀ ϕ ∈ H 1 Mn . (4.2) then lim n 1 h n u n -M f (x , •)dx M b(x , •)dx L 2 ( Mn) = 0 = lim n 1 h n u n -hnM f n (
                     u n ∈ H 1 Mn , 1 -δ 2 n Ĉn A y h n , x 3 D u n D ϕdy dx 3 + δ 2 n Mn A y h n , x 3 D u n D ϕdy dx 3 + Mn b y h n , x 3 u n ϕdy dx 3 = Mn f n ϕdy dx 3 , ∀ ϕ ∈ H 1 Mn . (4.11)
Let M be defined in (2.2) and for every n ∈ N let D hn be defined in (3.13),

f n : (x , x 3 ) ∈ M → f n (h n x , x 3 ), (4.12)
and u n : (x , x 3 ) ∈ M → u n (h n x , x 3 ).

D rn : ϕ ∈ H 1 (C) →      1 r n D x 1 ϕ 1 r n D x 2 ϕ D x 3 ϕ      ∈ (L 2 (C)) 3 , and
u n : (x , x 3 ) ∈ C → u n (r n x , x 3 ). (4.14)
Note that the pair (u n , u n ) satisfies the equality

u n (x , x 3 ) = u n (r n x , x 3 ) = u n h n r n h n x , x 3 = u n r n h n x , x 3 , so that if we set for every n ∈ N V n = (ϕ, ϕ) ∈ H 1 (M ) × H 1 (C) : ϕ(x , x 3 ) = ϕ r n h n x , x 3 , a.e. in C , (4.15)
then (u n , u n ) is the unique solution to

                   (u n , u n ) ∈ V n , (1 -δ 2 n ) r n h n 2 C A r n h n x , x 3 D rn u n D rn ϕdx + δ 2 n M AD hn u n D hn ϕdx + M bu n ϕdx = M f n ϕdx, ∀(ϕ, ϕ) ∈ V n . (4.16)
which can be also rewritten in the following way

                                                               (u n , u n ) ∈ V n , (1 -δ 2 n ) r n h n 2 C A αβ r n h n x , x 3 1 r n D x β u n + A α3 r n h n x , x 3 D x 3 u n 1 r n D xα ϕ + A 3α r n h n x , x 3 1 r n D xα u n + A 33 r n h n x , x 3 D x 3 u n D x 3 ϕ dx +δ 2 n M A αβ 1 h n D x β u n + A α3 D x 3 u n 1 h n D xα ϕ + A 3α 1 h n D xα u n + A 33 D x 3 u n D x 3 ϕ dx + M bu n ϕdx = M f n ϕdx, ∀(ϕ, ϕ) ∈ V n .
(4.17)

Step 2. This step is devoted to obtaining a priori estimates for problem (4.16). Choosing (ϕ, ϕ) = (u n , u n ) as test function in (4.16) and using (2.3), (2.4), (2.5), (3.4), and the Young inequality provide the existence of a positive constant c such that

u n L 2 (M ) ≤ c, ∀n ∈ N, (4.18) 1 h n D x u n (L 2 (C)) 2 ≤ c, ∀n ∈ N, (4.19)
r n h n D x 3 u n L 2 (C) ≤ c, ∀n ∈ N, (4.20)
δ n h n D x u n (L 2 (M )) 2 ≤ c, ∀n ∈ N, (4.21)
δ n D x 3 u n L 2 (M ) ≤ c, ∀n ∈ N. (4.22)
Step 3. This step is devoted to choosing suitable test functions in (4.17). Let w ∈ C ∞ (M ) be such that w is constant in a neighborhood of 0 and let v ∈ C ∞ 0 (]0, l[). For every n ∈ N set

ϕ n : (x , x 3 ) ∈ C → w r n h n x v(x 3 ). Then, (wv, ϕ n ) ∈ V n ,
) r n h n C A 3α r n h n x , x 3 1 h n D xα u n + A 33 r n h n x , x 3 r n h n D x 3 u n D x 3 v(x 3 )w r n h n x dx + M A αβ δ n h n D x β u n + δ n A α3 D x 3 u n δ n h n D xα w(x )v(x 3 ) + A 3α δ n h n D xα u n + A 33 δ n h n h n D x 3 u n δ n D x 3 v(x 3 )w(x ) dx + M bu n wvdx = M
f n wvdx, for n large enough.

(4.23) Step 4. This step is devoted to prove case i). Estimates (4.18), (4.21) and (4.22) combined with (4.5) provide that

h n D x 3 u n 0 weakly in L 2 (M ), (4.24)
and that there exist a subsequence of N, still denoted by N, and u in L 2 (0, l, H 1 m (M )) (which can depend on the subsequence) such that u n u weakly in L 2 (0, l, H 1 (M )). Consequently, by a density argument (compare the proof of Proposition 3.1, in the case p ≤ N -1, in [START_REF] Gaudiello | Asymptotic analysis of a class of minimization problems in a thin multidomain[END_REF]) one has Eventually, (4.6) follows from (4.27) by a changing of variable.

Step 5. This step is devoted to prove case ii). Estimate (4.18) provides the existence of a subsequence of N, still denoted by n, and u in L 2 (M ) (which can depend on the subsequence) such that

δ 2 n

 2 .45) Passing to the limit in (3.15) with ϕ = u n as test function, using (3.4) and (3.16), and choosing v = u as test function in (3.10) (which is an admissible test function thanks to (3.22)) provide the convergence of the energies lim n Cn AD hn u n D hn u n dx + In AD hn u n D hn u n dx + M b|u n | 2 dx = 2.3), (2.4) (3.16), (3.17), (3.21), (3.23), (3.25), (3.27), (3.39), (3.8), (3.7), and (3.46) provide lim sup n

  i.e. (3.45) holds true.Remark that the third last integral in the last formula tends to Lebesgue dominated convergence theorem, since (3.23) holds true and (2.3) and (3.39) ensure that

  C = C ×]0, l[, and for every n ∈ N set

2 M

 2 limit in (4.23), as n → +∞, and using (2.1), (2.3), (2.4), (2.5), (3.4), (4.5), (4.19), (4.20), (4.24), and (4.25) provide µ A αβ D x β uD xα wvdx + M buwvdx = M f wvdx, ∀w ∈ C ∞ (M ) : w is constant in a neighborhood of 0 , ∀v ∈ C ∞ 0 (]0, l[).

µ 2 M 3 D 2 MA

 232 A αβ D x β uD xα wvdx + M buwvdx = M f wvdx, ∀w ∈ H 1 (M ), ∀v ∈ C ∞ 0 (]0, l[),which implies that for a.e. x 3 in ]0, l[ u(•, x 3 ) solves (4.4), since H 1 (M ) is separable. Note that (4.25) holds true for the whole sequence since Problem (4.4) admits a unique solution. Passing to the limit in (4.16) with ϕ = (u n , u n ) as test function, using (3.4) and (4.25), and choosing w = u(•., x 3 ) as test function in (4.4) provide the convergence of the energies lim n rn u n D rn u n dx +δ 2 n M AD hn u n D hn u n dx + M b|u n | 2 dx = µ αβ D x β uD xα udx + M

3 DM b |u n -u| 2 dx 21 3 Dδ 2 nD

 32132 rn u n D rn u n dx+ M A δ n D hn u n -µ D x u 0 δ n D hn u n -µ D x u 0 dx + rn u n D rn u n dx + M AD hn u n D hn u n dx + M b|u n | 2 dx x u n , h n D x 3 u n -D x u 0 2 dx + γ M |u n -u| 2 dx = 0,(4.27)thanks to (4.5).

u n u weakly in L 2 (

 2 limit in (4.23), as n → +∞, and using (2.1), (2.3), (2.4), (2.5),(3.4), (4.7), (4.19), (4.20), (4.21), (4.22), and (4.28) provide Mbuwvdx = M f wvdx, ∀w ∈ C ∞ (M ) : w is constant in a neighborhood of 0 , ∀v ∈ C ∞ 0 (]0, l[).argument. Note that (4.29) ensures that (4.28) holds true for the whole sequence. Passing to the limit in (4.16) with ϕ = (u n , u n ) as test function, using (2.3), (2.4), (3.4),

  .12) Remark 3.3. Let us point out that the last convergence provides a corrector since it means that u n behaves as u(x 3 ) + h n

	du dx 3	z	y h n	, x 3 in I n .

  Since for a.e. x 3 in ]0, l[ this equation admits a unique solution in H 1 m (M ), one has that for a.e. x 3 in ]0, l[

	Moreover, multiplying both sides of (3.5) by	du dx 3	implies that for a.e. x 3 in ]0, l[	du dx 3	z solves
	(3.40). z =	du dx 3	z, a.e. in M ,
	i.e. (3.39).
						.40)
	Remark that (3.6) ensures that
	du dx 3	z ∈ L 2 (0, l, H 1 m (M )).

which admits a unique solution by virtue of the Lax-Milgram Theorem.

The goal of this section is to study the asymptotic behavior of this problem as n diverges. To this aim for a.e. x 3 in ]0, l[, consider the following problem

where A = (A αβ ) α,β=1,2 and ν denotes the unit exterior normal to ∂M . For a.e. x 3 in ]0, l[ the weak formulation to (4.3) is i) If

)

where for a.e.

(4.28), a l.s.c argument, and (4.29) provide

Consequently, thanks to (4.28) and (2.4),

which implies

which implies that

Note that (4.33) ensures that (4.32) holds true for the whole sequence.

Passing to the limit in (4.16) with ϕ = (u n , u n ) as test function, using (2.3), (2.4), (3.4), (4.32), a l.s.c argument, and (4.33) provide

Consequently, thanks to (4.32) and (2.4), √ bu n → √ bu strongly in L 2 (M ), which implies

thanks to (4.33)and (2.4), too. On the other side, (3.4) provides