Acta Applicandae Mathematicae

, Volume 125, Issue 1, pp 209–229

Periodic Homogenization of Parabolic Nonstandard Monotone Operators

Article

DOI: 10.1007/s10440-012-9788-x

Cite this article as:
Bogning, R.K. & Nnang, H. Acta Appl Math (2013) 125: 209. doi:10.1007/s10440-012-9788-x

Abstract

We study the periodic homogenization for a family of parabolic problems with nonstandard monotone operators leading to Orlicz spaces. After proving the existence theorem based on the classical Galerkin procedure combined with the Stone-Weierstrass theorem, the fundamental in this topic is the determination of the global homogenized problem via the two-scale convergence method adapted to this type of spaces.

Keywords

Global solution Periodic homogenization Two-scale convergence Nonstandard monotone operators Orlicz spaces 

Mathematics Subject Classification (2010)

35B27 35B40 46E30 74G25 

1 Introduction

Let B:[0,∞)→[0,∞) be a differentiable N-function and let \(\widetilde{B}\) be the Fenchel’s conjugate of B such that
$$ \widetilde{B}\in \varDelta ^{\prime},t^{2}\leq B ( \rho_{0}t )\quad\text{and}\quad 1<\rho_{1}\leq\frac{tb ( t ) }{B ( t ) }\leq \rho_{2}\ \text{for any}\ t>0, $$
(1)
where ρ0>0, ρ1, ρ2 are constants, and b is odd, increasing homeomorphism from ℝ onto ℝ such that \(B ( t ) =\int_{0}^{t}b ( s ) ds\) (t≥0). Let Ω be a smooth bounded open set in \(\mathbb{R}_{x}^{d}\) (the space of variables x=(x1,…,xd) (integer d≥1)). Let T be a positive real number. Let \(f\in L^{\widetilde{B}} ( 0,T;W^{-1}L^{\widetilde{B}} ( \varOmega; \mathbb{R} ) ) \equiv L^{B} ( 0,T;W_{0}^{1}L^{B} ( \varOmega; \mathbb{R} ) )^{\prime}\), where Q=Ω×(0,T). For each given ε>0, we consider the initial-boundary value problem
$$ \begin{array} {@{}l} \displaystyle\frac{\partial u_{\varepsilon}}{\partial t}-\operatorname{div}a \biggl( \frac{x}{ \varepsilon},\frac{t}{\varepsilon},Du_{\varepsilon} \biggr) =f\quad \text{in}\ Q, \\[4mm] u_{\varepsilon}=0\quad\text{on}\ \partial\varOmega\times ( 0,T ), \\[2mm] u_{\varepsilon} ( x,0 ) =0\quad\text{in}\ \varOmega, \end{array} $$
(2)
where D denotes the usual gradient operator in Ω, i.e., \(D=(\frac{\partial}{\partial x_{i}}) _{1\leq i\leq d}\), LB(Q) and \(W_{0}^{1}L^{B} ( \varOmega;\mathbb{R})\) are Orlicz and Orlicz-Sobolev spaces (respectively) which will be specified later, and where a is a function from ℝd×ℝ×ℝd into ℝd satisfying the following properties:
$$ \begin{array} {@{}l} \mathrm{for\ each}\ \lambda\in \mathbb{R}^{d}\text{, the function}\ ( y,\tau ) \rightarrow a ( y,\tau , \lambda ),\ \mathrm{denoted\ by} \\[2mm] a ( \cdot,\cdot,\lambda )\text{, is measurable from}\ \mathbb{R}^{d} \times \mathbb{R}\ \text{into}\ \mathbb{R}^{d} \end{array} $$
(3)
$$ \begin{array} {@{}l} a(y,\tau,\omega)=\omega\ \text{almost everywhere (a.e.) in}\ ( y,\tau ) \in \mathbb{R}_{y}^{d}\times \mathbb{R}, \\[2mm] \mathrm{where}\ \omega\text{ is the origin in }\mathbb{R}^{d} \end{array} $$
(4)
$$ \begin{array}{@{}l} \mathrm{there\ exist\ three\ constants}\ c_{0},c_{1},c_{2}>0\ \text{such that, a.e. in}\ \mathbb{R}_{y}^{d}\times\mathbb{R}{:} \\[2mm] \qquad\phantom{\mbox{i}}\mbox{(i)}\quad\bigl\vert a( y,\tau,\lambda) -a (y,\tau,\mu)\bigr\vert\leq c_{0}\widetilde{B}^{-1}\bigl[ B\bigl(c_{1}\vert\lambda-\mu\vert\bigr)\bigr] \\[2mm] \qquad\mbox{(ii)}\quad\bigl[ a( y,\tau,\lambda) -a (y,\tau ,\mu) \bigr] \cdot( \lambda-\mu) \geq c_{2}B\bigl( \vert\lambda-\mu\vert\bigr) \\[2mm] \mathrm{for\ all}\ \lambda,\mu\in \mathbb{R}^{d},\ \mathrm{where\ the\ dot\ denotes\ the\ usual\ Euclidean} \\ \mathrm{inner\ product\ in}\ \mathbb{R}^{d} \end{array} $$
(5)
$$ \begin{array}{@{}l} \mathrm{for\ each}\ \lambda\in \mathbb{R}^{d},a( \cdot,\cdot,\lambda)\ \text{is periodic, that is (i.e.), a.e. in}\ \mathbb{R}_{y}^{d}\times \mathbb{R}{:} \\[2mm] a( y+k,\tau+l,\lambda) =a( y,\tau,\lambda)\ \text{for all}\ ( k,l) \in \mathbb{Z}^{d}\times \mathbb{Z}. \end{array} $$
(6)
Provided the differential operator \(u\rightarrow\operatorname{div}a( \frac{x}{ \varepsilon},\frac{t}{\varepsilon},Du)\) is rigorously defined, and an existence and uniqueness result for (2) is sketched, we are interested in this paper to the homogenization of (2), i.e., the limiting behaviour, as ε→0, of uε (the solution of (2)) under the periodic assumption (6).

Concerning the aspect of classical Sobolev spaces, the homogenization problem for (2) has been studied by many authors. We refer, e.g., to [4, 10, 11, 15, 28] in the periodic setting. Concerning results beyond the periodic setting, we refer to [22] in the deterministic setting, and [8, 19, 24, 27] for almost periodic and/or stochastic cases. All the works cited above have at least one common point: they have been studied when test functions are taking values in classical Sobolev spaces.

In the framework of periodic homogenization problems, the two-scale reflexive theorem for Sobolev spaces proved by Nguetseng [20], was recently extended to Orlicz Sobolev spaces in [13] as follows: any bounded sequence (uε)ε>0inW1LΦ(Ω) (ΦbeingaN-functionofclassΔ2) admits a subsequence such that, asε→0, one has
$$ \left\{ \begin{array} {@{}l} u_{\varepsilon}\rightarrow u_{0}\quad\mbox{in}\ W^{1}L^{\varPhi} ( \varOmega ) \mbox{-weak}, \\[2mm] \int_{\varOmega}\frac{\partial u_{\varepsilon}}{\partial x_{i}} ( x ) \phi \bigl( x, \frac{x}{\varepsilon} \bigr) dx\rightarrow \int_{\varOmega}\int _{Y} \bigl( \frac{\partial u_{0}}{\partial x_{i}} ( x ) +\frac{\partial u_{1}}{\partial y_{i}} ( x,y ) \bigr) \phi ( x,y ) dxdy\end{array} \right. $$
for all \(\phi\in L^{\widetilde{\varPhi}} ( \varOmega;\mathcal{C}_{per} ( Y ) ) \), where Y=(0,1)d is the unit cube in \(\mathbb{R}_{y}^{d}\), and where \(u_{1}\in L^{\varPhi} ( \varOmega;W_{\#}^{1}L^{\varPhi } ( Y ) ) \) with \(W_{\#}^{1}L^{\varPhi} ( Y ) = \{ v\in W^{1}L_{per}^{\varPhi} ( Y ) :\int_{Y}v ( y ) dy=0 \} \) and \(W^{1}L_{per}^{\varPhi} ( Y ) = \{ v\in W^{1}L_{\ell oc}^{\varPhi} ( \mathbb{R}_{y}^{N} ) :v\text{ is }Y\text{-periodic} \}\).
The quasi-reflexivity of this result on Q=Ω×(0,T) is fundamental in the proof of the main result of this work. Precisely, we show that: under (6), the sequence of solutions of (2), (uε)ε>0, satisfies, as ε→0,
$$ \left\{ \begin{array} {@{}l} u_{\varepsilon}\rightarrow u_{0}\quad\mbox{in}\ L^{B} \bigl( 0,T;W_{0}^{1}L^{B} ( \varOmega ) \bigr)\mbox{-weak}, \\[2mm] \frac{\partial u_{\varepsilon}}{\partial t}\rightarrow\frac{\partial u_{0}}{\partial t}\quad\text{in}\ L^{\widetilde{B}} \bigl( 0,T;W^{-1}L^{ \widetilde{B}} ( \varOmega ) \bigr)\text{-weak}, \\[2mm] \int_{Q}\frac{\partial u_{\varepsilon}}{\partial x_{i}} ( x,t ) \phi \bigl( x,t, \frac{x}{\varepsilon},\frac{t}{\varepsilon} \bigr) dxdt\rightarrow\int\int _{Q\times Z} \bigl[ \bigl( \frac{\partial u_{0}}{\partial x_{i}}+\frac{\partial u_{1}}{\partial y_{i}} \bigr) \phi \bigr] ( x,t,y,\tau ) dxdtdyd\tau \end{array} \right. $$
for all \(\phi\in L^{\widetilde{B}} ( Q;\mathcal {C}_{per} ( Z ) )\), 1≤iN, Z=Y×(0,1), where the couple (u0,u1) is the solution of the variational problem
$$ \left\{ \begin{array} {@{}l} \int_{0}^{T} \bigl( \frac{\partial u_{0}}{\partial t} ( t ) ,v_{0} ( t ) \bigr) dt+\int _{0}^{1}\int_{Q\times Y}a ( y,\tau ,Du_{0}+Du_{1} ) \cdot ( Dv_{0}+Dv_{1} ) dxdtdyd\tau \\[2mm] \quad{}=\int_{0}^{T} \bigl( f ( t ) ,v_{0} ( t ) \bigr) dt \\[2mm] \quad\text{for all}\ ( v_{0},v_{1} ) \in L^{B} \bigl( 0,T;W_{0}^{1}L^{B} ( \varOmega ) \bigr) \times L^{B} \bigl( Q\times ( 0,1 ) ;W_{\#}^{1}L^{B} ( Y; \mathbb{R} ) \bigr), \end{array} \right. $$
where \(D_{y}= ( \frac{\partial}{\partial y_{i}} )_{1\leq i\leq d}\); the space \(L^{B} ( Q\times ( 0,1 ) ;W_{\#}^{1}L^{B} ( Y;\mathbb{R} ) ) \) will be specified later.

The layout of the paper is as follows: Sect. 2 deals with some preliminary results on Orlicz-Sobolev spaces. In Sects. 3 and 4, we sketch out the existence and uniqueness, for each ε, of the solution of (2). Finally, in Sects. 5 and 6, after extending the main two-scale convergence result in our setting, we study the periodic homogenization of problem for (2).

2 Preliminaries on Orlicz-Sobolev Spaces

2.1 Orlicz Spaces

All definitions and results recalled here are classical and can be found in [1, 16, 17]. Let Φ:[0,∞)→[0,∞) be a N-function, that is, Φ is continuous, convex, Φ(t)>0 for t>0, \(\frac{\varPhi(t)}{t}\rightarrow0\) as t→0, and \(\frac {\varPhi (t)}{t}\rightarrow\infty\) as t→∞. Then Φ is represented by \(\varPhi( t) =\int_{0}^{t}\phi(\tau)d\tau\) where ϕ:[0,∞)→[0,∞) is nondecreasing, right continuous, with ϕ(0)=0, ϕ(t)>0 if t>0 and ϕ(t)→∞ if t→∞; and one has \(\frac {t\phi (t)}{\varPhi(t)}\geq1\) (resp. >1 if ϕ is strictly increasing) for all t>0. We denote by \(\widetilde{\varPhi}\), the Fenchel’s conjugate (or the complementary function) of a N-function Φ, that is, \(\widetilde {\varPhi }(t)=\sup_{s\geq0}[ st-\varPhi(s)]\) (t≥0), then \(\widetilde{\varPhi}\) also is a N-function and one has \(\widetilde{\varPhi} ( \phi(t)) \leq t\phi( t) \leq\varPhi (2t)\) for all t≥0. Let us recall a result which will be of great interest in the sequel.

A N-function Φ belongs to the class Δ2, denoted by ΦΔ2, if there exist α>0 and t0≥0 such that Φ(2t)≤αΦ(t) for all tt0. A N-function Φ belongs to the class Δ′, denoted by ΦΔ′, if there exists β>0 such that Φ(ts)≤βΦ(t)Φ(s) for all t,s≥0. A N-function Φ dominates a N-function Ψ (denoted by ΦΨ or ΨΦ) if there exist k>1 and t0>0 such that Ψ(t)≤Φ(kt) for all t>t0.

Example 1

Functions Φ,Ψ:[0,∞)→[0,∞) defined by Ψ(t)=tp (p>1) and Φ(t)=tpln(e+t), (p≥1) are N-functions satisfying: Φ,ΨΔ2Δ′, and ΨΦ.

Remark 1

It follows by (1) that N-functions \(B,\widetilde {B}\in \varDelta _{2}\). We have \(B ( ht ) \leq h^{\rho_{2}}B ( t ) \) and \(\widetilde{B} ( ht ) \leq h^{\frac{\rho_{1}}{\rho _{1}-1}}\widetilde{B} ( t ) \) for all h≥1 and all t≥0 [9, Lemma C.3].

Let U be a bounded open set in ℝd (integer d≥1), and let Φ be a N-function. The Orlicz-space LΦ(U) is defined to be the vector space of all measurable functions v:U→ℂ such that \(\int_{U}\varPhi( \frac{\vert v(x)\vert }{\delta}) dx<+\infty\) for some δ=δ(v)>0. Endowed with the Luxemburg norm [1, 14]:
$$ \Vert v\Vert_{\varPhi,U}=\inf \biggl\{ \delta>0:\int_{U} \varPhi \biggl( \frac{\vert v(x)\vert }{\delta} \biggr) dx\leq1 \biggr\}\quad\bigl(v\in L^{\varPhi} ( U )\bigr), $$
LΦ(U) is a Banach space. For further well-known properties of Orlicz spaces, we refer to [1, 16, 17] (see also [13, Lemma 2.3]). It is worth recalling the following Fubini’s type result.

Lemma 1

LetΦΔ2, let\(U_{1}\subset \mathbb{R}^{d_{1}}\)and\(U_{2}\subset \mathbb{R}^{d_{2}}\)be two bounded open sets (withd1+d2=d), and letuLΦ(U1×U2). Then for almost allx1U1, u(x1,⋅)∈LΦ(U2). If in addition\(\widetilde{\varPhi}\in \varDelta ^{\prime}\)associated with a constantβ, then the functionubelongs toLΦ(U1;LΦ(U2)) with
$$ \Vert u\Vert_{L^{\varPhi} ( U_{1};L^{\varPhi} ( U_{2} ) ) }\leq\int\int_{U_{1}\times U_{2}}\varPhi \bigl( \bigl\vert u ( x_{1},x_{2} ) \bigr\vert \bigr) dx_{1}dx_{2}+\beta. $$

Proof

cf. [26, Theorem 5]. □

Moreover, we will be interested with properties below:

Lemma 2

LetBbe theN-function of (1).

IfvLB(U) then [18]:
  1. (i)

    vB,U>1 implies\(\Vert v\Vert_{B,U}^{\rho_{1}}\leq\int_{U}B ( \vert v ( x ) \vert ) dx\leq\Vert v\Vert_{B,U}^{\rho _{2}}\),

     
  2. (ii)

    vB,U<1 implies\(\Vert v\Vert_{B,U}^{\rho_{2}}\leq\int_{U}B ( \vert v ( x ) \vert ) dx\leq\Vert v\Vert_{B,U}^{\rho _{1}}\),

     
and if\(v\in L^{\widetilde{B}}( U) \)then [9, Lemma C.7]:
  1. (iii)

    \(\Vert v\Vert_{\widetilde{B},U}<1\)implies\(\int_{U}\widetilde{B} ( \vert v ( x ) \vert ) dx\geq \Vert v\Vert_{\widetilde{B},U}^{\frac{\rho_{1}}{\rho_{1}-1}}\).

     

2.2 Orlicz-Sobolev Spaces

The notations being those of the above subsection, we put
$$ W^{1}L^{\varPhi} ( U ) = \biggl\{ v\in\mathcal{D}^{\prime} ( U ) :v,\frac{\partial v}{\partial x_{i}}\in L^{\varPhi} ( U ) ,1\leq i\leq d \biggr\}, $$
where derivatives are taken in the distributional sense on U. Endowed with the norm
$$ \Vert u\Vert_{W^{1}L^{\varPhi}}=\Vert u\Vert_{\varPhi ,U}+\sum_{i=1}^{d}\biggl\Vert \frac{\partial u}{\partial x_{i}} \biggr\Vert_{\varPhi,U}\quad \bigl(u\in W^{1}L^{\varPhi} ( U ) \bigr), $$
W1LΦ(U) is a Banach space, termed Orlicz-Sobolev space. Assume ΦΔ2 and the boundary of U is lipschitzian, then the canonical embedding W1LΦ(U)⊂LΦ(U) is compact [1, 2]. On the other hand, denoting by \(W_{0}^{1}L^{\varPhi} ( U ) \), the set of functions in W1LΦ(U) with zero boundary condition, \(W_{0}^{1}L^{\varPhi} ( U ) \) coincides with the closure of \(\mathcal{D}(U)\) in W1LΦ(U), and the seminorm \(u\rightarrow \Vert u\Vert_{W_{0}^{1}L^{\varPhi}}\equiv\Vert Du\Vert_{\varPhi,U}\) defined by
$$ \Vert Du\Vert_{\varPhi,U}=\sum_{i=1}^{d} \biggl\Vert \frac{\partial u}{\partial x_{i}}\biggr\Vert_{\varPhi,U}\quad \bigl(u\in W^{1}L^{\varPhi} ( U )\bigr), $$
is a norm on \(W_{0}^{1}L^{\varPhi}( U) \) equivalent to the norm \(\Vert\cdot\Vert_{W^{1}L^{\varPhi}}\) (see, e.g., [12, Lemma 2.8 and Remark 3.8]).

3 An Abstract Existence and Uniqueness Result

Let V be a real reflexive Banach space, and H be a real Hilbert space such that
$$ V\subset H\ \mbox{with continuous embedding},\ V\ \mbox{dense in}\ H, V\ \mbox{is separable}. $$
(7)
Let (⋅,⋅) and |⋅| be the scalar product and the associated norm in H. Identifying H to its dual yields VHV′, and it is convenient to denote also by (⋅,⋅) the duality pairing between V and V′; norms in V and in V′ will be denoted ∥⋅∥.
Let 0<T<∞ and let ΦΔ2 which dominates the function tt2. For each t∈[0,T], let A(t):VV′ be an operator satisfying the assumption (C) below:
(C)
if uV,vLΦ(0,T;V),(vn)⊂LΦ(0,T;V) and ω is the origin in V, then
  1. (i)

    the mapping tA(t)u sends continuously [0,T] into V′;

     
  2. (ii)

    the function tA(t)u(t) belongs to \(L^{ \widetilde{\varPhi}}(0,T;V^{\prime})\);

     
  3. (iii)

    (A(t)ω,w)=0 for all t∈[0,T] and all wV;

     
  4. (iv)

    there exists a constant c>0 such that for all u,wV (A(t)uA(t)w,uw)≥(∥uw∥) (t∈[0,T]);

     
  5. (v)

    if vnv in LΦ(0,T;V)-weak, then for all χLΦ(0,T;V) one has \(\lim_{n\rightarrow\infty}\int_{0}^{T}( A(t)u_{n}(t),\chi(t)) dt=\int_{0}^{T}( A(t)u(t),\chi (t)) dt\).

     

Here is the main result of this section, and throughout the rest, we will sketch its proof. We will follow [23].

Theorem 1

Assume that (C) holds. Let\(f\in L^{\widetilde{\varPhi}}(0,T;V^{\prime})\)andu0H. There exists one and only one function\(u\in\mathcal{C} ( [ 0,T ] ;H ) \cap L^{\varPhi} ( 0,T;V ) \)satisfying
$$ \begin{array} {@{}l} u^{\prime} ( t ) +A ( t ) u ( t ) =f ( t ), \quad 0<t\leq T, \\[1mm] u ( 0 ) =u_{0}. \end{array} $$
(8)

3.1 A Priori Estimate

The first equation in (8) implies the variational formulation
$$ \bigl( u^{\prime} ( t ) ,v \bigr) + \bigl( A ( t ) u ( t ) ,v \bigr) = \bigl( f ( t ) ,v \bigr),\quad 0<t\leq T, $$
(9)
for all vV. Letting v=u(t), 0<t<T, further taking into account (iii)–(iv) of (C), we are led to
$$ \frac{d}{dt}\bigl\vert u ( t ) \bigr\vert^{2}+2c\varPhi \bigl( \bigl\Vert u ( t ) \bigr\Vert \bigr) \leq2\bigl\Vert f ( t ) \bigr \Vert \bigl\Vert u ( t ) \bigr\Vert ,\quad 0<t\leq T. $$
Since \(st\leq\varPhi ( s ) +\widetilde{\varPhi} ( t ) \) for any s,t>0 [1], it follows of Φ that:
$$ 2\bigl\Vert f ( t ) \bigr\Vert \bigl\Vert u ( t ) \bigr\Vert \leq \left\{\begin{array}{@{}l} \widetilde{\varPhi} \bigl( \frac{2}{c} \Vert f ( t ) \Vert \bigr) +\varPhi ( c\Vert u ( t ) \Vert )\quad \text{if}\ 0<c<1, \\[2mm] \widetilde{\varPhi} ( 2c\Vert f ( t ) \Vert ) +\varPhi \bigl( \frac{1}{c}\bigl\Vert u ( t ) \Vert \bigr)\quad \text{if}\ c\geq1. \end{array} \right. $$
Remarking that \(2c-\frac{1}{c}\geq2c-1\geq c\) when c≥1, and using the convexity of Φ yields
$$ \frac{d}{dt}\bigl\vert u ( t ) \bigr\vert^{2}+c\varPhi \bigl( \bigl\Vert u ( t ) \bigr\Vert \bigr) \leq\widetilde{\varPhi } \bigl( \gamma \bigl\Vert f ( t ) \bigr\Vert \bigr),\quad 0<t\leq T, $$
with \(\gamma=\frac{2}{c}\) if 0<c<1 and γ=2c if c≥1. Next, integrating from 0 to t we are derived to the following a priori estimate
$$ \bigl\vert u ( t ) \bigr\vert^{2}+c\int_{0}^{T} \varPhi \bigl( \bigl\Vert u ( t ) \bigr\Vert \bigr) dt\leq\int _{0}^{T} \widetilde{\varPhi} \bigl( \gamma\bigl \Vert f ( t ) \bigr\Vert \bigr) dt+\vert u_{0} \vert^{2},\quad 0<t\leq T. $$
(10)

Remark 2

(i) The inequality (10) suggests to look for a solution of (8) as a distribution \(u\in\mathcal{D}^{\prime} ( ( 0,T ) ;V ) \) with uL(0,T;H)∩LΦ(0,T;V).

(ii) Let uL1(0,T;V) such that \(u^{\prime}\in L^{\widetilde {\varPhi}} ( 0,T;V^{\prime} ) \). Then, with a possible modification on a subset of measure zero of [0,T], u is a continuous function from [0,T] into V with
$$ \sup_{0\leq t\leq T}\bigl\Vert u(t)\bigr\Vert \leq\alpha\bigl\Vert u^{\prime}\bigr\Vert_{L^{\widetilde{\varPhi}} ( 0,T;V^{\prime } ) }+\bigl\Vert u(0)\bigr\Vert \quad( \alpha>0). $$

3.2 Approximate Problem

As in [5], according to (7) combined with a result in [7], there is a separable Hilbert space W, dense and continuously embedded in V. Denoting by [⋅,⋅] the scalar product on W, we consider the spectral problem
$$ [ w_{i},v ] =\lambda_{i} ( w_{i},v )\quad\text{for all}\ v\in W, $$
and we put Wm=span(w1,…,wm). Let us define um(t)∈ W as
$$ \begin{array} {@{}l} \displaystyle u_{m}(t)=\sum _{i=1}^{m}\psi_{mi}(t)w_{i}, \\[4mm] \bigl( u_{m}^{\prime}(t),w \bigr) + \bigl( A(t)u_{m}(t),w \bigr) = \bigl( f(t),w \bigr)\quad\text{for all}\ w\in W_{m}, \\[2mm] u_{m}(0)=u_{0m}, \end{array} $$
(11)
where
$$ u_{0m}\rightarrow u_{0}\quad\text{strongly in}\ H\ \text{as}\ m\rightarrow \infty. $$
(12)
Assume that ψmi is a continuous mapping. Then, it follows from part (i) of (C) that the mapping tA(t)um(t) belongs to \(\mathcal{C} ( [ 0,T ] ;V^{\prime} ) \). Since \(\mathcal{C}( [ 0,T ] )\otimes V^{\prime}\) is dense in \(\mathcal{C} ( [ 0,T ] ;V^{\prime} ) \), we will consider two cases.

First case: A(⋅)um(⋅)=(θψm)σ(w), where \(\theta\circ\psi_{m}\in \mathcal{C}( [ 0,T ] )\) and σ(w)∈V′.

Step 1. Derivation of ODE: Putting K=ψm([0,T]) which defines a compact subset in ℝm, θ can be viewed as an element of \(\mathcal{C}(K)\). By Stone-Weierstrass theorem (see [6, Chapter X, p. 37]), for all ε>0 there is a polynomial function in λ∈ℝm, P(λ)=a+bλ+⋅⋅⋅ (a∈ℝ and b∈ℝm), such that \(\vert P(\lambda)-\theta(\lambda)\vert < \frac{\varepsilon}{2}\) for all λK. Let Q(λ)=a+bλ. There is a connexe compact subset KK with ψm(0)∈K, such that \(\vert P(\lambda)-Q(\lambda )\vert <\frac{\varepsilon}{2}\) for all λK. Hence |θ(λ)−Q(λ)|<ε for all λK. Setting ψm([0,τ])=K yields
$$ \bigl\vert \theta\bigl(\psi_{m}(t)\bigr)-Q\bigl( \psi_{m}(t)\bigr)\bigr\vert <\varepsilon\quad \text{for all}\ t\in [ 0,\tau ] ; $$
which suggests to define the approximate problem in [0,τ] of (11)–(12) as follows: where \(u_{0m}=\sum_{i=1}^{m}\delta_{mi}w_{i}\). But the wi’s (1≤im) are eigenvectors, then the matrix (wi,wj)i,j=1,…,m admits an inverse; further (13) is equivalent to the following initial value problem
$$ \begin{array}{@{}l} \psi_{m}^{\prime}(t)+A_{m}\psi_{m}(t)=F_{m}(t),\quad 0<t\leq\tau, \\ \psi_{m}(0)=\delta_{m}, \end{array} $$
where Am is a m-square matrix.
Step 2. Existence of a solution for (11) when \(f\in \mathcal{C}([ 0,T] ;V^{\prime})\). It is well known that (see, e.g., [25]): there exists 0<τmτ such that the previous problem admits one and only one solution in [0,τm], ψm, given by
$$ \psi_{m}(t)=e^{t}\delta_{m}+\int_{0}^{t}e^{(s-t)A_{m}}F_{m}(s)ds,\quad 0\leq t\leq\tau_{m}. $$
(14)
Hence ψm belongs to \(\mathcal{C}^{1}([ 0,\tau _{m}]; \mathbb{R}^{m}) \). Combining (11) with (13)–(14) yields
$$ \begin{array} {@{}l} \bigl\vert \bigl( u_{m}^{\prime}(t),w \bigr) + \bigl( A(t)u_{m}(t),w \bigr) - \bigl( f(t),w \bigr) \bigr \vert \\[1.5mm] \quad{} \leq\bigl\vert \theta\bigl(\psi_{m}(t)\bigr)-Q\bigl( \psi_{m}(t)\bigr)\bigr\vert \bigl\vert \bigl( \sigma(w),w \bigr) \bigr\vert\\[1.5mm] \quad{}\leq\varepsilon\bigl\vert \bigl( \sigma(w),w \bigr) \bigr\vert \quad\text{for all}\ t\in [ 0,\tau_{m} ]\ \text{and all}\ w\in W_{m}. \end{array} $$
Using the arbitrariness of ε, the solution of (13) implies the existence of a local solution in [0,τm] for (11).

It is a routine exercise to see that a priori estimate (10) combined with the local solution for (11) in [τm,τm+δ], δ>0, τm+δT, permits to replace the local interval [0,τm] by [0,T]. Therefore (11) admits at least one global solution.

Remark 3

If A(⋅)um(⋅)=∑finite(θψm)σ(w), where \(\theta^{\ell}\circ\psi_{m}\in\mathcal{C}([ 0,T] )\), σ(w)∈V′, and w=(w1,…,wm)∈Wm, then there exist 0<τm<T, and a function um from [0,τm] into Wm satisfying (11).

Step 3: The general case \(f\in L^{\widetilde{\varPhi}} ( 0,T;V^{\prime} ) \). Let (fn)n∈ℕ be a sequence in \(\mathcal{C} ( [0,T];V^{\prime} ) \) which converges to f in \(L^{\widetilde{\varPhi}} ( 0,T;V^{\prime} ) \). For each n∈ℕ, there is a sequence (umn)m∈ℕLΦ(0,T;V) such that
$$ \begin{array} {@{}l} u_{mn}(t)\in W_{m},0<t<T, \\[1mm] \bigl( u_{mn}^{\prime}(t),w \bigr) + \bigl( A(t)u_{mn}(t),w \bigr) \\[1mm] \quad{}= \bigl( f_{n}(t),w \bigr),\quad 0<t<T,\ \text{for all}\ w\in W_{m}, \\[1mm] u_{mn}(0)=u_{0mn}, \end{array} $$
(15)
where u0mnu0m in H as n→∞. By a priori estimate (10), for p,q∈ℕ one has
$$ \begin{array}{@{}l} \bigl\vert u_{mp}(t)-u_{mq}(t) \bigr\vert^{2}+c\int_{0}^{T}\varPhi \bigl( \bigl\Vert u_{mp}(t)-u_{mq}(t)\bigr\Vert \bigr) dt\\[2mm] \quad{}\leq\vert u_{0mp}-u_{0mq}\vert^{2}+\int _{0}^{T}\widetilde{\varPhi} \bigl( \gamma \bigl\Vert f_{p} ( t ) -f_{q} ( t ) \bigr\Vert \bigr) dt. \end{array} $$
Hence (umn)n∈ℕ is a Cauchy sequence in L(0,T;H)∩LΦ(0,T;V). There exists umL(0,T;H)∩LΦ(0,T;V), such that umnum in L(0,T;H)∩LΦ(0,T;V) when n→∞. Further, multiplying (15) by φ(t), \(\varphi\in\mathcal{D}( 0,T)\), 0<t<T, integrating by parts, passing to the limit when n→∞, and using part (v) of the assumption (C) yields
$$ -\int_{0}^{T} \bigl( u_{m}(t),w \bigr) \varphi^{\prime }(t)dt+\int_{0}^{T} \bigl( A(t)u_{m}(t),w \bigr) \varphi (t)dt=\int_{0}^{T} \bigl( f(t),w \bigr) \varphi(t)dt. $$
Since φ is arbitrarily fixed, we derive (11). There is no difficulty to obtain (12).
General case of the mappingA(⋅)um(⋅). Let ε>0 be fixed. There is a mapping A(⋅)um(⋅) of the form
$$ A_{\circ} ( t ) u_{m} ( t ) \equiv A_{\circ } ( t ) \psi_{m} ( t ) \cdot w=\sum_{\mathrm{finite}} \theta^{\ell} \bigl( \psi_{m} ( t ) \bigr) \sigma^{\ell} ( w ) \quad ( 0\leq t\leq T ), $$
where \(\theta^{\ell}\circ\psi_{m}\in\mathcal{C} ( [ 0,T ] ) \), σ(w)∈V′ and w=(w1,…,wm)∈Wm, such that
$$ \sup_{0\leq t\leq T}\bigl\Vert A ( t ) u_{m} ( t ) -A_{\circ} ( t ) u_{m} ( t ) \bigr\Vert_{V^{\prime }}<\varepsilon, $$
\(u_{m} ( t ) =\sum_{i=1}^{m}\psi_{mi} ( t ) w_{i}\) being a solution of the problem (11)–(12) when A(t) is replaced by A(t). Then
$$ \begin{array}{@{}l} \bigl\vert \bigl( u_{m}^{\prime} ( t ) ,w \bigr) + \bigl( A ( t ) u_{m} ( t ) ,w \bigr) - \bigl( f ( t ) ,w \bigr) \bigr\vert \\[1.5mm] \quad{}\leq\bigl\vert \bigl( A ( t ) u_{m} ( t ) -A_{\circ} ( t ) u_{m} ( t ) ,w \bigr) \bigr\vert \leq \varepsilon \Vert w\Vert \quad\text{for all}\ w\in W_{m}, \end{array} $$
which, by the arbitrariness of ε, yields the existence of a solution of (11)–(12).

3.3 Existence and Uniqueness of a Solution for (8)

Let (um)m∈ℕ be the sequence defined in (11)–(12). According to a priori estimate (10) we have
$$ \sup_{m\in \mathbb{N}} \bigl( \Vert u_{m}\Vert_{L^{\infty} ( 0,T;H ) } \bigr) < \infty\quad\text{and}\quad\sup_{m\in \mathbb{N} } \bigl( \Vert u_{m} \Vert_{L^{\varPhi} ( 0,T;V ) } \bigr) <\infty. $$
There exists a subsequence \(( u_{k_{m}}) _{m\in \mathbb{N} }\) of (um)m∈ℕ such that, as m→∞
$$ u_{k_{m}}\rightarrow u\ \text{in}\ L^{\infty} ( 0,T;H )\text{-weak}\ \ast,\quad\text{and}\quad u_{k_{m}}\rightarrow u\ \text{in}\ L^{\varPhi } ( 0,T;V )\text{-weak}. $$
(16)
Taking into account (v) of the assumption (C); further, multiplying (11) by φ(t), \(\varphi\in\mathcal{D}( 0,T)\), integrating by parts from 0 to T, and passing to the limit when km→∞, we get
$$ -\int_{0}^{T} \bigl( u ( t ) ,w_{j} \bigr) \varphi^{\prime } ( t ) dt+\int_{0}^{T} \bigl( A ( t ) u ( t ) ,w_{j} \bigr) \varphi ( t ) dt=\int _{0}^{T} \bigl( f ( t ) ,w_{j} \bigr) \varphi(t)dt $$
for all j∈ℕ. Let wW and (wm)m∈ℕW such that wmWm and ∥wmwW→0 when m→∞. We may replace wj by wm in the above equality. Next we pass to the limit when m→∞ and obtain
$$ \frac{d}{dt} \bigl( u ( t ) ,w \bigr) + \bigl( A(t)u(t),w \bigr) = \bigl( f(t),w \bigr)\quad\text{for all}\ w\in W. $$
(17)
According to (ii) of the assumption (C) and the density, we may replace W by V in (17). Next the function tf(t)−A(t)u(t) from (0,T) into V′ belongs to \(L^{\widetilde{\varPhi}} ( 0,T;V^{\prime } )\). Hence \(u^{\prime}\in L^{\widetilde{\varPhi}} ( 0,T;V^{\prime } ) \) with the first equation in (8). It remains to show that u defined by (17) satisfies the initial condition in (8). But this is a routine exercise thanks to point (ii) of Remark 2 and the fact that \(u^{\prime}\in L^{\widetilde{\varPhi}} ( 0,T;V^{\prime} ) \).

For the uniqueness, assume u0=f=0. By (10), it follows that |u(t)|=0 a.e. in t∈[0,T]. Since u(0)=0 then u(t)=0 for all t∈[0,T].

4 Application: Existence and Uniqueness for (2)

Let \(\mathbf{v}= ( v_{i} ) \in\mathcal{C} ( \overline{Q}; \mathbb{R} )^{d}=\mathcal{C} ( \overline{Q};\mathbb{R} ) \times\cdots\times\mathcal{C} ( \overline{Q}; \mathbb{R} ) \) (d times). By (4)–(5), there is no difficulty in checking that the function (x,t,y,τ)→a(y,τ,v(x,t)) belongs to \(\mathcal {C} ( \overline{Q};L^{\infty} ( \mathbb{R}_{y,\tau}^{d+1} ) )^{d}\). Hence for each ε>0, the function \(x\rightarrow a ( \frac{x}{\varepsilon},\frac {t}{\varepsilon},\mathbf{v} ( x,t ) ) \) of Q into ℝd, denoted by aε(−,v), is well defined as an element of L(Q;ℝ)d (cf. [21, Lemma 2.1]); and we have the following proposition and corollary.

Proposition 1

Let hypotheses be (1), (3)(5). Givenε>0, the transformationvaε(⋅,v) of\(\mathcal{C}( \overline{Q};\mathbb{R} ) ^{d}\)intoL(Q;ℝ)dextends by continuity to a mapping, still denoted byvaε(−,v), ofLB(Q;ℝ)dinto\(L^{\widetilde{B}}( Q; \mathbb{R} ) ^{d}\)and for allv,wLB(Q;ℝ)d, we have
$$ \bigl\Vert a^{\varepsilon} ( -,\mathbf{v} ) -a^{\varepsilon } ( -,\mathbf{w} ) \bigr\Vert_{L^{\widetilde{B}} ( Q ) ^{d}}\leq c\Vert \mathbf{v}-\mathbf{w}\Vert_{L^{B} ( Q ) ^{d}}^{\rho}, $$
(18)
where\(\rho\in\{ \rho_{1},\rho_{2},\rho_{1}-1,\frac{\rho _{2}}{\rho_{1}}( \rho_{1}-1)\} \), andc=c(c0,c1,ρ)>0.

Proof

Putting
$$ u ( x,t,y,\tau ) =c_{0}\widetilde{B}^{-1} \bigl( B \bigl( c_{1}\bigl\vert \mathbf{v} ( x,t ) -\mathbf{w} ( x,t ) \bigr\vert \bigr) \bigr) -\bigl\vert a\bigl(y,\tau,\mathbf{v} ( x,t ) \bigr)-a\bigl(y,\tau, \mathbf{w} ( x,t ) \bigr)\bigr\vert , $$
\(\mathbf{v},\mathbf{w}\in\mathcal{C}( \overline{Q}; \mathbb{R} ) ^{d}\), and for each ε>0, it follows by applying Lemma 2.2 in [21] that
$$ \bigl\vert a^{\varepsilon}(-,\mathbf{v})-a^{\varepsilon}(-,\mathbf{w} )\bigr \vert \leq c_{0}\widetilde{B}^{-1} \bigl( B \bigl( c_{1}\vert \mathbf{v}-\mathbf{w}\vert \bigr) \bigr)\quad\text{in}\ Q. $$
Since \(\widetilde{B}\) is increasing, for δ>0 we have
$$ \int_{Q}\widetilde{B} \biggl( \frac{\vert a^{\varepsilon}(-,\mathbf {v})-a^{\varepsilon}(-,\mathbf{w})\vert }{\delta} \biggr) dxdt \leq \int_{Q}\widetilde{B} \biggl( \frac{c_{0}\widetilde{B}^{-1} ( B ( c_{1}\vert \mathbf{v}-\mathbf{w}\vert ) ) }{\delta}\biggr) dxdt. $$
If \(\Vert a^{\varepsilon}(-,\mathbf{v})-a^{\varepsilon}(-,\mathbf {w})\Vert_{\widetilde{B},Q}<1\), then the point (iii) of Lemma 2 combined with the above inequality when δ=1 and the fact that \(\widetilde{B}\in \varDelta ^{\prime}\) (with constant η) imply
$$ \begin{array}{@{}rcl} \bigl\Vert a^{\varepsilon}(-, \mathbf{v})-a^{\varepsilon}(-,\mathbf{w} )\bigr\Vert_{\widetilde{B},Q}^{\frac{\rho_{1}}{\rho_{1}-1}}& \leq& \displaystyle\int_{Q} \widetilde{B} \bigl( c_{0} \widetilde{B}^{-1} \bigl( B \bigl( c_{1}\vert \mathbf{v}- \mathbf{w}\vert \bigr) \bigr) \bigr) dxdt \\[4mm] &\leq&\displaystyle\eta\widetilde{B} ( c_{0} ) \int_{Q}B \bigl( c_{1}\vert \mathbf{v}-\mathbf{w}\vert \bigr) dxdt. \end{array} $$
Applying points (i)–(ii) of Lemma 2 yields (with \(\rho\in \{\rho_{1}-1,\frac{\rho_{2}}{\rho_{1}}( \rho_{1}-1)\} \))
$$ \bigl\Vert a^{\varepsilon}(-,\mathbf{v})-a^{\varepsilon}(-,\mathbf{w} )\bigr \Vert_{\widetilde{B},Q}\leq \bigl[ \eta\widetilde{B} ( c_{0} ) \bigr]^{\frac{\rho_{1}-1}{\rho_{1}}}c_{1}^{\rho}\Vert \mathbf{v}-\mathbf{w} \Vert_{B,Q}^{\rho}. $$
Let us assume now that \(\Vert a^{\varepsilon}(-,\mathbf{v} )-a^{\varepsilon}(-,\mathbf{w})\Vert_{\widetilde{B},Q}\geq1\). Since
$$ \biggl\{ \delta>0:\int_{Q}\widetilde{B} \biggl( \frac{c_{0}\widetilde{B}^{-1} ( B ( c_{1}\vert \mathbf{v}-\mathbf{w}\vert ) ) }{\delta} \biggr) dxdt\leq1 \biggr\} $$
is a subset of \(\{ \delta>0:\int_{Q}\widetilde{B} ( \frac{ \vert a^{\varepsilon}(-,\mathbf{v})-a^{\varepsilon}(-,\mathbf{w} )\vert }{\delta} ) dxdt\leq1 \} \), we get
$$ \bigl\Vert a^{\varepsilon}(-,\mathbf{v})-a^{\varepsilon}(-,\mathbf{w})\bigr\Vert_{\widetilde{B},Q}\leq c_{0}\bigl\Vert \widetilde {B}^{-1} \bigl( B \bigl( c_{1}\vert \mathbf{v}-\mathbf{w} \vert \bigr) \bigr) \bigr\Vert_{\widetilde{B},Q}. $$
If \(\Vert \widetilde{B}^{-1} ( B ( c_{1}\vert \mathbf {v}-\mathbf{w}\vert ) ) \Vert_{\widetilde{B},Q}<1\) then
$$ \bigl\Vert \widetilde{B}^{-1} \bigl( B \bigl( c_{1}\vert \mathbf{v}-\mathbf{w}\vert \bigr) \bigr) \bigr\Vert_{\widetilde{B},Q} \leq \biggl( \int_{Q}B \bigl( c_{1}\vert \mathbf{v}-\mathbf{w}\vert \bigr) dxdt \biggr)^{\frac{\rho_{1}-1}{\rho_{1}}}\leq c_{1}^{\rho}\Vert \mathbf{v}-\mathbf{w}\Vert_{B,Q}^{\rho}, $$
where \(\rho\in \{ \rho_{1}-1,\frac{\rho_{2}}{\rho_{1}} ( \rho_{1}-1 ) \} \). Suppose now \(\Vert \widetilde {B}^{-1} ( B ( c_{1}\vert \mathbf{v}-\mathbf{w}\vert ) ) \Vert_{\widetilde{B},Q}>1\). Hence there is an integer P0 such that \(\Vert \widetilde{B}^{-1} ( B ( c_{1}\vert \mathbf{v}-\mathbf{w}\vert ) ) \Vert_{\widetilde {B},Q}-\frac{1}{n}>1\) for all integers nP0 and
$$ 1<\int_{\varOmega}\widetilde{B} \biggl( \biggl( \bigl\Vert \widetilde{B}^{-1} \bigl( B \bigl( c_{1}\vert \mathbf{v}-\mathbf{w}\vert \bigr) \bigr) \bigr\Vert_{\widetilde{B},Q}- \frac{1}{n} \biggr)^{-1}\widetilde{B}^{-1} \bigl( B \bigl( c_{1}\vert \mathbf{v}-\mathbf{w}\vert \bigr) \bigr) \biggr) dxdt. $$
Using the convexity of \(\widetilde{B}\) yields
$$ \bigl\Vert \widetilde{B}^{-1} \bigl( B \bigl( c_{1}\vert \mathbf{v}- \mathbf{w}\vert \bigr) \bigr) \bigr\Vert_{\widetilde {B},Q}- \frac{1}{n}<\int_{Q} \bigl( B \bigl( c_{1} \vert \mathbf{v}-\mathbf{w}\vert \bigr) \bigr) dxdt\quad\text{for all}\ n\geq P_{0}. $$
Passing to the limit (as n→∞) and taking into account (i)–(ii) of Lemma 2, we are led to
$$ \bigl\Vert \widetilde{B}^{-1} \bigl( B \bigl( c_{1}\vert \mathbf{v}-\mathbf{w}\vert \bigr) \bigr) \bigr\Vert_{\widetilde{B},Q} \leq c_{1}^{\rho}\Vert \mathbf{v}-\mathbf{w} \Vert_{B,Q}^{\rho },\quad\text{with}\ \rho=\rho_{1}\text{ or }\rho_{2}. $$
Therefore (18) follows for all \(\mathbf{v},\mathbf{w}\in\mathcal {C}( \overline{Q};\mathbb{R} ) ^{d}\); then for all v,wLB(Q;ℝ)d by density and continuity. □

Corollary 1

Let hypotheses be (1), (3)(5). GivenvLB(0,T;W1LB(Ω;ℝ)), the function\(( x,t) \rightarrow a( \frac{x}{\varepsilon},\frac{t}{\varepsilon},Dv( x,t)) \)fromQintod, denoted byaε(−,Dv), is well defined as an element of\(L^{\widetilde{B}}( Q; \mathbb{R}) ^{d}\). Moreover we have
$$ \begin{array} {@{}l} a^{\varepsilon} ( -,\omega ) =\omega\quad \mathit{in}\ Q\ \bigl(\omega\ \mathit{being\ the\ origin\ in}\ \mathbb{R}^{d}\bigr), \\[1mm] \bigl[ a^{\varepsilon} ( -,Dv ) -a^{\varepsilon} ( -,Dw ) \bigr] \cdot ( Dv-Dw ) \geq c_{2}B \bigl( \vert Dv-Dw\vert \bigr)\ \mathit{in}\ Q \end{array} $$
(19)
for allv,wLB(0,T;W1LB(Ω;ℝ)).
Let \(\theta\in\mathcal{D} ( \mathbb{R} ) \) with 0≤θ≤1, ∫θ(t)dt=1, θ supports in the closed interval [−1,1]. For each integer n≥1, we put θn(t)=(nt), t∈ℝ, so that the sequence (θn)n≥1 is a mollifier on ℝ. For fixed xΩ and λ∈ℝd, let \(\overline{a}^{\varepsilon} ( x,\cdot,\lambda ) \) and \(\overline{f} ( x,\cdot ) \) be functions on ℝ defined by
$$ \overline{a}_{i}^{\varepsilon} ( x,t,\lambda ) =\left\{ \begin{array} {@{}l} a_{i}^{\varepsilon} ( x,t,\lambda )\quad \text{if}\ t\in ( 0,T ) \\ 0\quad\text{if}\ t\in \mathbb{R} \diagdown ( 0,T ),\end{array} \right.\quad\text{and}\quad\overline{f} ( x,t ) =\left\{ \begin{array} {@{}l} f ( x,t )\quad\text{if}\ t\in ( 0,T ) \\ 0\quad\text{if}\ t\in \mathbb{R} \diagdown ( 0,T ) .\end{array} \right. $$
For n≥1, let \(a_{in}^{\varepsilon} ( x,\cdot,\lambda ) \), 1≤id, and fn(x,⋅) be restrictions in (0,T) of functions
$$ t\rightarrow\int\theta_{n} ( \tau ) \overline{a}_{i}^{\varepsilon} ( x,t-\tau,\lambda ) d\tau\quad\text{and}\quad t\rightarrow\int \theta_{n} ( \tau ) \overline{f} ( x,t-\tau ) d\tau\ ( \text{respectively}). $$
Then \(f_{n} ( x,\cdot ) \in\mathcal{C} ( [ 0,T ] ) \) and \(a_{n}^{\varepsilon} ( x,\cdot,\lambda ) = ( a_{in}^{\varepsilon} ( x,\cdot,\lambda ) ) \in \mathcal{C}( [ 0,T ] )^{d}\). Next, it is obvious that
$$ \begin{array} {@{}l} a_{n}^{\varepsilon} ( -,\omega ) = \omega, \\[1mm] \bigl\vert a_{n}^{\varepsilon} ( -,\lambda ) -a_{n}^{\varepsilon } ( -,\mu ) \bigr\vert \leq c_{0}\widetilde{B}^{-1} \bigl( B \bigl( c_{1}\vert \lambda-\mu\vert \bigr) \bigr), \\[1mm] \bigl( a_{n}^{\varepsilon} ( -,\lambda ) -a_{n}^{\varepsilon } ( -,\mu ) \bigr) \cdot ( \lambda-\mu ) \geq c_{2}B \bigl( \vert \lambda-\mu\vert \bigr) \end{array} \quad\mbox{a.e. in}\ Q $$
(20)
for all λ,μ∈ℝN. Let us consider the problem
$$ \begin{array} {@{}l} \displaystyle\frac{\partial u_{\varepsilon n}}{\partial t}- \operatorname{div}a_{n}^{\varepsilon } ( -,Du_{\varepsilon n} ) =f_{n}\quad\text{in}\ Q, \\[4mm] u_{\varepsilon n} ( x,0 ) =0\quad\text{in}\ \varOmega, \\[2mm] u_{\varepsilon n} ( x,t ) =0\quad\text{in}\ \partial\varOmega\times ( 0,T ). \end{array} $$
(21)
Given uLB(0,T;W1LB(Ω;ℝ)), the function \(a_{n}^{\varepsilon}( -,Du) \) belongs to \(L^{\widetilde{B}}( Q; \mathbb{R} ) ^{d}\). Next, by (20) the transformation \(u\rightarrow a_{n}^{\varepsilon}( -,Du) \) from LB(0,T;W1LB(Ω;ℝ)) into \(L^{\widetilde{B}}( Q; \mathbb{R} ) ^{d}\) is continuous with
$$ \begin{array} {@{}l} \bigl\Vert a_{n}^{\varepsilon} ( -,Du ) -a_{n}^{\varepsilon } ( -,Dv ) \bigr\Vert_{L^{\widetilde{B}} ( Q; \mathbb{R} ) ^{d}}\leq c\Vert Du-Dv\Vert_{L^{B} ( Q; \mathbb{R} ) ^{d}}^{\rho}, \\[1mm] \bigl[ a_{n}^{\varepsilon} ( -,Du ) -a_{n}^{\varepsilon} ( -,Dv ) \bigr] \cdot ( Du-Dv ) \geq c_{2}B \bigl( \vert Du-Dv\vert \bigr)\quad\text{a.e. in}\ Q, \end{array} $$
(22)
for all u,vLB(0,T;W1LB(Ω;ℝ)). Let \(A_{n}^{\varepsilon}( t) :W_{0}^{1}L^{B}( \varOmega; \mathbb{R} ) \rightarrow W^{-1}L^{\widetilde{B}}( \varOmega;\mathbb{R} ) \) be the operator defined by \(A_{n}^{\varepsilon}( t) u=-\!\operatorname{div} a_{n}^{\varepsilon}( \cdot,t,Du)\). Then (21) is equivalent to the problem
$$ \begin{array} {@{}l} u_{\varepsilon n}^{\prime} ( t ) +A_{n}^{\varepsilon} ( t ) u_{\varepsilon n} ( t ) =f_{n} ( t ),\quad 0<t\leq T, \\[1mm] u_{\varepsilon n} ( 0 ) =u_{0}. \end{array} $$
(23)
For \(V=W_{0}^{1}L^{B}( \varOmega; \mathbb{R} ) \), H=L2(Ω;ℝ), and \(A( t) =A_{n}^{\varepsilon}( t)\) (0≤tT), the assumption (C) is satisfied. Thanks to Theorem 1 we deduce the following one:

Theorem 2

For eachn∈ℕand eachε>0, there exists one and only one functionuεnfrom [0,T] into\(W_{0}^{1}L^{B}( \varOmega;\mathbb{R} ) \)defined by (23) and satisfying\(u_{\varepsilon n}\in L^{B}( 0,T;W_{0}^{1}L^{B}( \varOmega; \mathbb{R})) \cap L^{\infty}( 0,T;L^{2}( \varOmega; \mathbb{R} )) \).

Let us pass to the limit in (23) when n→∞. Equation (23) takes the variational formulation
$$ \bigl( u_{\varepsilon n}^{\prime} ( t ) ,v \bigr) +\int_{\varOmega }a_{n}^{\varepsilon} \bigl( \cdot,t,Du_{\varepsilon n} ( t ) \bigr) \cdot Dvdx= \bigl( f_{n} ( t ) ,v \bigr) , $$
(24)
0<tT, for all \(v\in W_{0}^{1}L^{B}( \varOmega; \mathbb{R} )\). Letting v=uεn(t), 0<t<T, and arguing as in (10) yields
$$ \bigl\Vert u_{\varepsilon n} ( t ) \bigr\Vert_{L^{2} ( \varOmega ) }^{2}+c \int_{0}^{T}B \bigl( \bigl\Vert Du_{\varepsilon n} ( t ) \bigr\Vert_{L^{B} ( \varOmega ) ^{d}} \bigr) dt\leq \int_{0}^{T} \widetilde{B} \bigl( \gamma\bigl\Vert f_{n} ( t ) \bigr \Vert_{W^{-1}L^{\widetilde{B}} ( \varOmega; \mathbb{R} ) } \bigr) dt. $$
(25)
Recalling that there is an integer P≥1 such that \(\Vert f_{n}-f\Vert_{L^{\widetilde{B}}( 0,T;W^{-1}L^{\widetilde{B} }( \varOmega;\mathbb{R} )) }\leq1\) for all integer nP; we are derived to
$$ \sup_{n\in \mathbb{N}} \bigl( \Vert u_{\varepsilon n}\Vert_{L^{\infty} ( 0,T;L^{2} ( \varOmega ) ) } \bigr) < \infty\quad\text{and}\quad \sup_{n\in \mathbb{N}} \bigl( \Vert Du_{\varepsilon n} \Vert_{L^{B} ( 0,T;L^{B} ( \varOmega ) ) ^{d}} \bigr) <\infty. $$
There exists a subsequence \(( u_{\varepsilon k_{n}} )_{n\in \mathbb{N} ^{\ast}}\) of \(( u_{\varepsilon n} )_{n\in \mathbb{N} ^{\ast}}\) such that, as n→∞,
$$ \begin{array} {@{}l} \displaystyle u_{\varepsilon k_{n}}\rightarrow u_{\varepsilon}\quad\text{in}\ L^{\infty } \bigl( 0,T;L^{2} ( \varOmega ) \bigr)\text{-weak*}, \\[4mm] \displaystyle\frac{\partial u_{\varepsilon k_{n}}}{\partial x_{i}}\rightarrow \frac{\partial u_{\varepsilon}}{\partial x_{i}}\quad\text{in}\ L^{B} ( Q )\text{-weak}\ (1\leq i\leq d). \end{array} $$
(26)
This being so, there is no difficulty in proving that \(a_{n}^{\varepsilon }( -,Du_{\varepsilon}) \rightarrow a^{\varepsilon}( -,Du_{\varepsilon}) \) in \(L^{\widetilde{B}}( Q) ^{d}\) when n→∞. Besides, according to (22) and (25), \(a^{\varepsilon}( -,Du_{\varepsilon k_{n}}) \rightarrow a^{\varepsilon}( -,Du_{\varepsilon}) \) in \(L^{\widetilde{B} }( Q) ^{d}\)-weak when n→∞. Therefore, fixing ε>0, v in \(W_{0}^{1}L^{B} ( \varOmega;\mathbb{R} ) \) and φ in \(\mathcal{D}( 0,T) \) with \(\Vert v\varphi\Vert_{L^{B}( 0,T;W_{0}^{1}L^{B}( \varOmega )) }>0\), there exists P∈ℕ such that
$$ \begin{array} {@{}l} \displaystyle\bigl\Vert a_{k_{n}}^{\varepsilon} ( -,Du_{\varepsilon} ) -a^{\varepsilon} ( -,Du_{\varepsilon} ) \bigr \Vert_{L^{ \widetilde{B}} ( Q ) ^{d}}\leq\frac{\varepsilon}{4\Vert v\varphi\Vert_{L^{B} ( 0,T;W_{0}^{1}L^{B} ( \varOmega ) )}}\quad\mathrm{and} \\[4mm] \displaystyle\biggl\vert \int_{Q} \bigl( a^{\varepsilon} ( -,Du_{\varepsilon k_{n}} ) -a^{\varepsilon} ( -,Du_{\varepsilon} ) \bigr) \cdot Dv \varphi dxdt\biggr\vert \leq\frac{\varepsilon}{2} \end{array} $$
for all integer nP. Since
$$ \begin{array} {@{}l} \displaystyle\biggl\vert \int _{Q} \bigl( a_{k_{n}}^{\varepsilon} ( -,Du_{\varepsilon k_{n}} ) -a^{\varepsilon} ( -,Du_{\varepsilon} ) \bigr) \cdot Dv\varphi dxdt\biggr \vert \\[4mm] \displaystyle\quad{}\leq\int_{-1}^{1} \biggl( \theta_{k_{n}} ( \tau ) \biggl\vert \int_{Q} \bigl( a^{\varepsilon} ( \cdot,t-\tau,Du_{\varepsilon k_{n}} ) -a^{\varepsilon} ( \cdot,t- \tau,Du_{\varepsilon } ) \bigr) \cdot Dv\varphi dxdt\biggr\vert \biggr) d\tau \\[4mm] \displaystyle\qquad{}+\biggl\vert \int_{Q} \bigl( a_{k_{n}}^{\varepsilon} ( -,Du_{\varepsilon } ) -a^{\varepsilon} ( -,Du_{\varepsilon} ) \bigr) \cdot Dv\varphi dxdt\biggr\vert \\[4mm] \displaystyle\quad{}\leq\int_{-1}^{1}\biggl\vert \int_{Q} \bigl( a^{\varepsilon} ( \cdot ,t- \tau,Du_{\varepsilon k_{n}} ) -a^{\varepsilon} ( \cdot ,t-\tau ,Du_{\varepsilon} ) \bigr) \cdot Dv\varphi dxdt\biggr\vert d\tau \\[4mm] \displaystyle\qquad{}+2\bigl\Vert a_{k_{n}}^{\varepsilon} ( -,Du_{\varepsilon} ) -a^{\varepsilon} ( -,Du_{\varepsilon} ) \bigr \Vert_{L^{\widetilde{B}} ( Q ) ^{d}} \Vert v\varphi\Vert_{L^{B} ( 0,T;W_{0}^{1}L^{B} ( \varOmega ) )}, \end{array} $$
and \(a^{\varepsilon}( \cdot,t-\tau,Du_{\varepsilon}) =a^{\varepsilon}( \cdot,t-\tau,Du_{\varepsilon k_{n}}) =0\) if tτ∉(0,T), we are led to
$$ \biggl\vert \int_{Q} \bigl( a_{k_{n}}^{\varepsilon} ( -,Du_{\varepsilon k_{n}} ) -a^{\varepsilon} ( -,Du_{\varepsilon} ) \bigr) \cdot Dv \varphi dxdt\biggr\vert \leq\varepsilon\quad\text{for all integer}\ n\geq N. $$
Therefore, as n→∞, we have
$$ a_{k_{n}}^{\varepsilon} ( -,Du_{\varepsilon k_{n}} ) \rightarrow a^{\varepsilon} ( -,Du_{\varepsilon} )\quad\text{in}\ L^{\widetilde{B} } ( Q )^{d} \text{-weak}. $$
(27)
Multiplying (24) by φ(t), 0<t<T, \(\varphi\in \mathcal{D}( 0,T) \), and integrating from 0 to T (where n is replaced by kn), it follows
$$ -\int_{Q}u_{\varepsilon k_{n}}(t)v\varphi^{\prime} ( t ) dxdt+\int_{Q}a_{k_{n}}^{\varepsilon} \bigl( \cdot,t,Du_{\varepsilon k_{n}}(t) \bigr) \cdot Dv\varphi ( t ) dxdt=\int _{Q}f_{k_{n}}(t)v\varphi ( t ) dxdt. $$
Passing to the limit, as n→∞, and using (26)–(27) yields
$$ -\int_{Q}u_{\varepsilon}(t)v\varphi^{\prime} ( t ) dxdt+\int_{Q}a^{\varepsilon} \bigl( \cdot,t,Du_{\varepsilon}(t) \bigr) \cdot Dv\varphi ( t ) dxdt=\int_{Q}f(t)v\varphi ( t ) dxdt, $$
which drives us to
$$ u_{\varepsilon}^{\prime} ( t ) +A^{\varepsilon} ( t ) u_{\varepsilon} ( t ) =f ( t ),\quad 0<t\leq T, $$
(28)
where \(A^{\varepsilon}( t) u=-\operatorname{div}a^{\varepsilon}( \cdot ,t,Du) \). It is obvious to prove that
$$ u_{\varepsilon} ( 0 ) =0, $$
(29)
and (28)–(29) are equivalent to (2). Assuming that f=0, it follows by (10) that \(\Vert u_{\varepsilon}^{\prime }(t)\Vert_{L^{2} ( \varOmega ) }\leq0\) a.e. in (0,T), that is, uε=0. Clearly, we have the following

Theorem 3

For eachε>0, there exists a unique functionuεfrom [0,T] into\(W_{0}^{1}L^{B}( \varOmega;\mathbb{R} ) \)defined by (2) and satisfying\(u\in\mathcal{W}( 0,T;W_{0}^{1}L^{B}( \varOmega; \mathbb{R} )) \cap L^{\infty}( 0,T;L^{2}( \varOmega; \mathbb{R} )) \).

Remark 4

We denote by \(\mathcal{W} ( 0,T;W_{0}^{1}L^{B} ( \varOmega;\mathbb{R} ) ) \) the Banach space of functions \(v\in L^{B} ( 0,T;W_{0}^{1}L^{B} ( \varOmega; \mathbb{R} ) ) \) such that \(v^{\prime}\in L^{\widetilde{B}} ( 0,T;W^{-1}L^{\widetilde{B}} ( \varOmega; \mathbb{R} ) ) \) for the norm \(v\rightarrow\Vert v\Vert_{L^{B} ( 0,T;W_{0}^{1}L^{B} ( \varOmega; \mathbb{R} ) ) }+\Vert v^{\prime}\Vert_{L^{\widetilde{B} } ( 0,T;W^{-1}L^{\widetilde{B}} ( \varOmega; \mathbb{R} ) ) }\). By (1) the embedding \(\mathcal{W} ( 0,T; W_{0}^{1}L^{B} ( \varOmega; \mathbb{R} ) ) \subset\mathcal{C} ( [ 0,T ] ;L^{2} ( \varOmega; \mathbb{R} ) ) \) is continuous.

The periodic homogenization of problems (2) amounts to find a homogenized problem, by using the two-scale convergence method (see [3, 20]), such that the sequence of solutions uε converges to a limit u, which is precisely the solution of the homogenized problem. For this purpose, we need to recall the notion of two-scale convergence in LB(Q) (for details, see [13]).

5 Fundamentals of Periodic Homogenization

In all that follows, B is the N-function in (1), Y=(0,1)d, Θ=(0,1) and Z=Y×Θ. The letter ε throughout will denote a family of positive real numbers admitting 0 as an accumulation point. When ε=(εn)n∈ℕ with 0<εn≤1 and εn→0 as n→∞, we will refer to ε as a fundamental sequence. We put
$$ \begin{array} {@{}l} \mathcal{C}_{per} ( Z ) = \bigl\{ v\in \mathcal{C} \bigl( \mathbb{R}_{y,\tau}^{d+1} \bigr){:}\ v\text{ is }Z \text{-periodic} \bigr\}, \\[1mm] L_{per}^{B} ( Z ) = \bigl\{ v\in L_{\ell oc}^{B} \bigl( \mathbb{R}_{y,\tau}^{d+1} \bigr){:}\ v\text{ is }Z \text{-periodic} \bigr\} ,\end{array} $$
then \(L_{per}^{B}(B)\) is a Banach space under the Luxemburg norm ∥⋅∥B,Z, and \(\mathcal{C}_{per} ( Z ) \) is dense in \(L_{per}^{B} ( Z ) \). For \(v\in L_{per}^{B} ( Z ) \), letting
$$ v^{\varepsilon}(x,t)=v \biggl( \frac{x}{\varepsilon},\frac {t}{\varepsilon} \biggr)\quad \bigl( ( x,t ) \in \mathbb{R}^{d}\times \mathbb{R}\bigr), $$
we have vε→∫Zv(y,τ)dydτ in LB(Q)-weak as ε→0. Given \(v\in L_{\ell oc}^{B} ( Q\times \mathbb{R}_{y,\tau}^{d+1} ) \) and ε>0, we put
$$ v^{\varepsilon}(x,t)=v \biggl( x,t,\frac{x}{\varepsilon},\frac{t}{ \varepsilon} \biggr) \quad\bigl( ( x,t ) \in Q\bigr) $$
(30)
when \(v\in\mathcal{C} ( Q\times \mathbb{R}_{y,\tau}^{d+1} ) \). Since \(\mathcal{C} ( \overline {Q};\mathcal{B}( \mathbb{R}_{y,\tau}^{d+1} ) ) \) will legitimately be viewed as a subspace of \(\mathcal{C} ( Q\times \mathbb{R}_{y,\tau}^{d+1} ) \), as in [13] given \(v\in\mathcal{C}( \overline{Q};\mathcal{B} ( \mathbb{R}_{y,\tau}^{d+1} ) ) \), the trace \(( x,t ) \rightarrow v ( x,t,\frac{x}{\varepsilon},\frac{t}{\varepsilon} ) \), denoted by vε, is an element of \(\mathcal{B} ( Q ) \). Moreover the transformation vvε of \(\mathcal {C} ( \overline{Q};\mathcal{B} ( \mathbb{R}_{y,\tau}^{d+1} ) ) \) into \(\mathcal{B} ( Q ) \) extends by continuity to a unique linear mapping, still denoted vvε, of \(L^{B} ( Q;\mathcal{B} ( \mathbb{R}_{y,\tau}^{d+1} ) ) \) into LB(Q) with \(\Vert v^{\varepsilon}\Vert_{B,Q}\leq\Vert v\Vert_{L^{B} ( Q;\mathcal{B} ( \mathbb{R} _{y,\tau}^{d+1} ) ) }\) for all \(v\in L^{B} ( Q;\mathcal {B} ( \mathbb{R}_{y,\tau}^{d+1} ) ) \).

The right-hand side of (30) also makes sense when \(v\in\mathcal {C}( \overline{Q};L^{\infty}( \mathbb{R}_{y,\tau}^{d+1})) \). Moreover the mapping vvε sends linearly and continuously \(\mathcal{C}( \overline{Q};L^{\infty}( \mathbb{R}_{y,\tau}^{d+1} ) ) \) to L(Q) with \(\Vert v^{\varepsilon}\Vert_{L^{\infty} ( Q ) }\leq \sup_{ ( x,t ) \in\overline{Q}}\Vert v ( x,t ) \Vert_{L^{\infty} ( \mathbb{R} ^{d} ) }\) for all \(v\in\mathcal{C} ( \overline{Q};L^{\infty } ( \mathbb{R}_{y,\tau}^{d+1} ) ) \).

This being so, the following result will be useful in the sequel.

Proposition 2

For every\(v\in L^{B}( Q;\mathcal{C}_{per}( Z)) \)we have
$$ v^{\varepsilon}\rightarrow\int_{Z}v ( -,y,\tau ) dyd\tau \quad \mathit{in}\ L^{B}(Q)\mbox{\textit{-weak as}}\ \varepsilon \rightarrow0, $$
(31)
and
$$ \lim_{\varepsilon\rightarrow0}\bigl\Vert v^{\varepsilon}\bigr\Vert_{B,Q}=\Vert v\Vert_{B,Q\times Z}. $$
(32)

Proof

cf. [13]. □

Let us recall that, putting we have the identification \(L^{B} ( Q\times Z_{per} ) \equiv L^{B} ( Q;L_{per}^{B} ( Z ) ) \) (see Lemma 1), and the embedding \(L^{B}(Q;\mathcal{C}_{per}(Z))\hookrightarrow L^{B} ( Q;L_{per}^{B} ( Z ) ) \) is continuous with density.

Definition 1

A sequence (vε)εLB(Q) is said to:
  1. (i)
    weakly two-scale converge in LB(Q) to some \(v_{0}\in L^{B} ( Q;L_{per}^{B} ( Z ) ) \) if, as ε→0, we have
    $$ \int_{\varOmega}v_{\varepsilon}\varphi^{\varepsilon}dx\quad\rightarrow\quad \int\int_{Q\times Z}v_{0}\varphi dxdtdyd\tau $$
    (33)
    for every \(\varphi\in L^{\widetilde{B}} ( Q;\mathcal{C}_{per} ( Z ) ) \), where φε is defined as in (30);
     
  2. (ii)

    strongly two-scale converge in LB(Q) to some \(v_{0}\in L^{B} ( Q;L_{per}^{B} ( Z ) ) \) if the following property is verified:

     
(SSC)

Given η>0 and \(\varphi\in L^{\widetilde{B}} ( Q;\mathcal{C} _{per} ( Z ) ) \) with \(\Vert v_{0}-\varphi \Vert_{B,Q\times Z}\leq\frac{\eta}{2}\), there exists some α>0 such that ∥vεφεB,Qη provided 0<εα.

We express the case (i) by vεv0 in LB(Q)-weak 2s, and the case (ii) by vεv0 in LB(Q)-strong 2s.

Remark 5

Suppose a sequence (vε)εLB(Q) is weakly 2s-convergent in LB(Q) to some \(v_{0}\in L^{B} ( Q;L_{per}^{B} ( Z ) ) \), then vε→∫Zv0(−,y,τ)dydτ in LB(Q)-weak as ε→0, and (vε)ε is bounded in LB(Q) whenever ε is a fundamental sequence.

Since \(\widetilde{B}\in \varDelta _{2}\), we recall the following fundamental results (see [13]):

Theorem 4

Supposeεis a fundamental sequence, and (vε)εis a bounded sequence inLB(Q). Then, a subsequence can be extracted, still denotedε, such that the sequence (vε)εis weakly 2s-convergent inLB(Q).

Proposition 3

Letεbe a fundamental sequence, and suppose a sequence (vε)εLΦ(Ω) is weakly 2s-convergent inLB(Q) to some\(v_{0}\in L^{B} ( Q;L_{per}^{B} ( Z ) ) \) (Definition 1). Then (33) holds for\(\varphi\in\mathcal{C}(\overline {Q};L_{per}^{\infty} ( Z ) )\).

We define the periodic Orlicz-Sobolev space
$$ W^{1}L_{per}^{B} ( Y ) = \biggl\{ u\in W^{1}L_{\ell oc}^{B} \bigl( \mathbb{R}_{y}^{d} \bigr) :u,\frac{\partial u}{\partial y_{i}}(1\leq i\leq d)\ \text{are}\ Y\text{-periodic} \biggr \} $$
(the derivative \(\frac{\partial u}{\partial y_{i}}\) is taken in the distributional sense on \(\mathbb{R} _{y}^{d}\)) for the norm
$$ \Vert u\Vert_{W^{1}L_{per}^{B}}=\Vert u\Vert_{B,Y}+\sum _{i=1}^{d}\biggl\Vert \frac{\partial u}{\partial y_{i}} \biggr \Vert_{B,Y}\quad\bigl(u\in W^{1}L_{per}^{B} ( Y ) \bigr); $$
and its useful subspace
$$ W_{\#}^{1}L_{per}^{B} ( Y ) = \biggl\{ u\in W^{1}L_{per}^{B} ( Y ) :\int_{Y}u(y)dy=0 \biggr\} . $$
Provided with the gradient norm
$$ \Vert u\Vert_{W_{\#}^{1}L_{per}^{B}}= \sum_{i=1}^{d} \biggl\Vert \frac{\partial u}{\partial y_{i}}\biggr\Vert_{\varPhi,Y}\quad\bigl(u\in W_{\#}^{1}L_{per}^{B} ( Y )\bigr) $$
\(W_{\#}^{1}L_{per}^{B} ( Y ) \) is a Banach space. Besides, putting \(\mathcal{C}_{per}^{\infty} ( Y ) =\mathcal{C}_{per} ( Y ) \cap\mathcal{C}^{\infty} ( \mathbb{R}^{d} ) \), the space \(\mathcal{C}_{per}^{\infty} ( Y;\mathbb{R} ) /\mathbb{C} = \{ u\in\mathcal{C}_{per}^{\infty} ( Y;\mathbb{R} ) :\int_{Y}u(y)dy=0 \} \) is dense in \(W_{\# }^{1}L_{per}^{B} ( Y ) \).

Theorem 5

Assume thatεis a fundamental sequence, and that\(\left( u_{\varepsilon}\right) _{\varepsilon}\)is a bounded sequence in\(L^{B}\left( 0,T;W^{1}L^{B}\left( \varOmega\right) \right) \). Then, a subsequence can be extracted, still denotedε, such that asε→0,
$$ \begin{array} {@{}l} u_{\varepsilon}\rightarrow u_{0}\quad \mathit{in}\ L^{B} \bigl( 0,T;W^{1}L^{B} ( \varOmega ) \bigr)\mbox{\textit{-weak}}, \\[2mm] u_{\varepsilon}\rightarrow u_{0}\quad\mbox{\textit{in}}\ L^{B} ( Q ) \mbox{\textit{-weak}}\ 2s, \\[4mm] \displaystyle\frac{\partial u_{\varepsilon}}{\partial x_{i}}\rightarrow\frac {\partial u_{0}}{\partial x_{i}}+\frac{\partial u_{1}}{\partial y_{i}} \quad\mathit{in}\ L^{B} ( Q )\mbox{\textit{-weak}}\ 2s\ (1\leq i\leq d), \end{array} $$
(34)
whereu0LB(0,T;W1LB(Ω)) and\(u_{1}\in L^{B}(Q\times\varTheta;W_{\#}^{1}L_{per}^{B} ( Y ) )\). Furthermore if\(( u_{\varepsilon} )_{\varepsilon}\subset L^{B} ( 0,T;W_{0}^{1}L^{B} ( \varOmega ) ) \)then the weak limitu0lies in\(L^{B} ( 0,T;W_{0}^{1}L^{B} ( \varOmega ) ) \),

Proof

It is an easy adaptation of the proof in [13, Theorem 4.2]. □

6 Homogenization Result for (2)

Our main purpose here is to investigate the limiting behaviour, as ε→0, of the sequence (uε)ε>0 defined by Theorem 3 under the periodic hypothesis (6). Let us begin by some preliminaries.

6.1 Preliminaries

Given \(\mathbf{v}= ( v_{i} ) \in\mathcal{C}_{per} ( Z; \mathbb{R} )^{d}\), it is a routine exercise, thanks to (3)–(6), that the function (y,τ)→a(y,τ,v(y,τ)), denoted by a(−,v), belongs to \(L_{per}^{\infty} ( Z; \mathbb{R} )^{d}\). Next, for every \(\mathbf{v}\in\mathcal{C} ( \overline{Q};\mathcal{C}_{per} ( Z;\mathbb{R} ) )^{d}\) the function (x,t)→a(−,v(x,t,−)) from \(\overline{Q}\) to \(L_{per}^{\infty} ( Z; \mathbb{R} )^{d}\) (still denoted by a(−,v)) lies in \(\mathcal{C}(\overline{Q};L_{per}^{\infty} ( Z; \mathbb{R} ) )^{d}\) with
$$ \begin{array} {@{}l} \bigl\vert a ( -,\mathbf{v} ) -a ( -,\mathbf{w} ) \bigr\vert \leq c_{0}\widetilde{B}^{-1} \bigl( B \bigl( c_{1}\vert \mathbf{v}-\mathbf{w}\vert \bigr) \bigr), \\[1.5mm] \bigl( a ( -,\mathbf{v} ) -a ( -,\mathbf{w} ) \bigr) \cdot ( \mathbf{v}- \mathbf{w} ) \geq c_{2}B \bigl( \vert \mathbf{v}-\mathbf{w}\vert \bigr)\quad\text{in}\ \overline{Q}\times Z \end{array} $$
for all \(\mathbf{v},\mathbf{w}\in\mathcal{C}( \overline {Q};\mathcal{C}_{per}( Z;\mathbb{R} ) ) ^{d}\). Therefore, for fixed ε>0, one defines the function \(( x,t) \rightarrow a( \frac {x}{\varepsilon}, \frac{t}{\varepsilon},\mathbf{v}( x,t,\frac{x}{\varepsilon},\frac {t}{\varepsilon})) \) of Q into ℝd (denoted aε(−,vε)) as an element of L(Q;ℝ)d. Finally, the following results are a routine exercise.

Proposition 4

Let hypotheses be (1), (3)(6). For all\(\mathbf{v}\in\mathcal{C}( \overline{Q};\mathcal{C}_{per}( Z; \mathbb{R} )) ^{d}\)one has
$$ a^{\varepsilon} \bigl( -,\mathbf{v}^{\varepsilon} \bigr) \rightarrow a ( -, \mathbf{v} )\quad\mathit{in}\ L^{\widetilde{B}} ( Q )^{d}\mbox{\textit{-weak}}\ 2s\ \mathit{when}\ \varepsilon\rightarrow0. $$
(35)
Furthermore, the mappingva(−,v) of\(\mathcal{C} ( \overline{Q};\mathcal{C}_{per} ( Z; \mathbb{R} ) )^{d}\)into\(L^{\widetilde{B}} ( Q;L_{per}^{\widetilde{B}}(Z; \mathbb{R} ) )^{d}\)extends by continuity to a mapping, still denoted byva(−,v), of\(L^{B} ( Q;L_{per}^{B} ( Z; \mathbb{R} ) )^{d}\)into\(L^{\widetilde{B}} ( Q;L_{per}^{\widetilde{B}}(Z; \mathbb{R} ) )^{d}\)satisfying
$$ \bigl\Vert a ( -,\mathbf{v} ) -a ( -,\mathbf{w} ) \bigr\Vert_{L^{\widetilde{B}} ( Q\times Z ) ^{d}} \leq c\Vert \mathbf{v}-\mathbf{w}\Vert_{L^{B}(Q\times Z)^{d}}^{\rho} $$
(36)
and
$$ \bigl( a ( -,\mathbf{v} ) -a ( -,\mathbf{w} ) \bigr) \cdot ( \mathbf{v}- \mathbf{w} ) \geq c_{2}B \bigl( \vert \mathbf{v}-\mathbf{w}\vert \bigr)\quad\mathit{in}\ \overline{Q}\times Z $$
(37)

for allv,\(\mathbf{w}\in L^{B} ( Q;L_{per}^{B} ( Z; \mathbb{R} ) )^{d}\).

Proof

By [13, Proposition 4.4], for all \(\mathbf{\varphi}\in L^{B} ( Q;\mathcal{C}_{per} ( Z; \mathbb{R} ) )^{d}\), \(\mathbf{\varphi}^{\varepsilon}\rightarrow \mathbf{\varphi}\) in LB(Q)d-weak 2s when ε→0. Since a(−,v) belongs to \(\mathcal{C}(\overline{Q};L_{per}^{\infty} ( Z; \mathbb{R} ) )^{d}\) for all \(\mathbf{v}\in\mathcal{C} ( \overline{Q}; \mathcal{C}_{per} ( Z; \mathbb{R} ) )^{d}\), by Proposition 3 we get, as ε→0,
$$ \int_{Q}a^{\varepsilon} \bigl( -,\mathbf{v}^{\varepsilon} \bigr) \boldsymbol{\varphi}^{\varepsilon}dxdt\rightarrow\int\int_{Q\times Z}a ( -, \mathbf{v} ) \boldsymbol{\varphi}dxdtdyd\tau $$
for all \(\mathbf{\varphi}\in L^{B} ( Q;\mathcal{C}_{per} ( Z;\mathbb{R} ) )^{d}\); and (35) follows.

On the other hand, taking into account (5) and arguing as in the proof of Proposition 1, there exists a constant c such that (36) and (37) hold true for all v,\(\mathbf{w}\in \mathcal{C} ( \overline{Q};\mathcal{C}_{per} ( Z; \mathbb{R} ) )^{d}\). We end the proof by continuity and density arguments. □

Corollary 2

Let
$$ \phi_{\varepsilon} ( x,t ) =\psi_{0} ( x,t ) +\varepsilon \psi_{1}^{\varepsilon} ( x,t )\quad \bigl(\varepsilon >0,\ ( x,t ) \in Q \bigr), $$
where\(\psi_{0}\in\mathcal{D} ( Q; \mathbb{R} ) \)and\(\psi_{1}\in\mathcal{D} ( Q; \mathbb{R} ) \otimes\mathcal{C}_{per}^{\infty}(Z; \mathbb{R} )\). Then, asε→0,
$$ a^{\varepsilon} ( -,D\phi_{\varepsilon} ) \rightarrow a ( -,D \psi_{0}+D_{y}\psi_{1} )\quad\mathit{in}\ L^{\widetilde{B}} ( Q )^{d}\mbox{\textit{-weak}}\ 2s. $$
Furthermore, given a sequence (vε)εLB(Q)dsuch thatvεv0inLB(Q)d-weak 2sasε→0, one has
$$ \lim_{E\ni\varepsilon\rightarrow0}\int_{Q}a^{\varepsilon} ( -,D \phi_{\varepsilon} ) \cdot v_{\varepsilon}dxdt=\int\int_{Q\times Z}a ( -,D\varphi_{0}+D_{y}\varphi_{1} ) \cdot v_{0}dxdtdyd\tau. $$

Proof

Let \(\mathbf{\varphi}\in L^{B}\left( Q;\mathcal{C}_{per}\left( Z; \mathbb{R} \right) \right) ^{d}\) and put
$$ I ( \varepsilon ) =\int_{Q}a^{\varepsilon} ( -,D \phi_{\varepsilon} ) \boldsymbol{\varphi}^{\varepsilon}dxdt-\int \int _{Q\times Z}a ( -,D\psi_{0}+D_{y} \psi_{1} ) \boldsymbol{\varphi}dxdtdyd\tau,\varepsilon>0. $$
We have I(ε)=I1(ε)+I2(ε), where
$$ \begin{array} {@{}l} \displaystyle I_{1} ( \varepsilon ) =\int _{Q} \bigl[ a^{\varepsilon} ( -,D\phi_{\varepsilon} ) -a^{\varepsilon} \bigl( -,D\psi_{0}+ ( D_{y} \psi_{1} )^{\varepsilon} \bigr) \bigr] \boldsymbol{\varphi}^{\varepsilon}dxdt \quad\mathrm{and} \\[4mm] \displaystyle I_{2} ( \varepsilon ) =\int_{Q}a^{\varepsilon} \bigl( -,D\psi_{0}+ ( D_{y}\psi_{1} )^{\varepsilon} \bigr) \boldsymbol{\varphi}^{\varepsilon}dxdt \\[4mm] \displaystyle\phantom{I_{2} ( \varepsilon ) =} {} -\int\int _{Q\times Z}a ( -,D\psi_{0}+D_{y} \psi_{1} ) \boldsymbol{\varphi}dxdtdyd\tau. \end{array} $$
By Proposition 3, we get aε(−,0+(Dyψ1)ε)→a(−,0+Dyψ1) in \(L^{\widetilde{B}} ( Q )^{d}\)-weak 2s when ε→0; that is, limε→0I2(ε)=0. For I1(ε), since ε=0+(Dyψ1)ε+ε(1)ε, applying the Hölder’s inequality and taking into account Proposition 1 yields
$$ \bigl\vert I_{1} ( \varepsilon ) \bigr\vert \leq 2c\varepsilon \Vert \varphi\Vert_{L^{B} ( Q;\mathcal{C}_{per} ( Z; \mathbb{R} ) ) ^{d}}\Vert D\psi_{1} \Vert_{L^{B} ( Q;\mathcal{C}_{per} ( Z; \mathbb{R} ) ) ^{d}}^{\sigma}\rightarrow0\quad\text{as}\ \varepsilon \rightarrow0. $$
Therefore limε→0I(ε)=0 and, due to the arbitrariness of \(\mathbf{\varphi}\), the first limit follows.
For the second limit, we repeat a similar decomposition where φε is replaced by vε. Precisely, on letting
$$ I^{\prime} ( \varepsilon ) =\int_{Q}a^{\varepsilon} ( -,D\phi_{\varepsilon} ) v_{\varepsilon}dxdt-\int\int_{Q\times Z}a ( -,D\psi_{0}+D_{y}\psi_{1} ) v_{0}dxdtdyd \tau, $$
one has \(I^{\prime} ( \varepsilon ) =I_{1}^{\prime} ( \varepsilon ) +I_{2}^{\prime} ( \varepsilon ) \). Since a(−,0+Dyψ1) lies in \(\mathcal {C}(\overline{Q};L_{per}^{\infty} ( Z; \mathbb{R} ) )^{d}\), by Proposition 3 one gets \(\lim_{\varepsilon \rightarrow0}I_{2}^{\prime} ( \varepsilon ) =0\). By Proposition 1, we have
$$ \bigl\Vert a^{\varepsilon} ( -,D\phi_{\varepsilon} ) -a^{\varepsilon} \bigl( -,D\psi_{0}+ ( D_{y}\psi_{1} )^{\varepsilon} \bigr) \bigr\Vert_{\widetilde{B},Q}\leq c\varepsilon \Vert D\varphi_{1} \Vert_{L^{B} ( Q;\mathcal {C}_{per} ( Z;\mathbb{R} ) ) ^{d}}^{\rho}. $$
Hence, \(\lim_{\varepsilon\rightarrow0}I_{1}^{\prime} ( \varepsilon ) =0\), and the proof is completed. □
Let
$$ \mathbb{F}_{0}^{1}L^{B}=\mathcal{W}_{0} \bigl( 0,T;W_{0}^{1}L^{B} ( \varOmega; \mathbb{R} ) \bigr) \times L^{B} \bigl( Q\times\varTheta ;W_{\#}^{1}L_{per}^{B} ( Y; \mathbb{R} ) \bigr), $$
where \(\mathcal{W}_{0} ( 0,T;W_{0}^{1}L^{B} ( \varOmega; \mathbb{R} ) ) = \{ v\in\mathcal{W} ( 0,T;W_{0}^{1}L^{B} ( \varOmega; \mathbb{R} ) ) :v ( 0 ) =0 \} \) (see Remark 4), then \(\mathbb{F}_{0}^{1}L^{B}\) is a Banach space under the norm
$$ \Vert \mathbf{u}\Vert_{\mathbb{F}_{0}^{1}L^{B}}=\Vert Du_{0} \Vert_{B,Q}+\Vert D_{y}u_{1}\Vert_{B,Q\times Z} \quad \bigl(\mathbf{u}=(u_{0},u_{1})\in\mathbb{F}_{0}^{1}L^{B} \bigr). $$
Besides, thanks to the density of \(\mathcal{D} ( Q; \mathbb{R} ) \) in \(\mathcal{W} ( 0,T;W_{0}^{1}L^{B} ( \varOmega; \mathbb{R} ) ) \) and that of \(\mathcal{C}_{per}^{\infty} ( Y; \mathbb{R} ) /\mathbb{C}\) in \(W_{\#}^{1}L_{per}^{B} ( Y ) \), the space
$$ F_{0}^{\infty}=\mathcal{D} ( Q; \mathbb{R} ) \times \bigl[ \mathcal{D} ( Q; \mathbb{R} ) \otimes\mathcal{C}_{per}^{\infty} ( \varTheta; \mathbb{R} ) \otimes\mathcal{C}_{per}^{\infty} ( Y; \mathbb{R} ) / \mathbb{C} \bigr] $$
is dense in \(\mathbb{F}_{0}^{1}L^{B}\). Finally, for \(\mathbf{v}=( v_{0},v_{1}) \in\mathbb{F}_{0}^{1}L^{B}\) denote \(\mathbb {D}\mathbf{v}=Dv_{0}+D_{y}v_{1}\), hypotheses (1), (3)–(6) drive to

Lemma 3

There is at most a function\(\mathbf{u}= ( u_{0},u_{1} ) \in\mathbb{F}_{0}^{1}L^{B}\)such that
$$ \int_{0}^{T} \bigl( u_{0}^{\prime} ( t ) ,v_{0} ( t ) \bigr) dt+\int\int_{Q\times Z}a ( -, \mathbb{D}\mathbf{u} ) \cdot \mathbb{D}\mathbf{v}dxdtdyd\tau=\int _{0}^{T} \bigl( f ( t ) ,v_{0} ( t ) \bigr) dt $$
for all\(\mathbf{v}= ( v_{0},v_{1} ) \in\mathbb{F}_{0}^{1}L^{B}\).

6.2 Main Result

Theorem 6

Under hypotheses (1), (3)(6), and for eachε>0, letuεbe the unique solution of (2) (see Theorem 3). Then, asε→0,
$$ u_{\varepsilon}\rightarrow u_{0}\quad\mathit{in}\ L^{B} \bigl( 0,T;W_{0}^{1}L^{B} ( \varOmega; \mathbb{R} ) \bigr)\mbox{\textit{-weak}}, $$
(38)
$$ \frac{\partial u_{\varepsilon}}{\partial t}\rightarrow\frac{\partial u_{0}}{\partial t}\quad\mathit{in}\ L^{\widetilde{B}} \bigl( 0,T;W^{-1}L^{\widetilde {B}} ( \varOmega;\mathbb{R} ) \bigr)\mbox{\textit{-weak}}, $$
(39)
$$ Du_{\varepsilon}\rightarrow\mathbb{D}\mathbf{u}=Du_{0}+D_{y}u_{1} \quad \mathit{in}\ L^{B} ( Q )^{d}\mbox{\textit{-weak}}\ 2s $$
(40)
where\(\mathbf{u}= ( u_{0},u_{1} ) \in\mathbb {F}_{0}^{1}L^{B}\)is the unique solution of the variational problem in Lemma 3.

Proof

For 0<ε≤1 arbitrarily fixed, and by a mere routine, (2) implies the variational formulation
$$ \bigl( u_{\varepsilon}^{\prime} ( t ) ,v \bigr) +\int_{\varOmega }a^{\varepsilon} ( -,Du_{\varepsilon} ) \cdot Dvdx=\int_{\varOmega }f ( t ) vdx $$
(41)
for all \(v\in W_{0}^{1}L^{B}( \varOmega; \mathbb{R} ) \). Taking v=uε(t) (0<t<T) and using properties (3)–(5), it follows as in a priori estimate (10): where \(\gamma=\frac{2}{c_{2}}\) if 0<c2<1 and γ=2c2 if c2≥1. Hence, by Lemma 2, one has
$$ \bigl\Vert u_{\varepsilon}(t)\bigr\Vert_{L^{2}(\varOmega )}^{2}+c_{2} \Vert u_{\varepsilon}\Vert_{L^{B} ( 0,T;W_{0}^{1}L^{B} ( \varOmega; \mathbb{R} ) ) }^{\rho}\leq\int _{0}^{T}\widetilde{B} \bigl( \gamma \bigl\Vert f ( t ) \bigr\Vert_{W^{-1}L^{\widetilde{B}} ( \varOmega;\mathbb{R} ) } \bigr) dt $$
(0<tT) with ρ=ρ1 or ρ2. Thus
$$ \sup_{0<\varepsilon\leq1} \bigl( \Vert u_{\varepsilon}\Vert_{L^{B} ( 0,T;W_{0}^{1}L^{B} ( \varOmega; \mathbb{R} ) ) } \bigr) < \infty, $$
which, combined with (4) and (18), implies
$$ \sup_{0<\varepsilon\leq1} \bigl( \bigl\Vert a^{\varepsilon} ( -,Du_{\varepsilon} ) \bigr\Vert_{L^{\widetilde{B}} ( Q ) ^{d}} \bigr) <\infty. $$
This being so, we rapidly deduce that
$$ \sup_{0<\varepsilon\leq1} \biggl( \biggl\Vert \frac{\partial u_{\varepsilon} }{\partial t}\biggr \Vert_{L^{\widetilde{B}} ( 0,T;W^{-1}L^{\widetilde{B} } ( \varOmega; \mathbb{R} ) ) },\bigl\Vert \operatorname{div}a^{\varepsilon} ( -,Du_{\varepsilon} ) \bigr\Vert_{L^{\widetilde{B}} ( 0,T;W^{-1}L^{\widetilde{B}} ( \varOmega; \mathbb{R} ) ) } \biggr) <\infty. $$
Therefore, the sequence (uε)0<ε≤1 is bounded in \(\mathcal{W}_{0} ( 0,T;W_{0}^{1}L^{B} ( \varOmega;\mathbb{R} ) ) \). Then, given an arbitrary fundamental sequence, there are a subsequence ε and \(\mathbf{u}= ( u_{0},u_{1} ) \in \mathbb{F}_{0}^{1}L^{B}\) such that (38)–(40) hold as ε→0 (see in particular Theorem 5). Let us verify that u=(u0,u1) is a solution of the variational problem of Lemma 3. Let \(\boldsymbol{\phi}= ( \psi_{0},\psi_{1} ) \in F_{0}^{\infty}\) with \(\psi_{0}\in\mathcal {D}( Q; \mathbb{R} ) \), \(\psi_{1}\in\mathcal{D} ( Q; \mathbb{R} ) \otimes\mathcal{C}_{per}^{\infty} ( \varTheta; \mathbb{R} ) \otimes\mathcal{C}_{per}^{\infty} ( Y; \mathbb{R} ) / \mathbb{C}\), and let \(\phi_{\varepsilon}=\psi_{0}+\varepsilon\psi_{1}^{\varepsilon}\). Then \(\phi_{\varepsilon}\in\mathcal{D} ( Q; \mathbb{R} ) \), so take v=ϕε(t) (0<t<T) in (41), integrate from 0 to T and then use (19) to have
$$ \int_{0}^{T} \bigl( f ( t ) -u_{\varepsilon}^{\prime} ( t ) ,u_{\varepsilon} ( t ) -\phi_{\varepsilon} ( t ) \bigr) dt-\int _{Q}a^{\varepsilon} ( -,D\phi_{\varepsilon } ) \cdot ( Du_{\varepsilon}-D\phi_{\varepsilon} ) dxdt\geq0. $$
Since \(\int_{0}^{T} ( u_{\varepsilon}^{\prime}(t),u_{\varepsilon }(t) ) dt=\frac{1}{2}\Vert u_{\varepsilon} ( T ) \Vert_{L^{2} ( \varOmega ) }^{2}\), the preceding inequality is equivalent to Recalling that \(\frac{\partial\phi_{\varepsilon}}{\partial x_{i}}\rightarrow\frac{\partial\psi_{0}}{\partial x_{i}}+\frac{\partial \psi_{1}}{\partial y_{i}}\) in LB(Q)-weak 2s (1≤id), and \(\frac{\partial\phi_{\varepsilon}}{\partial t}\rightarrow \frac{\partial\psi_{0}}{\partial t}+\frac{\partial\psi_{1}}{\partial\tau}\) in LB(Q)-weak 2s, hence ϕεψ0 in \(L^{B} ( 0,T;W_{0}^{1}L^{B} ( \varOmega; \mathbb{R} ) ) \)-weak, up to the subsequence extracted above (as ε→0) we have
$$ \begin{array} {@{}l} \displaystyle\int_{0}^{T} \bigl( f ( t ) ,u_{\varepsilon} ( t ) -\phi_{\varepsilon} ( t ) \bigr) dt \rightarrow\int_{0}^{T} \bigl( f ( t ) ,u_{0} ( t ) -\psi_{0} ( t ) \bigr) dt \\[4mm] \displaystyle\int_{0}^{T} \bigl( u_{\varepsilon}^{\prime} ( t ) ,\phi_{\varepsilon} ( t ) \bigr) dt=-\int _{Q}u_{\varepsilon}\frac {\partial\phi_{\varepsilon}}{\partial t}dxdt\rightarrow-\int _{Q}u_{0} \frac{\partial\psi_{0}}{\partial t}dxdt \\[4mm] \displaystyle\phantom{\int_{0}^{T} \bigl( u_{\varepsilon}^{\prime} ( t ) ,\phi_{\varepsilon} ( t ) \bigr) dt} {}=\int _{0}^{T} \bigl( u_{0}^{\prime } ( t ) ,\psi_{0} ( t ) \bigr) dt \\[4mm] \displaystyle\int_{Q}a^{\varepsilon} ( -,D\phi_{\varepsilon} ) \cdot ( Du_{\varepsilon}-D\phi_{\varepsilon} ) dxdt\rightarrow\int \int _{Q\times Z}a ( -,D\boldsymbol{\phi} ) \cdot\mathbb{D} ( \mathbf{u}- \mathbf{\phi} ) dxdtdyd\tau. \end{array} $$
Next, deducing by (38)–(39) that uεu0 in \(\mathcal{W}_{0} ( 0,T;W_{0}^{1}L^{B} ( \varOmega; \mathbb{R} ) ) \)-weak, and noting that the transformation \(v\rightarrow \Vert v ( T ) \Vert_{L^{2} ( \varOmega ) }\) from \(\mathcal{W}_{0} ( 0,T;W_{0}^{1}L^{B} ( \varOmega;\mathbb{R} ) ) \) into ℝ is continuous, we are led to
$$ \bigl\Vert u_{0} ( T ) \bigr\Vert_{L^{2} ( \varOmega ) }^{2}\leq \lim\inf_{\varepsilon\rightarrow0}\bigl\Vert u_{\varepsilon } ( T ) \bigr \Vert_{L^{2} ( \varOmega ) }^{2}. $$
Therefore, passing to the liminf in (42), when ε→0, and taking into account the arbitrariness of \(\boldsymbol{\phi} \in F_{0}^{\infty}\), and the density of \(F_{0}^{\infty}\) in \(\mathbb {F} _{0}^{1}L^{B}\), yields
$$ \int_{0}^{T} \bigl( f ( t ) -u_{0}^{\prime} ( t ) ,u_{0} ( t ) -v_{0} ( t ) \bigr) dt-\int\int _{Q\times Z}a ( -,D\mathbf{v} ) \cdot\mathbb{D} ( \mathbf{u}-\mathbf {v} ) dxdtdyd\tau\geq0 $$
for all \(\mathbf{v}= ( v_{0},v_{1} ) \in\mathbb{F}_{0}^{1}L^{B}\). The end of the proof is a routine exercise (see [21, Theorem 4.1]). □

6.3 Macroscopic Homogenized Problem

The variational problem in Lemma 3 is referred to as the global homogenized problem for (2) under the periodic hypothesis (6), and implies the following variational equation: By mere routine (see [21, page 18]) we get u1=π(Du0), where for each r∈ℝd, π(r) is the unique solution of the variational equation
$$ \int_{Z}a \bigl( -,r+D_{y}\pi ( r ) \bigr) \cdot D_{y}\theta dyd\tau=0\quad\text{for all}\ \theta\in L^{B} \bigl( \varTheta ;W_{\#}^{1}L_{per}^{B} ( Y; \mathbb{R} ) \bigr). $$
Replacing u1 by π(Du0) in Lemma 3 and taking v=(v0,v1) with v1=0, we are led to the macroscopic homogenized problem
$$ \begin{array} {@{}l} \displaystyle\frac{\partial u_{0}}{\partial t}-\text{div}q ( Du_{0} ) =f\quad\text{in}\ Q, \\[4mm] u_{0}=0\quad\text{on}\ \partial\varOmega\times ( 0,T ), \\[2mm] u_{0}( x,0) =0\quad\text{in}\ \varOmega, \end{array} $$
where q(r)=∫Za(−,r+Dyπ(r))dydτ (r∈ℝd). Clearly q(ω)=ω, \(\vert q ( \lambda ) -q ( \mu ) \vert \leq c_{0}\widetilde {B}^{-1}[ B ( c_{1}\vert \lambda-\mu\vert ) ] \), [q(λ)−q(μ)]⋅(λμ)≥c2B(|λμ|) for all λ,μ∈ℝd.

Acknowledgements

The authors would like to thank the anonymous referee for his/her pertinent remarks, comments and suggestions.

Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  1. 1.Faculty of Sciences, Department of MathematicsUniversity of Yaounde IYaoundeCameroon
  2. 2.École Normale Supérieure de YaoundéUniversity of Yaounde IYaoundeCameroon

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