and Jos´ e D. Mart´ın-G´ omez

ESAIM: Control, Optimisation and Calculus of Variations

DOI:10.1051/cocv/2014034 www.esaim-cocv.org

A CORRECTOR FOR A WAVE PROBLEM WITH PERIODIC COEFFICIENTS IN A 1D BOUNDED DOMAIN

Juan Casado-D´ıaz

, Julio Couce-Calvo

, Faustino Maestre

and Jos´ e D. Mart´ın-G´ omez

Abstract.We consider a wave problem posed in a bounded open interval ofR, where the coefficients, the initial conditions and the right-hand side are highly oscillating, periodic in the space variable and almost periodic in the time one. Our purpose is to find not only the corresponding limit equation but a corrector,i.e.a strong approximation in theH¹ topology, which for the wave equation is known to be non-local. In a previous paper we have studied this problem in the wholeR^N, here we consider the case of a bounded domain in dimension one. Thus the novelty in this paper is the analysis of the boundary conditions.

Mathematics Subject Classification. 35B27, 35L20.

Received March 5, 2014. Revised June 12, 2014.

Published online March 4, 2015.

1. Introduction

The homogenization of a wave problem with oscillating coefficients in a bounded open setΩ⊂R^N such as

⎧⎪

⎪⎨

⎪⎪

⎩

∂_t(ρ_ε(x)∂_tu_ε)−div_x(A_ε(x)∇xu_ε) =f_ε in (0, T)×Ω u_ε= 0 on (0, T)×∂Ω

u_ε|t=0=u⁰_ε, ∂_tu_ε|t=0=ϑ_ε inΩ,

(1.1)

has been studied in several papers ([5,9,11]). Assuming the coefficientsρ_ε,A_εuniformly elliptic and bounded, the right-hand side converging strongly inL¹(0, T;L²(Ω)) to some functionf and the initial datau⁰_ε,ϑ_εconverging weakly in H₀¹(Ω) and L²(Ω) respectively to some functions u⁰, ϑ, it is known that the solution u_ε of (1.1) converges inL^∞(0, T;H₀¹(Ω)) weak-∗ to the solutionuof

⎧⎪

⎪⎨

⎪⎪

⎩

∂_t(ρ(x)∂_tu)−div_x(A(x)∇xu) =f in (0, T)×Ω u= 0 on (0, T)×∂Ω

u_|t=0=u⁰, ∂_tu_|t=0=ϑ in Ω.

Keywords and phrases.Wave equation, highly oscillating coefficients, homogenization, corrector, boundary conditions.

1 Dpto. de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Sevilla, Spain.jcasadod@us.es; couce@us.es; fmaestre@us.es;jdmartin@us.es

Article published by EDP Sciences c EDP Sciences, SMAI 2015

Here the coefficientρis the weak-∗limit of ρ_ε inL^∞(Ω) and the coefficient matrixA is the limit ofA_ε in the sense of the elliptic homogenization ([13,17,18]). This homogenization result provides a weak approximation of the derivatives of u_ε. It is also interesting to get an approximation of these derivatives in the strong topology ofL²((0, T)×Ω). This is called a corrector result in homogenization. SinceAis the limit ofA_ε in the sense of the elliptic homogenization one could expect that the elliptic corrector also provides a corrector for the wave problem. However it has been proved in [5] that this only holds true if the initial data are “well posed”.

A corrector result for problem (1.1) in the case of periodic coefficients andΩ=R^N has been obtained in [6]

and [10]. For the elliptic or parabolic problems the corrector in every point is just obtained from the value of the derivative of the limit function in such point. In particular initial and boundary conditions do not affect to the corrector. However for the wave equation the corrector is non-local. In general, its value in a point depends on the value of the limit function and the right-hand side in the whole domain (or more exactly in a certain cone of dependence) and of the initial conditions. This is due to the dispersion of the waves in an heterogeneous domain. Namely, it has been considered in [10] the wave problem

∂_t(ρ_ε(t, x)∂_tu_ε)−div_x(A_ε(t, x)∇xu_ε) +B_ε(t, x)· ∇t,xu_ε=f_ε in (0, T)×R^N

u_ε|t=0=u⁰_ε, ∂_tu_ε|t=0=ϑ_ε in R^N, (1.2)

where the functionsρ_ε,A_ε,B_ε,f_ε,u⁰_εandϑ_ε have the following structure ρ_ε(t, x) =ρ⁰

x ε

+ερ¹ t, x,t ε,x

ε

, A_ε(t, x) =A⁰ x

ε

+εA¹ t, x,t ε,x

ε

, B_ε(t, x) =B t, x,t ε,x

ε

f_ε(t, x) =f t, x,t ε,x

ε

, u⁰_ε(x) =u⁰(x) +εu¹

x,x ε

, ϑ_ε(x) =ϑ

x,x ε

,

and they are periodic iny =x/εof period the unitary cube Y in R^N and almost-periodic ins=t/ε. Then it has been proved the corrector result

u_ε(t, x)−u₀(t, x)−εu₁ t, x,t ε,x

ε

→0 inH¹((0, T)×R^N), (1.3) whereu₀,u₁ (and someu₂) solve

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩

ρ⁰∂²_ssu₁−div_y(A⁰(∇xu₀+∇yu₁)) = 0

ρ⁰∂²_ttu₀−div_x(A⁰∇xu₀) + 2ρ⁰∂_st²u₁−div_x(A⁰∇yu₁)−div_y(A⁰∇xu₁) +∂_s

ρ¹(∂_tu₀+∂_su₁)

−div_y

A¹(∇xu₀+∇yu₁) +B·(∇_t,xu₀+∇_s,yu₁) +ρ⁰∂_ss²u₂−div_y(A⁰∇_yu₂) =f

u₁(t, x, s, y), u₂(t, x, s, y) periodic of periodY iny and almost periodic ins u_0|t=0=u⁰,

∂_tu₀+∂_su₁

|t=s=0=ϑ, ∂_yu_1|t=s=0=∂_yu¹.

(1.4)

Here the second equation is just formal. The corresponding variational formulation consider test functions ψ=ψ(t, x, s, y) satisfying the wave equation

ρ⁰∂_ss²ψ−div_y(A⁰∇yψ) = 0,

and then the terms containing u₂ disappear. Therefore, in this formulation we get a system of two equations for the two functionsu₀,u₁ which appear in (1.3).

We observe that the second equation in (1.4) contains derivatives ofu₁not only in the microscopic variables (s, y) but also in the macroscopic ones (t, x). This is completely different to the elliptic and parabolic cases and

as we mentioned above shows that the corrector is non-local for the wave equation. The behavior of u₁ in a point (t, x, s, y) does not only depend on the value ofu₀in (t, x). As a consequence of this non-local behavior, it is proved in [10] that the presence of the first order termB_ε(t, x)· ∇u_ε in (1.2) provides a non-local problem for the limitu₀ ofu_ε. On the other hand, we observe that due to the presence of derivatives int foru₁ in the second equation of (1.4) we have needed to introduce initial conditions for u₁ int= 0.

An interesting question is if a result like (1.3) holds also true for a bounded domainΩ. In such case a formal calculus shows thatu₀,u₁ must also satisfy (1.4) where now it would be necessary to introduce some boundary conditions foru₁ on∂Ω. These boundary conditions must depend on the boundary conditions imposed foru_ε in (1.2) and probably on the geometry ofΩ. For this reason the problem in a bounded domain seems to be much more difficult than the problem inR^N. In the present paper we analyze this question in the simplest case whereΩis a bounded one-dimensional interval (α, β). Our formulation considers Dirichlet, Neumann and mixed boundary conditions. We show that (1.3) holds still true, but just for a subsequence because the behavior of the derivatives ofu_ε depends on the relative position of the extremes of the interval (α, β) with respect to the periodicity cell. The main difficulty is to find the good boundary conditions foru₁. They describe the shocks of the wave with the walls. Namely, we show that u₁ can be decomposed as

u₁(t, x, s, y) = ˆu₁(t, x, y) +

j∈J1

˜u_j(t, x)Φ_j(y)e^iλjs+

j∈J2

˜u_j1(t, x)Φ_j1(y) + ˜u_j2(t, x)Φ_j2(y) e^iλjs,

where ˆu₁ is the classical elliptic corrector, the numbersλ_j are the (negative and positive) square roots of the positive eigenvalues μ_j of the following problem, with a⁰ the above function A⁰ which is now denoted by a lowercase letter to emphasize that it is a scalar function.

⎧⎪

⎪⎨

⎪⎪

⎩

−d

dy a⁰dΦ dy

=μ_jρ⁰Φ in R Φ, a⁰dΦ

dy periodic of periodY.

(1.5)

The sets J₁ and J₂ correspond to the index j such that μ_j has multiplicity one or two respectively and the functionsΦ_j,Φ_j1,Φ_j2are a basis of the corresponding eigenfunctions spaces. The boundary conditions foru₁are in fact given for the functions ˜u_j1, ˜u_j2associated to the eigenvalues of multiplicity two while boundary conditions for the functions ˜uj are not needed. This different behavior corresponds to the fact that the perturbations associated to simple eigenvalues do not propagate in space while the corresponding to eigenvalues of multiplicity two travel in space.

We observe that (1.5) corresponds to the eigenvalue problem (2) in [6] for l = 0, where a corrector for problem (1.1) withΩ=R^N and coefficients periodic of periodεis obtained by using the Bloch theory. This is not surprising because we are studying very related problems but in our case this is the only eigenvalue problem we need, while in [6] it is necessary to introduce the second parameterl∈Z^N.

2. Notation

The functions will be assumed valued in the complex fieldC, withi the imaginary unit. The conjugate of a vectoru∈C^k is denoted byu. The real part of a complex numberzis denoted byRe(z).

In order to write shorter expressions, we will only write the arguments of the functions when it is essential.

Forα, β∈R,α < β, we denote byI the intervalI= (α, β)⊂R.

ForT >0, we denote byQ_T the open set (0, T)×I.

We denote byO_εan arbitrary sequence which converges to zero whenεtends to zero and which can change from line to line.

We denote byY the unitary interval (0,1).

For functions defined inY, we use the indexto note periodicity with respect toY (thus the functions are in fact defined in the whole ofR). For exampleL¹(Y) denotes the space of functions inL¹_loc(R) which are periodic of periodY. The integral inY of a functionuinL¹(Y) will be denoted byM_y(u) (mean value).

For functions defined in R, the index denotes almost-periodicity. Namely, we will use the following spaces of almost-periodic functions.

1. We denote byC(R) the space of almost-periodic functions in the Bohr sense,i.e.the closure of the trigono- metric polynomialsuof the form

u(s) = m j=1

c_pje^ipjs, ∀s∈R, withm∈N, pj∈R, cpj ∈C, 1≤j≤m, (2.1) with respect to the uniform convergence topology.

2. We denote by L^p(R), 1 ≤ p < +∞ the space of Besicovitch defined as the closure of the trigonometric polynomials as above with respect to the norm

u_L^p_(R)=

R→∞lim 1 2R

_R

−R|u(s)|^pds ¹_p

.

Indeed, · _L^p_(R)is not a norm but a seminorm. In order to have a structure of normed space inL^p(R) it is necessary to work with a quotient space,i.e. to identify functionsu, v∈L^p(R) such thatu−v_L^p_(R)= 0.

It is well known that every functionuin the spaceL¹(R) has a mean valueM_s(u)∈C, which is defined by

ε→0lim

Eu s

ε

ds=M_s(u)|E|, ∀E⊂R, bounded and measurable.

3. We denote byH^k(R),k∈R(in the paper we just usek=−1,0,1) by H^k(R) =

⎧⎨

⎩

p∈R

c_pe^ips: c_p∈C, ∀p∈R,

p∈R\{0}

|cp|²|p|^2k<∞

⎫⎬

⎭. It is a Hilbert space endowed with the norm

p∈R

c_pe^ips H^k(R)

=

⎛

⎝|c₀|²+

p∈R\{0}

|c_p|²|p|^2k

⎞

⎠

12

.

We remark thatH⁰(R) =L²(R). Moreover, we haveH^k(R) =H^−k(R).

InH^k(R) we define the derivative operator _ds^d :H^k(R)→H^k−1(R) just by d

ds

⎛

⎝

p∈R

c_pe^ips

⎞

⎠=

p∈R

ipc_pe^ips.

We will also use the indexfor functions defined inR×Y to denote periodicity with respect toY and almost periodicity with respect toR. Namely, we will work with the spacesC(R×Y),L^p(R×Y),H^k(R×Y), which are constructed as above but starting from functions of the form

u(s, y) = m j=1

c_pjnje^{i(pjs+2πnjy)}, ∀(s, y)∈R×Y, m∈N, pj ∈R, nj ∈Z, cpjnj ∈C, 1≤j≤m.

The mean value of a functionuinL¹(R×Y) will be denoted byM_s,y(u) =M_s(M_y(u)) =M_y(M_s(u)).

Along the paper we will consider two fixed periodic real functionsρ⁰∈L^∞ (Y) anda⁰∈L^∞ (Y), such that infy∈Yessρ⁰(y)>0, inf

y∈Yessa⁰(y)>0. (2.2)

Then, we introduce the spacesW^k,k∈Rby W^k =

⎧⎨

⎩

p∈R

c_p(y)e^ips∈H^k(R×Y) : c₀≡0, ρ⁰∂_ss²w−∂_y a⁰∂_yw

= 0

⎫⎬

⎭. (2.3)

A spectral decomposition for the elements ofW^k can be obtained in the following way: we introduceμ₀= 0<

μ₁< μ₂<· · · as the numbers (eigenvalues)μ_j∈[0,+∞) satisfying that the space W_j =

Φ∈H¹(Y) : − d

dy a⁰dΦ dy

=μ_jρ⁰Φ inR

(2.4) does not reduce to the null space. For an eigenvalueμ_j, we denote

W_−j =W_j, λ_j=√

μ_j, λ_−j=−√μ_j. (2.5)

Then,

W^k =

⎧⎪

⎨

⎪⎩ ∞

j=−∞j=0

c_j(y)e^iλjs: c_j∈W_j, ∀j ∈Z, ^∞

j=−∞j=0

cj²_L2

(Y)|λj|^2k<∞

⎫⎪

⎬

⎪⎭. (2.6)

We remark that with our definitions, the spaceH¹(R×Y) is not contained inL²(R×Y) since

p∈R\{0}

c_p²_L2

(Y)|p|²<∞ ⇒

p∈R\{0}

c_p²_L2

(Y)<∞,

but W¹is contained inW⁰ because ∞

j=−∞j=0

c_j²_L2

(Y)≤ 1 λ²₁

∞

j=−∞j=0

c_j²_L2

(Y)|λ_j|².

We also recall the following well known result: IfΦ¹,Φ²are two solutions of the differential equation

− d

dy a⁰dΦ dy

=μ_jρ⁰Φ inR (2.7)

then

a⁰ Φ¹dΦ²

dy −Φ²dΦ¹ dy

is constant inR, (2.8)

where the constant is zero if and only ifΦ¹, Φ² are linearly dependent.

Taking into account that the spacesW_j can be of dimension one or two, we denote J₁={j∈Z\ {0}: dim(W_j) = 1}, J₂={j∈Z\ {0}: dim(W_j) = 2}, and

J₁⁺=J₁∩Z⁺, J₂⁺=J₂∩Z⁺.

For a function

u= ∞

j=−∞j=0

c_j(y)e^iλjs∈ W⁰, we defineP₁u,P₂uby

P₁u=

j∈J1

c_j(y)e^iλjs, P₂u=

j∈J2

c_j(y)e^iλjs.

Besides of the functionsρ⁰,a⁰, we will also consider functionsρ¹∈L^∞(Q_T;C(R×Y)),a¹∈L^∞(Q_T;C(R×

Y)),ρ¹,a¹ real, such that

∂_tρ¹, ∂_sρ¹, ∂_ta¹, ∂_sa¹∈L^∞(Q_T;C(R×Y)), (2.9) and a functionB∈L^∞(Q_T;C(R×Y))².

With these functions, andε >0, we will defineρ_ε, a_ε∈W^1,∞(0, T;L^∞(I)),B_ε∈L^∞(Q_T)² by a_ε(t, x) =a⁰

x ε

+εa¹ t, x,t ε,x

ε

(2.10)

ρ_ε(t, x) =ρ⁰ x

ε

+ερ¹ t, x,t ε,x

ε

(2.11)

B_ε(t, x) =B t, x,t ε,x

ε

, (2.12)

a.e. (t, x)∈Q_T.

We denote byc_α, d_α, c_β, d_β complex constants such that

|cα|+|dα|>0, |cβ|+|dβ|>0.

With these constants, we define the spaceV by V =

v∈H¹(I) :v(α) = 0 ifc_α= 0 andv(β) = 0 ifc_β= 0 . The spaceV is endowed with theH¹(I) norm.

3. Main results

Foru⁰_ε,ϑ_ε andf_εbounded inV,L²(I) andL²(Q_T) respectively, we consider the wave problem

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

∂_t(ρ_ε∂_tu_ε)−∂_x(a_ε∂_xu_ε) +B_ε· ∇t,xu_ε=f_ε in Q_T −c_αa_ε∂_xu_ε+d_αu_ε

|x=α= 0,

c_βa_ε∂_xu_ε+d_βu_ε

|x=β= 0 in (0, T) u_ε|t=0=u⁰_ε, ∂_tu_ε|t=0=ϑ_ε inI

u_ε∈L^∞(0, T;V), ∂_tu_ε∈L^∞(0, T;L²(I)).

(3.1)

Our purpose in the present paper is to study the asymptotic behavior of u_ε whenε tends to zero. First we remark that since forε >0 small enough,ρ_ε,a_ε, are uniformly elliptic and bounded inW^1,∞(0, T;L^∞(I)),B_ε is bounded inL^∞(Q_T)², andu⁰_ε, ϑ_ε,f_ε bounded inV,L²(I) andL²(Q_T) respectively, we immediately have Theorem 3.1. Forε >0 small enough, problem(3.1)has a unique solution. Moreover, there existsC >0such that

uε_L^∞_(0,T;V)+∂tu_εL^∞_(0,T;L²_(I))≤C. (3.2)

Theorem 3.1implies that, at least for a subsequence, there exists the limitu₀ ofu_εin the weak-∗ topology ofL^∞(0, T;V). In order to characterize this limit and to obtain a corrector result for problem (3.1), let us use the two-scale convergence theory.

Definition 3.2. We say that a sequencev_ε∈L²(I) two-scale converges to a functionv∈L²(I;L²(Y)) and we writev_ε v, if for every^2e ψ∈C_c(I;C(Y)) we have

ε→0lim

Iv_εψ

x,x ε

dx=

IM_y(vψ) dx. (3.3)

Analogously, we say that a sequence v_ε∈L²(Q_T) two-scale converges to a function v∈L²(Q_T;L²(R×Y)) and we writev_ε v, if for every^2e ψ∈C_c(Q_T;C(R×Y)) we have

ε→0lim

QT

v_εψ t, x,t ε,x

ε

dtdx=

QT

M_s,y(vψ) dtdx. (3.4)

The interest of the two-scale convergence follows from the classical compactness results which establishes the existence of a two-scale limit for bounded sequences inL²(see [1,7,8,14–16]). As a consequence of these results andu⁰_ε,ϑ_ε,f_ε bounded inV,L²(I) andL²(Q_T), there existu⁰∈V,u¹∈L²(I;H¹(Y)),ϑ∈L²(I;L²(Y)) and f ∈L²(Q_T;L²(R×Y)) such that, up to a subsequence, we have

u⁰_ε u⁰ inV, ∂_xu⁰_ε ∂^2e _xu⁰+∂_yu¹, ϑ_ε ϑ,^2e f_ε f.^2e (3.5) The main result of the paper is given by Theorem3.4below which provides the limit functionu₀of the solution u_εof (3.1) and the two-scale limit of the sequence∇_t,xu_ε. Related results have been obtained in [2–6,9,11,12].

Here the novelty is the analysis of the boundary conditions which (because the waves travel and shock with the walls) influence the behavior ofu_εat the interior ofΩ.

Although the problem is periodic, we will prove that the two-scale limit for the whole sequence∇t,xu_εdoes not exist, instead we will need to consider a subsequence ofεsuch that there existα^∗, β^∗∈R/Zsatisfying

α

ε →α^∗, β

ε →β^∗ in R/Z. (3.6)

Associated to this subsequence, we define the following spaces which will appear in Theorem3.4.

Definition 3.3. For a subsequence ofεsuch that (3.6) holds, we define the spacesV1 andV2 by V1=H₀¹

0, T;L² I;P₁

W¹

(3.7)

V2=

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

ψ∈H¹(Q_T;P₂(W¹)) : ψ_|t=0=ψ_|t=T = 0,

(a⁰∂_yψ)_|y=α^x=α_∗ = 0 ifc_α= 0, ψ_|x=α,y=α∗ = 0 ifc_α= 0 (a⁰∂_yψ)_|x=β

y=β∗ = 0 ifc_β= 0 ψ_|x=β,y=β∗ = 0 ifc_β= 0.

⎫⎪

⎪⎪

⎪⎪

⎬

⎪⎪

⎪⎪

⎪⎭

. (3.8)

Theorem 3.4. We consider a subsequence of ε, still denoted by ε such that (3.5) and (3.6) hold. Then, the sequence u_εsatisfies

u_ε u^∗ ₀ in L^∞(0, T;V) (3.9)

∇_t,xu_ε^2e ∇_t,xu₀+∇_s,yu₁, (3.10)

whereu₀,u₁ are the unique solutions of the variational system

u₀∈L^∞(0, T;V), ∂_tu₀∈L^∞(0, T;L²(I)) (3.11)

u₁∈L^∞(0, T;L²(I;H¹(R×Y))), M_s,y(u₁) = 0a.e. in Q_T (3.12) u_0|t=0=u⁰,

∂_tu₀+∂_su₁

|t=s=0=ϑ, ∂_yu_1|t=s=0=∂_yu¹ (3.13)

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩

QT

M_s,y

−ρ⁰(∂_tu₀∂_tv₀+∂_su₁∂_sv₁) +a⁰(∂_xu₀+∂_yu₁)(∂_xv₀+∂_yv₁) dtdx +d_α

c_α

{x=α}u₀v₀dt+d_β c_β

{x=β}u₀v₀dt +

QT

M_s,y

B·(∇t,xu₀+∇s,yu₁)v₀ dtdx=

QT

M_s,y(f v₀ dtdx

∀v₀∈W₀^1,1(0, T;L²(I))∩L¹(0, T;V), ∀v₁∈L²(Q_T;H¹(R×Y))

(3.14)

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

QT

M_s,y

−2ρ⁰∂_su₁∂_tψ−ρ¹(∂_tu₀+∂_su₁)∂_sψ+a¹(∂_xu₀+∂_yu₁)∂_yψ dtdx +

QT

M_s,y

B·(∇_t,xu₀+∇_s,yu₁)ψ dtdx=

QT

M_s,y(f ψ)dtdx, ∀ψ∈ V1

(3.15)

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

QT

M_s,y

−2ρ⁰∂_su₁∂_tψ+a⁰∂_yu₁∂_xψ−a⁰∂_x,y² ψ u₁ dtdx +

QT

M_s,y

−ρ¹(∂_tu₀+∂_su₁)∂_sψ+a¹(∂_xu₀+∂_yu₁)∂_yψ dtdx +

QT

M_s,y

B·(∇t,xu₀+∇s,yu₁)ψ dtdx=

QT

M_s,y(f ψ)dtdx, ∀ψ∈ V2.

(3.16)

Remark 3.5. In the case wherec_α= 0, the term d_α c_α

{x=α}u₀v₀dt

which appears in (3.14) must be understood as zero. Observe that in this case, the functionsu₀and v₀ vanish atx=α. Analogously, ifc_β = 0, the term

d_β c_β

{x=β}

u₀v₀dt is defined as zero.

Remark 3.6. In order to have uniqueness foru₁we have taken it satisfyingM_s,y(u₁) = 0. Clearly, (3.10) still holds adding tou₁ any function independent of the variables (s, y).

Remark 3.7. Observe that the two-scale limit of the solutionu_εof (3.1) depends on the position of the extremes ofI with respect to the periodic cell, which correspond to the pointsα^∗ andβ^∗ given by (3.6).

Remark 3.8. In equations (3.15) and (3.16) the functionψ applies Q_T into the spacesP₁(W¹) and P₂(W¹) respectively. This shows that the behavior of elementary waves associated to simple or multiple eigenvalues of (2.4) is different.

Remark 3.9. From (3.14), withv₀= 0, we deduce the equation

ρ⁰∂_ss²u₁−∂_y(a⁰(∂_xu₀+∂_yu₁)) = 0 in R×R, a.e. inQ_T. (3.17) A solution of this problem independent ofsis given by the elliptic corrector ˆu₁(seee.g.[1,4,14]) solution of

uˆ₁∈L^∞(0, T;L²(I;H¹(Y))), M_y(ˆu₁) = 0 a.e. (t, x)∈Q_T

−∂_y(a⁰(∂_xu₀+∂_yuˆ₁)) = 0 in R, a.e. inQ_T, (3.18) or equivalently

∂_yuˆ₁(t, x, y) =a⁰_h 1 a⁰(y)− 1

a⁰_h

∂_xu₀(t, x), a.e. (t, x, y)∈Q_T,×R, M_y(ˆu₁) = 0, (3.19) witha⁰_h the homogenized coefficient corresponding toa⁰,i.e.

a⁰_h= 1

M_y(_a¹0)· (3.20)

From (3.17) and (3.18), we deduce thatu₁can be decomposed as

u₁= ˆu₁+ ˜u₁, (3.21)

with

˜

u₁∈L^∞(0, T;L²(I;W¹)). (3.22)

Remark 3.10. Taking v₁ = 0 in (3.14) and using (3.21), (3.19) and M_s(˜u₁) = 0, which is a consequence of (3.22), we deduce thatu₀ satisfies

⎧⎪

⎪⎨

⎪⎪

⎩

M_y(ρ⁰)∂_tt²u₀−a⁰_h∂_xx² u₀+B_h· ∇t,xu₀+M_s,y(B· ∇s,yu˜₁) =M_s,y(f) in Q_T (−cαa⁰_h∂_xu₀+d_αu₀)_|x=α= 0, (c_βa⁰_h∂_xu₀+d_βu₀)_|x=β= 0 in (0, T) u_0|t=0=u⁰,

(3.23)

where, denoting byb_t, b_xthe two components of B, the functionB_h is given by B_h= (b_h,t, b_h,x) withb_h,t=M_s,y(b_t), b_h,x =M_s,ybx

a⁰

M_y1

a⁰

· (3.24)

If we assume thatB satisfies

M_s,y(B· ∇s,yv) = 0, ∀v∈ W¹, a.e. inQ_T, (3.25) then the fourth term in the PDE in (3.23) vanishes providing an equation for u₀ (the limit equation). This holds, for example, if B does not depend on s. In general (3.25) is not satisfied and then, since ˜u₁ does not depend locally onu₀, we get a non-local equation foru₀. An example of this situation is given in [10], where Q_T = (0, T)×R^N.

Remark 3.11. Equation (3.23) for u₀ must be completed with initial conditions for u₀ and ∂_tu₀. The first one is contained in (3.13). The second one can be obtained as follows: Using (3.21), we can write the second

equation in (3.13) as

∂_tu₀+∂_su˜₁

|t=s=0=ϑ, (3.26)

where ∂_su˜₁ is a combination of eigenfunctions of problem (2.7) corresponding to non-vanishing eigenvalues.

ThusM_y(ρ⁰∂_su˜₁) = 0 and therefore, multiplying (3.26) byρ⁰ and taking the mean value iny we get

∂_tu_0|t=0=M_y(ρ⁰ϑ)

M_y(ρ⁰) · (3.27)

Equation (3.16) implicitly provides boundary conditions for the function ˜u₁ given by (3.21). This is given by the following Proposition which shows thatP₂u˜₁ satisfies the boundary conditions imposed to ψin (3.16).

Proposition 3.12. In the conditions of Theorem3.4, and defining u˜₁ by (3.21), we have (a⁰∂_yP₂u˜₁)_|x=α,y=α∗ = 0 if c_α= 0, P₂u˜_1|x=α,y=α∗ = 0 ifc_α= 0,

(a⁰∂_yP₂u˜₁)_|x=β,y=β∗ = 0 ifc_β= 0, P₂u˜_1|x=β,y=β∗ = 0 if c_β= 0. (3.28) From Theorem 3.4 it is also possible to obtain a corrector result for problem (3.1). For this purpose, it is necessary to assume that in (3.5) the two-scale convergence holds in the following strong sense:

⎧⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎩

ε→0lim

I∂_xu⁰_εψ_εdx=

IM_y

(∂_xu⁰+∂_yu¹)ψ dx

ε→0lim

Iϑ_εψ_εdx=

IM_y ϑψ

dx

∀ψ_ε bounded inL²(I)^N, ψ∈L²(I;L²(Y)), Ψ_ε ψ,^2e

(3.29)

which is possible to prove (see e.g. [1]) that it is equivalent to assume that the second and third assertions in (3.5) hold and

ε→0lim

I|∂_xu⁰_ε|²dx=

IM_y

|∂_xu⁰|²+|∂_yu¹|²

dx, lim

ε→0

I|ϑ_ε|²dx=

IM_y

|ϑ|² dx.

The corrector result is given by Theorem 3.13 below. Its proof is very similar to the one of Theorem 3.7 in [10] and then it will not be given here. The main idea consists in using Theorem3.4to pass to the limit in the energy identity for problem (3.1).

Theorem 3.13. We assume that in Theorem 3.4, the sequences u⁰_ε and ϑ_ε satisfy (3.29), then, for every sequence Γ_ε∈L²(Q_T)² which two-scale converges strongly to ∇_s,yu₁, we have

ε→0lim

QT

|∇t,x(u_ε−u₀)−Γ_ε|²dxdt= 0. (3.30) Remark 3.14. If the function∇s,yu₁ is inL²(Q_T;C⁰(R×Y)²),then we can take in (3.30)

Γ_ε=∇s,yu₁ t, x,t ε,x

ε

· In this case Theorem3.13asserts

∇_t,xu_ε− ∇_t,xu₀− ∇_s,yu₁ t, x,t ε,x

ε

→0 inL²(Q_T)²,

which assuming further regularity inu₁ reads as u_ε−u₀−εu₁ t, x,t

ε,x ε

→0 inH¹(Q_T).

We finish this section with a simplified model of (3.1) where we can explicitly obtain the function u₁ which appears in Theorem3.4 fromu₀, the initial data and the right-hand side. For this purpose, let us consider an orthonormal basis of the spacesW_j composed by real functions. Namely:

Ifj∈J₁, we consider a real eigenfunctionΦ_j∈W_j such that

M_y(ρ⁰|Φ_j|²) = 1. (3.31)

Ifj∈J₂, we consider two real eigenfunctions Φ_j1, Φ_j2∈W_j such that M_y

ρ⁰Φ_jkΦ_jl

=δ_kl, k, l∈ {1,2}. (3.32) SinceW_j =W_−j for everyj∈Z\ {0}, we can also assume

Φ_j =Φ_−j, ∀j ∈J₁, Φ_j1 =Φ_−j1, Φ_j2=Φ_−j2, ∀j∈J₂. (3.33) Moreover, we remark that these functionsΦ_j, Φ_j1,Φ_j2 withj∈Z⁺ are a basis ofL²(Y)/RandH¹(Y)/R.

Taking into account that f belongs to L²(Q_T;C(R×Y)), and then to the spaceL²(Q_T;L²(R×Y)), that u¹belongs toL²(I;H¹(Y)), and can be chosen with zero mean value iny, and thatϑbelongs toL²(I;L²(Y)), we can decompose these functions as

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

f(t, x, s, y) ρ⁰(y) =

j∈J1f_j(t, x)Φ_j(y)e^iλjs

+

j∈J2

f_j1(t, x)Φ_j1(y)+f_j2(t, x)Φ_j2(y)

e^iλjs+f^∗(t, x, s, y)

(3.34)

u¹(x, y) =

j∈J₁⁺

u¹_j(x)Φ_j(y) +

j∈J₂⁺

u¹_j1(x)Φ_j1(y) +u¹_j2(x)Φ_j2(y)

(3.35)

ϑ(x, y) =

j∈J₁⁺

ϑ_j(x)Φ_j(y) +

j∈J₂⁺

ϑ_j1(x)Φ_j1(y) +ϑ_j2(x)Φ_j2(y)

+M_y(ϑ), (3.36)

with

M_s,y(ρ⁰f^∗v) = 0 a.e. inQ_T, ∀v∈ W⁰ (3.37)

j∈J1

f_j²_L2(QT)+

j∈J2

f_j1²_L2(QT)+f_j2²_L2(QT)

<+∞ (3.38)

j∈J⁺₁

λ²_ju¹_j²_L2(I)+

j∈J₂⁺

λ²_j

u¹_j1²_L2(I)+u¹_j2²_L2(I)

<+∞ (3.39)

j∈J₁⁺

ϑ_j²_L2(I)+

j∈J₂⁺

ϑ_j1²_L2(I)+ϑ_j2²_L2(I)

<+∞. (3.40)