Homogenization of evolution problems for a composite medium with very small and heavy inclusions

April 2005, Vol. 11, 266–284 DOI: 10.1051/cocv:2005007

HOMOGENIZATION OF EVOLUTION PROBLEMS FOR A COMPOSITE MEDIUM WITH VERY SMALL AND HEAVY INCLUSIONS

Michel Bellieud

Abstract. We study the homogenization of parabolic or hyperbolic equations like ρε∂ⁿuε

∂tⁿ −div(aε∇uε) =f in Ω×T+ boundary conditions, n∈ {1,2}, when the coefficientsρε, aε (defined in Ω) take possibly high values on aε-periodic set of grain-like inclusions of vanishing measure. Memory effects arise in the limit problem.

Mathematics Subject Classification. 35K05, 35L05, 73B27.

Received April 2, 2004. Revised September 13, 2004.

1. Introduction

We are concerned with the homogenization of parabolic or hyperbolic boundary-value problems of the type





ρ_ε∂ⁿu_ε

∂tⁿ −div(a_ε∇uε) =f in Ω×T , n∈ {1,2}, + boundary conditions,

(1.1)

when the coefficientsρ_ε,a_ε do not satisfy the assumptions of uniform ellipticity and boundedness like 0< α <

a_ε(x), ρ_ε(x)< β <+∞which guarantee a classical asymptotic behaviour. It is well known that in this case, homogenization may lead to unusual models such as non-local ones [7, 9]. As far as scalar elliptic equations are concerned, it actually seems that the effective equation obtained by the homogenization of grain-like inclusions should be of local type (see Rem. 2.2 (v)). With regard to evolution equations on the contrary, we bring to the fore in this paper the possible presence of memory terms in the limit equation, when inclusions of high mass density and vanishing measure are considered.

In Section 2 we fix the notations and state the result (Th. 2.1). The proof, based on the argument developed in [1], is situated in Section 4. Section 3 is devoted toa prioriestimates.

Keywords and phrases. Homogenization, memory effects, grain-like inclusions.

1 D´epartement de Math´ematiques, Universit´e de Perpignan, 52 av. de Villeneuve, 66100 Perpignan, France;

[email protected]

c EDP Sciences, SMAI 2005

Figure 1. The composite medium.

2. Notations and main results

Let Ω be a bounded open domain ofR³with smooth boundary. Given a sequence of positive real numbers (r_ε), theε-periodic distributionB_εof balls of radiusr_εis described as follows (see Fig. 1): we introduce

Y :=

−1 2, 1

2 ₃

; B_r:=

x∈R³,

x²₁+x²₂+x²₃< r

; Y_εⁱ:=ε({i}+Y); B_rⁱ :=εi+B_r; I_ε:= i∈Z³, Y_εⁱ⊂Ω

, (2.1)

and set

B_ε:=

i∈Iε

B_rⁱ_ε. (2.2)

Our aim is to study the asymptotic behaviour of the sequence of evolution problems ρ_ε∂ⁿu_ε

∂tⁿ −div(a_ε∇u_ε) =f in Ω×T , u_ε∈ Dn, (2.3) with

n∈ {1,2},f ∈L²(Ω×T),U₀∈C₀¹(Ω), V₀∈C(Ω), 0< T <+∞, (2.4) and

D1:={u∈L²(0, T;H₀¹(Ω))∩C([0, T], L²(Ω)), u(0) =U₀ in Ω}, D2:=

u∈C([0, T];H₀¹(Ω))∩C¹([0, T];L²(Ω)), u(0) =U₀, ∂u

∂t(0) =V₀ in Ω

. (2.5)

The sequencesρ_ε, a_ε are defined by

ρ_ε(x) =ρ_1ε if x∈B_ε, ρ_ε(x) =ρ₀>0 if x∈Ω\B_ε,

a_ε(x) =a_1ε if x∈B_ε, a_ε(x) = 1 if x∈Ω\B_ε, (2.6)

where (ρ_1ε) and (a_1ε) are two sequences of positive real numbers such that ρ_1ε, a_ε> c >0, lim

ε→0

4π 3

r_ε³

ε³ρ_1ε=ρ₁, ρ₁∈[0,+∞[,

ε→0lima_1ε= +∞, sup

ε>0

Ωa_ε|∇U0|²dx <+∞. (2.7)

The functionρ_ε is allowed to take very high values on the subsetB_ε of disconnected balls, while at the same time the measure ofB_εtends to 0. More precisely we assume

0< r_εε. (2.8)

The limit problem depends onρ₁ defined by (2.7), and on the parameter γ:= lim

ε→0

r_ε

ε³ ∈[0,+∞]. (2.9)

It is expressed in terms of the limituof the sequenceu_εof solutions of (2.3) and the limitvof the sequence (v_ε) defined by

v_ε:= |Ω|

|B_ε|u_ε1_Bε×(0,T), (2.10) which describes the average behaviour of the restriction ofu_εto the subset of ballsB_ε. The effective boundary conditions are given by (u, v)∈ D^eff_n , where

D_n^eff :=

(u, v)∈(L²(0, T;H₀¹(Ω))×L²(0, T;L²(Ω)))∩(Cn(ρ₀)× Cn(ρ₁))

, (2.11)

with

Cn(0) :=L²(0, T;L²(Ω)), C_n(r) :=

g∈Cⁿ⁻¹([0, T];L²(Ω)),

g(0) =U₀ ifn= 1, g(0) =U₀ and ∂g

∂t(0) =V₀ ifn= 2

ifr >0. (2.12) Notice that the function ualways satisfies the initial condition, whilev only satisfies it ifρ₁>0.

Theorem 2.1. Assume (2.6), (2.7) and (2.8), then consider γ defined in (2.9):

(i) if γ >0, the sequence (u_ε)of solutions of (2.3) converges weakly in L²(0, T;H₀¹(Ω)) (resp. star-weakly in L^∞(0, T;H₀¹(Ω)) if n = 2) to u and the sequence v_ε defined by (2.10) converges star-weakly in M(Ω×T)(resp. star-weakly in L^∞(0, T;M(Ω)) if n= 2) to a function v, where (u, v)is the unique solution in D^eff_n of







 ρ₀∂ⁿu

∂tⁿ −∆u+ 4πγ(u−v) =f in Ω×T , ρ₁∂ⁿv

∂tⁿ + 4πγ(v−u) = 0 in Ω×T ,

if 0< γ <+∞,

v=u; (ρ₀+ρ₁)∂ⁿu

∂tⁿ −∆u=f in Ω×T , if γ= +∞. (2.13)

(ii) If γ= 0, the sequence (u_ε)of solutions of (2.3) converges weakly in L²(0, T;H₀¹(Ω)) (resp. star-weakly in L^∞(0, T;H₀¹(Ω)) ifn= 2) to the solution of

ρ₀∂ⁿu

∂tⁿ −∆u=f in Ω×T .

Remark 2.2.

i) The average value of the coefficient a_ε(x) on the set B_ε has no influence upon the limit problem, unlike in the case whereB_εis assumed to be a periodic distribution of fibers (see [1, 2]).

ii) Ifγ = 0 and ρ₁ >0 the sequence (v_ε) converges star-weakly in M(Ω×T) to the solutionv of the second line of (2.13) which satisfies the initial conditions given by (2.11), namelyv =U₀ ifn= 1 andv =U₀+V₀tif n= 2 (this is proved at the end of Sect. 4). Of course, in this case the variablesuandvare independent.

iii) The auxiliary variablev can be expressed in terms ofuby solving the second equation in (2.13). Assuming ρ₁>0 and settingω²=^4πγ_ρ

1 , we obtain











v(x, t) = _t

0

ω²exp(ω²(τ−t))u(x, τ) dτ+U₀(x) exp(−ω²t), ifn= 1, v(x, t) =

_t

0 ωsinω(t−τ)u(x, τ) dτ+V₀(x)sinωt

ω +U₀(x) cosωt, ifn= 2, yielding after substitution in the first equation

























 ρ₀∂u

∂t −∆u+ρ₁ω²

u−ω² _t

0exp(ω²(t−τ))u(τ) dτ

=ρ₀f+ρ₁ω²U₀(x) exp(−ω²t), ifn= 1, ρ₀∂²u

∂t² −∆u+ρ₁ω²

u−ω _t

0sin(ω(t−τ))u(τ) dτ

=ρ₀f+ρ₁ω

V₀(x) sin(ωt) +ωU₀(x) cos(ωt)

, ifn= 2, bringing to the fore the presence of memory terms in the limit problem.

iv) The asymptotic behaviour of ρ_ε∂ⁿu_ε

∂tⁿ −div(a_ε∇u_ε) =ρ_εf in Ω×T , u_ε∈ Dn, (2.14) can be studied in the same way, provided we assume

γ >0, f ∈C(Ω×T), (2.15)

(if γ= 0, the relative compactness of (u_ε) may fail). The effective equations read (u, v)∈ D^eff_n ,





 ρ₀∂ⁿu

∂tⁿ −∆u+ 4πγ(u−v) =ρ₀f in Ω×T , ρ₁∂ⁿv

∂tⁿ + 4πγ(v−u) =ρ₁f in Ω×T ,

if 0< γ <+∞,

v=u; (ρ₀+ρ₁)∂ⁿu

∂tⁿ −∆u= (ρ₀+ρ₁)f on Ω×T , if γ= +∞. (2.16) v) The homogenization of the sequence of elliptic problems

−div(a_ε∇u_ε) =ρ_ε(x)f in Ω, u_ε∈H₀¹(Ω), (2.17)

can be obtained by the same method under the asumptions (2.6), (2.7) (of course, with no assumption related to the initial condition), (2.8), (2.15). The effective equation satisfied by the weak limit u in H₀¹(Ω) of the sequence (u_ε) of solutions of (2.17), deduced from (2.13) by substituting 0 for the time derivatives, reads (see a sketch of the proof at the end of Sect. 4)

−∆u= (ρ₀+ρ₁)f in Ω, u∈H₀¹(Ω). (2.18)

This colour of simplicity covers the interesting behaviour of the sequence (v_ε), which converges star-weakly in the sense of measures to the functionv∈L²(Ω) such that (u, v) is the solution inH₀¹(Ω)×L²(Ω) of

−∆u+ 4πγ(u−v) =ρ₀f in Ω,

4πγ(v−u) =ρ₁f in Ω, if 0< γ <+∞,

v=u; −∆u= (ρ₀+ρ₁)f on Ω, if γ= +∞. (2.19)

The couple of equations (2.19) is the Euler-Lagrange system associated with

(u,v)∈(Lmin²(Ω))²Φ(u, v)−

Ωρ₀f udx−

Ωρ₁f vdx, (2.20)

where the functional Φ, defined on (L²(Ω))² by Φ(u, v) := 1

2

Ω|∇u|²dx+1 2

Ω4πγ (v−u)²dx if u∈H₀¹(Ω), Φ(u, v) := +∞ otherwise,

describes the energy associated with the couple (u, v). By fitting the argument developed in [2], we can prove that the sequence of functionals defined onL²(Ω) byF_ε(u) := ¹₂

Ωa_ε|∇u|²dx−

Ωρ_ε(x)f.udx,ifu∈H₀¹(Ω), F_ε(u) := +∞otherwise, Γ-converges (see [6]) strongly inL²(Ω) to the functional

F(u) := min

v∈L²(Ω,R³)Φ(u, v)−

Ωρ₀f udx−

Ωρ₁f vdx

= 1 2

Ω|∇u|²dx−

Ω(ρ₀+ρ₁)f udx− 1 8πγ

Ω(ρ₁f)²dx. (2.21) Classically, the solutions u_ε of (2.17), which minimize F_ε, converge to the solution of min_uF(u) equivalent to (2.18).

vi) The results of [1, 2] corresponding to fiber structures have recently been extended to the framework of linear elasticity (see [3, 4]). It is likely possible to do the same for grain-like inclusions, although there occurs an additional difficulty relating to the calculation of the exact solution of a three dimensional elasticity elementary problem (corresponding to (4.5)).

vii) Inclusions of high mass density with a radiusr_εof the same order of magnitude as the periodεhave been considered in [10].

3. A

estimates

In the sequel, the letter “C” represents a suitable positive constant, independent ofε, which may vary from line to line. We introduce the following measures on Ω defined by

m_ε:= 3 4π

ε³ r_ε³L³_Bε

= |Y_εⁱ|

|Bⁱ_ε|L³_Bε

. (3.1)

We havem_ε(Ω) =| ∪i∈IεY_εⁱ| ≤ |Ω|, hence the sequence (m_ε) is bounded inM(Ω). Fixingϕ∈ D(Ω) and setting

ϕ_ε(x) :=

i∈Iε

−Y_εⁱϕ(s)ds

1_Yi ε(x) (here the usual notation

−_Efdν= _ν(E)¹

Efdν is employed), noticing that

ϕ_εdm_ε=

i∈Iε

|Y_εⁱ|

|B_εⁱ|

Bεⁱ

ϕ_εdx=

∪i∈IεYεⁱ

ϕdx,

and that the inequality

||ϕ−ϕ_ε||_L^∞_(Ω)≤Cε,

holds as soon as the support of ϕis included in ∪i∈IεY_εⁱ (i.e. forε < ^√1₃dist(∂Ω,Supportϕ)), we deduce

ε→0lim

ϕdm_ε= lim

ε→0

ϕ_εdm_ε= lim

ε→0

∪i∈IεY_εⁱϕdx=

Ωϕdx, and

m_ε L³Ω, star-weakly inM(Ω). (3.2)

Let us fix a sequence of positive real numbers (R_ε) such that

r_εR_εε and ε³R_ε. (3.3)

For a given sequence (u_ε) inH₀¹(Ω), we introduce the following functions

˜

u_ε(x) :=

i∈Iε

−∂Bⁱ_Rεu_ε(s) dH²(s)

1_Yi ε(x),

˜

v_ε(x) :=

i∈Iε

−∂Bⁱ_rεu_ε(s) dH²(s)

1_Yi

ε(x), (3.4)

whereBⁱ_Rε,Bⁱ_rε,Y_εⁱ,I_εare defined by (2.1). Their asymptotic behaviour is characterized by

Lemma 3.1. Let u_ε be a sequence in H₀¹(Ω) andm_ε,(˜u_ε),(˜v_ε) be defined by (3.1), (3.4). Then the following estimates hold:

Ω|u_ε−u˜_ε|²dx≤C ε³

R_ε +ε²³

Ω|∇u_ε|²dx,

Bε

|u_ε−v˜_ε|²dx≤Cr_ε²

Bε

|∇u_ε|²dx,

Ω|˜u_ε−v˜_ε|²dx≤Cε³ r_ε

Ω|∇uε|²dx,

Ω|˜u_ε|²dx=

|˜u_ε|²dm_ε ,

Ω|˜v_ε|²dx=

|˜v_ε|²dm_ε. (3.5)

Proof. For a given couple (r₁, r₂) of positive real numbers such that r₁ < r₂, the elementary minimization problem

Γ(r₁, r₂) := min

ϕ∈H¹([r1,r2])

_r2

r1

|ϕ(r)|²r²dr, ϕ(r₁) = 1, ϕ(r₂) = 0

, (3.6)

is achieved atϕ(r) := ^r_r¹_r^r2−r

2−r1, yielding

Γ(r₁, r₂) = r₁r₂

r₂−r₁· (3.7)

In accordance with (3.6) and Cauchy-Schwartz’s inequality, for anyu∈H¹(B_r2 \B_r1) (see (2.1)) we have (in spherical coordinates)

Br2\Br1

|∇u|²dx≥ _π

θ1=0

_2π

θ2=0

_r2

r1

∂u

∂r

²r²sinθ₁drdθ₁dθ₂

≥Γ(r₁, r₂) _π

θ1=0

_2π

θ2=0|u(r₂, θ₁, θ₂)−u(r₁, θ₁, θ₂)|²sinθ₁dθ₁dθ₂

≥4πΓ(r₁, r₂) _π

θ1=0

_2π

θ2=0

(u(r₂, θ₁, θ₂)−u(r₁, θ₁, θ₂))sinθ₁

4π dθ₁dθ₂ ²

= 4πΓ(r₁, r₂)

−∂Br2

udH²−

−∂Br1

udH²

2

. (3.8)

Applying (3.8) on each subset (B_Rⁱ_ε\B_rⁱ_ε), we deduce from (3.4), (3.7)

Ω|˜u_ε−˜v_ε|²dx=

i∈Iε

Y_εⁱ

−∂B_Rεⁱ u_εdH²−

−∂Bⁱ_rεu_εdH²

2

dx

≤

i∈Iε

Y_εⁱ

1 4πΓ₁(R_ε, r_ε)

Bⁱ_Rε\B_rεⁱ |∇uε|²dx

dx

≤

i∈Iε

ε³

1 4πΓ₁(R_ε, r_ε)

B_Rεⁱ \Bⁱ_rε|∇uε|²dx

≤ ε³ 4πΓ₁(R_ε, r_ε)

Ω|∇uε|²dx= ε³(R_ε−r_ε) 4πR_εr_ε

Ω|∇uε|²dx

≤ ε³ 4πr_ε

Ω|∇uε|²dx,

and the third line of (3.5). Next we establish (see below) forR >0 andα∈]0,1]

BR

u(x)−

−∂BαR

u(s) dH²(s)

² dx≤CR² α

BR

|∇u|²dx, ∀u∈H¹(B_R). (3.9)

By applying (3.9) tou_ε(possibly extended toR³by settingu_ε(x) = 0 ifx∈R³\Ω) on each ballB^i√₃

2 ε(see (2.1)) with R= ^√₂³εand α= √²

3Rε

ε , noticing that Y_εⁱ⊂B^i√_3ε

2

and

i∈Iε1_Bi√ 23ε

≤2 onR³ (because the ballsB^i√₃

2 ε

intersect each other no more than two times), we infer

∪i∈IεY|u_εⁱε−u˜_ε|²dx≤

i∈Iε

Y_εⁱ

u_ε−

−∂B_Rεⁱ u_εdH²

2

dx

≤

i∈Iε

B^i√₃

2 ε

u_ε−

−∂Bⁱ_Rεu_εdH²

2

dx

≤C ε³ R_ε

i∈Iε

B^i√₃

2 ε

|∇u_ε|²dx≤Cε³ R_ε

Ω|∇u_ε|²dx. (3.10)

On the other hand, settingN_ε:= Ω\ ∪i∈IεY_εⁱ (notice that N_ε⊂ {x∈Ω,dist(x, ∂Ω)≤√

3ε} hence there holds

|N_ε| ≤Cε, because ∂Ω is smooth) by (3.4), H¨older’s inequality and the continuous inbeddingH₀¹(Ω)⊂L⁶(Ω) we have

Nε

|u_ε−u˜_ε|²dx=

Nε

|u_ε|²dx≤ ||u_ε||²_L6(Ω)|N_ε|²³ ≤Cε²³

Ω|∇u_ε|²dx. (3.11) We deduce the first line of (3.5) from (3.10) and (3.11). By repeating the argument on each ball B_rⁱ_ε (with R=r_ε,α= 1), we get

Bε

|uε−v˜_ε|²dx=

i∈Iε

B_rεⁱ

u_ε−

−∂B_rεⁱ u_εdH²

2

dx

≤Cr_ε²

i∈Iε

B_rεⁱ |∇u_ε|²dx=Cr²_ε

Bε

|∇u_ε|²dx ,

and the second line of (3.5). Finally by (3.1) and (3.4) there holds

|˜u_ε|²dm_ε=

i∈Iε

|Y_εⁱ|

|Bⁱ_ε|

Bⁱ_ε

−∂Bⁱ_Rεu_ε(s) dH²(s)

dx=

i∈Iε

Y_εⁱ

−∂B_Rεⁱ u_ε(s) dH²(s)

dx

=

Ω|u˜_ε|²dx,

resp.

|v˜_ε|²dm_ε=

Ω|v˜_ε|²dx

,

yielding the fourth line of (3.5).

Proof of (3.9). We prove the inequality forR= 1, the general case is inferred by making the change of variable y=Rx. Noticing that by (3.7) and (3.8), for anyr∈]0,1[ we have

−∂Br

udH²−

−∂Bα

udH^{2 2}≤ 1

4π

|r−α|

rα

B1

|∇u|²dx, (3.12)

we deduce from Cauchy-Schwartz’s inequality and (3.12) that

−B1

udx−

−∂Bα

udH^{2 2}=

₁

0

−∂Br

udH²

3r²dr− ₁

0

−∂Bα

udH²

3r²dr ²

≤ ₁

0

−∂Br

udH²−

−∂Bα

udH²

² 3r²dr

≤ ₁

0

B1|∇u|²dx 4π

|r−α|

rα 3r²dr≤C α

B1

|∇u|²dx, (3.13)

then, from the Poincar´e-Wirtinger’s inequality

B1|u(x)−

−_B1u(s) ds|²dx≤C

B1|∇u|²dx,that

B1

u−

−∂Bα

udH²

² dx≤ C α

B1

|∇u|²dx.

We recall (see [2] Lem. A2, p. 431).

Lemma 3.2. Let ν_ε andν be Radon measures on a compact K ⊂R^N such thatν_ε ν. Let (f_ε)a sequence of ν_ε-measurable functions such that sup_ε

|fε|²dν_ε <+∞. Then the sequence f_εν_ε is sequentially relatively compact in the weak-star topologyσ(M(K), C(K))and every cluster pointmis of the formm=f νwithf ∈L²_ν. Moreover, if f_εν_ε f ν, then liminf

ε

|f_ε|²dν_ε≥

|f|²dν.

The following proposition particularizes the asymptotic behaviour of the sequence (u_ε) of solutions of (2.3) and of the sequence (v_ε) defined by (2.10).

Proposition 3.3. The sequence (u_ε) of solutions of (2.3) is bounded in L²(0, T;H₀¹(Ω)) (resp. bounded in L^∞(0, T;H₀¹(Ω)) ifn= 2). More precisely, the following estimates hold

Ω×T|∇u_ε|²dxdt < C (resp.

Ω|∇u_ε(x, τ)|²dx < C, ∀τ∈[0, T], if n= 2),

Ω×Ta_ε|∇uε|²dxdt < C (resp.

Ωa_ε|∇uε(x, τ)|²dx < C, ∀τ∈[0, T], if n= 2), (3.14) and up to a subsequence, there exists a functionusuch that

u_ε u weakly in L²(0, T;H₀¹(Ω)), (resp. star-weakly in L^∞(0, T;H₀¹(Ω)), ifn= 2). (3.15) Besides the sequence (˜u_ε)defined by (3.4) satisfies

˜

u_ε−u_ε→0 strongly in L²(Ω×T) (resp. in L^∞(0, T;L²(Ω)), ifn= 2). (3.16) Assume moreover that γ >0. Then the following estimate holds

Ω×T|uε|²dm_ε(x)dt≤C, if n= 1,

resp.

Ω|u_ε(x, τ)|²dm_ε(x)≤C, ∀τ ∈[0, T], if n= 2

, (3.17)

and up to a subsequence, there exists v∈L²(Ω×T)such that u_εdm_εdt v dxdt star-weakly inM(Ω×T),

(resp. star-weakly in L^∞(0, T,M(Ω)) if n= 2). (3.18) In addition, the sequence(˜v_ε)defined by (3.4) satisfies

˜

v_ε v weakly in L²(Ω×T) (resp. star-weakly in L^∞(0, T;L²(Ω)), if n= 2). (3.19)

Proof. Assuming first n= 1, we multiply the equation (2.3) byu_ε and integrate it over Ω×(0, τ) for a fixed real numberτ∈[0, T]. After integration by parts we obtain

1 2

Ωρ_ε(x)|uε(x, τ)|²dx−1 2

Ωρ_ε(x)|U0(x)|²dx+ _τ

0

Ωa_ε|∇uε|²dxdt= _τ

0

Ωf u_εdxdt. (3.20) Choosingτ=T in (3.20), thanks to the Poincar´e’s inequality and the fact that (a_ε) can be bounded from below by a positive constant (see (2.7)), we infer

||uε||²_L2(Ω×T)≤C _T

0

Ω|∇uε|²dxdt≤C _T

0

Ωa_ε|∇uε|²dxdt

≤C||f||_L²_(Ω×T)||uε||_L²_(Ω×T)+C

Ωρ_ε(x)|U0(x)|²dx, (3.21) yielding the boundedness of (u_ε) in L²(0, T;H₀¹(Ω)), (3.14) and (3.15) (the sequence of real numbers ₁

2

Ωρ_ε(x)|U₀(x)|²dx

is bounded because by (2.7) the sequence of measures (ρ_ε(x) dx) is bounded in M(Ω) andU₀ is continuous).

Ifn= 2, by the standard regularity results (see [8] or [5], p. 222) we have u_ε∈C([0, T], H₀¹(Ω))∩C¹([0, T], L²(Ω)), ∂²u_ε

∂t² ∈L²(0, T;H⁻¹(Ω)). (3.22) Fixing t ∈ [0, T], multiplying the first line of (2.3) by ^∂u_∂t^ε and integrating over Ω, after integration by parts according to (3.22) with respect to the space variables, we obtain

d dt

1 2

Ωρ_ε(x) ∂u_ε

∂t (x, t)

² dx+1 2

Ωa_ε|∇u_ε|²dx

=

Ωf(x, t)∂u_ε(x, t)

∂t dx. (3.23)

Fixingτ∈[0, T] and integrating (3.23) with respect totover [0, τ], thanks to the initial conditions given in (2.3) we get

1 2

Ωρ_ε(x) ∂u_ε

∂t (x, τ) ² dx+

Ωa_ε|∇uε|²dx

= 1

2

Ωρ_ε(x)|V₀(x)|²dx+

Ωa_ε|∇U₀(x)|²dx

+

Ω×(0,τ)f(x, t)∂u_ε(x, t)

∂t dxdt. (3.24)

By (2.7), the sequence of real numbers 1

2

Ωρ_ε(x)|V0(x)|²dx+¹₂

Ωa_ε|∇U0(x)|²dx

is bounded, hence we deduce from (3.24) that

1 2

Ωρ_ε(x) ∂u_ε

∂t (x, τ) ² dx+

Ωa_ε|∇u_ε|²dx

≤C



1 +

Ω×T

∂u_ε

∂t

² dxdt



, ∀τ∈[0, T], (3.25)

and then, after integration over (0, T) with respect toτ, 1

2

Ω×Tρ_ε(x) ∂u_ε

∂t

² dxdt+1 2

Ω×Ta_ε|∇u_ε|²dxdt≤C



1 +

Ω×T

∂u_ε

∂t

² dxdt



.

We infer that

Ω×Tρ_ε(x)^∂uε

∂t ² dxdt is bounded (because by (2.7), ρ_ε is bounded from below by a positive constant) and then, coming back to (3.25), that

Ωρ_ε(x) ∂u_ε

∂t (x, τ) ² dx+

Ωa_ε|∇uε(x, τ)|²dx≤C, ∀τ ∈[0, T], (3.26) yielding (3.14), the boundedness of (u_ε) inL^∞(0, T;H₀¹(Ω)), and up to a subsequence the convergence (3.15).

The assertion (3.16) follows from (3.3), the first line of (3.5) and (3.14).

Let us assume thatγ >0. Then, by (2.9), (3.14), (3.15), (3.16) and the third line of (3.5), the sequence (˜v_ε) is bounded inL²(Ω×T) (resp. inL^∞(0, T;L²(Ω)) ifn= 2) thus converges weakly (resp. star-weakly), up to a subsequence, to somev∈L²(Ω×T) (resp. L^∞(0, T;L²(Ω)) ifn= 2), that is (3.19). We infer from the fourth line of (3.5)

Ω×T|˜v_ε|²dm_εdt≤C,

resp.

|˜v_ε(x, τ)|²dm_ε≤C, ∀τ ∈[0, T]

(3.27) hence we can apply Lemma 3.2 withν_ε:=m_ε⊗ L¹(0,T)(which by (3.2) converges star-weakly to the Lebesgue measure on Ω×T) and f_ε := ˜v_ε: we deduce that the sequence of measures (˜v_εm_ε⊗ L¹_(0,T)) is bounded in M(Ω×T) (resp. in L^∞(0, T;M(Ω))) and weak-star converges, up to a subsequence, to a measure of the typewdxdt withw∈L²(Ω×T). By passing to the limit asε→0 in the following estimate

Ω×T ϕv˜_εdm_εdt−

Ω×Tϕv˜_εdxdt

≤Cε, (3.28)

holding in view of the definitions (3.1) and (3.4), for anyϕ∈C(Ω×T) we deducew=v, hence

˜

v_εdm_εdt v dxdt,star-weakly inM(Ω×T),

(resp. star-weakly inL^∞(0, T,M(Ω)) ifn= 2). (3.29) By (3.1) , (3.14) and the second line of (3.5) we have

Ω×T|u_ε−˜v_ε|²dm_εdt= 3 4π

ε³ r³_ε

Bε×(0,T)|u_ε−˜v_ε|²dxdt≤Cε³ r_ε³r²_ε

Bε×(0,T)|∇u_ε|²dxdt

=Cε³ r_ε

1 a_1ε

Bε×(0,T)a_1ε|∇uε|²dxdt≤Cε³ r_ε

1 a_1ε,

resp.

Ω|u_ε(x, τ)−˜v_ε(x, τ)|²dm_ε(x)≤Cε³ r_ε

1

a_1ε, ∀τ ∈[0, T] ifn= 2

. (3.30)

The estimate (3.17) follows from (3.27), (3.30) and the inequality

Ω×T|u_ε|²dm_εdt≤2

Ω×T|u_ε−v˜_ε|²dm_εdt+ 2

Ω×T|˜v_ε|²dm_εdt,



resp.

Ω|u_ε(x, τ)|²dm_ε≤2

Ω|u_ε(x, τ)−˜v_ε(x, τ)|²dm_ε(x)+ 2

Ω|˜v_ε(x, τ)|²dm_ε(x),

∀τ ∈[0, T] ifn= 2



. (3.31)

By (3.17) we can apply Lemma 3.2 withν_ε:=m_ε⊗ L¹_(0,T)andf_ε:=u_ε. We infer, up to a subsequence, u_εdm_εdt g dxdt,star-weakly inM(Ω×T),

(resp. star-weakly inL^∞(0, T,M(Ω)) ifn= 2), (3.32) for someg∈L²(Ω×T). Finally, using the last line of Lemma 3.2 withν_ε:=m_ε⊗ L¹_(0,T)and f_ε:=u_ε−v˜_ε, taking into account (2.7), (3.29), (3.30), (3.32) and the assumptionγ >0, we obtain

Ω×T|g−v|²dxdt≤liminf

ε→0

Ω×T|u_ε−˜v_ε|²dm_εdt≤Climinf

ε→0

ε³ r_ε

1 a_1ε = 0,

henceg=v and (3.18) follows from (3.32).

4. Proof of Theorem 2.1

As in [1], the key point of the proof consists of the construction of an appropriate sequence of oscillating test functions (Φ_ε) by which we will multiply (2.3), then pass to the limit as ε → 0 in accordance with the convergences stated in Proposition 3.3 to get a weak formulation of the limit problem. We assume first that γ > 0 (the case γ = 0, much easier, is commented at the end of the section). Fixing two regular functions ϕ, ψ∈C^∞(Ω×(0, T)) such that ϕ=ψ= 0 on∂Ω×[0, T]∪Ω× {T}and (if n= 2) ^∂ϕ_∂t = ^∂ψ_∂t = 0 on Ω× {T}, we set

d_ε(x) := dist(x,{εi, i∈Z³}), C_ε:=

i∈Iε

Bⁱ_Rε\Bⁱ_rε ={x∈Ω, r_ε< d_ε(x)< R_ε}, (4.1)

and define the test functions (Φ_ε) by

Φ_ε(x, t) := (1−θ_ε(x))ϕ(x, t) +θ_ε(x)(ϕ_ε+ψ_ε)(x, t), (4.2) where

ϕ_ε(x, t) :=

i∈Iε

−B_rεⁱϕ(s, t) ds

1_Yi ε(x), ψ_ε(x, t) :=

i∈Iε

−B_rεⁱψ(s, t) ds

1_Yi

ε(x), (4.3)