Cent. Eur. J. Phys. 8(4) 2010 555-561 [610106]

Cent. Eur. J. Phys. • 8(4) • 2010 • 555-561
DOI: 10.2478/s11534-009-0141-6
Central European Journal of Physics
Multiscale analysis in nonlinear thermal diffusion
problems in composite structures
Research Article
Claudia Timofte∗
Faculty of Physics, University of Bucharest, P .O. Box MG-11, Bucharest-Magurele, Romania
Received 2 April 2009; accepted 10 September 2009
Abstract: The aim of this paper is to analyze the asymptotic behavior of the solution of a nonlinear problem arising in
the modelling of thermal diffusion in a two-component composite material. We consider, at the microscale,
a periodic structure formed by two materials with different thermal properties. We assume that we have
nonlinear sources and that at the interface between the two materials the flux is continuous and depends
in a dynamical nonlinear way on the jump of the temperature field. We shall be interested in describing
the asymptotic behavior of the temperature field in the periodic composite as the small parameter which
characterizes the sizes of our two regions tends to zero. We prove that the effective behavior of the solution
of this system is governed by a new system, similar to Barenblatt’s model, with additional terms capturing
the effect of the interfacial barrier, of the dynamical boundary condition, and of the nonlinear sources.
PACS (2008): 02.60.Lj, 44.35.+c
Keywords: homogenization • composite materials • interfacial barrier • dynamical boundary conditions
©Versita Sp. z o.o.
1. Introduction and posing of the
problem
The purpose of this paper is to analyze the effective be-
havior of the solution of a nonlinear problem arising in
the modelling of thermal diffusion in a periodic structure
formed by two materials with different thermal properties,
separated by an active interface. We assume that we have
nonlinear sources acting in one component and that at the
interface between our two materials the flux is continuous
and depends in a nonlinear way on the jump of the tem-
perature field.
∗E-mail: [anonimizat] theoretical and advanced computational studies
investigating the effective behavior of composite materi-
als are based on a model which considers the composite
material, despite of its discrete structure, as the coupling
of two continuous superimposed domains. Homogeniza-
tion theory will allow us to justify such a model and to
give a meaning to the effective properties of highly het-
erogeneous materials, modeled by equations with rapidly
oscillating periodic coefficients.
LetΩbe a bounded domain in R/n.math(/n.math≥3), with the bound-
ary∂Ωbeing a Lipschitz manifold, consisting of a finite
number of connected components. We assume that Ωis a
periodic structure formed by two components, ΩεandΠε,
representing two materials with different thermal proper-
ties, separated by an interface Sε. We further assume
that both ΩεandΠε= Ω\Ωεare connected, but only
555

Multiscale analysis in nonlinear thermal diffusion problems in composite structures
Ωεreaches the external fixed boundary of the domain Ω.
Hereεrepresents a small parameter related to the char-
acteristic size of the two regions.
More precisely, following H. I. Ene and D. Polisevski [12],
letY1be a Lipschitz open connected subset of the unit
cubeY= (0/comma.math1)/n.math. LetY2=Y\Y1. We suppose that Y2
has a local Lipschitz boundary. Moreover, we assume that
the intersections of the boundary of Y2with the boundary
of the unit cell Yare identically reproduced on opposite
faces of the cube, which are denoted, for any 1≤/i.math≤/n.math,
by
Σ/i.math={/y.math∈∂Y|/y.math/i.math= 1}
and
Σ−/i.math={/y.math∈∂Y|/y.math/i.math= 0}/period.math
We suppose that periodically repeating Y, the union of all
the setsY1is connected and has a locally C2boundary.
Also, we assume that the origin of the coordinate system
is set in a ball contained in this union.
Let
Zε={/k.math∈Z/n.math|ε/k.math+εY⊆Ω}
and
Iε=/braceleftbig
/k.math∈Zε|ε/k.math±ε/e.math/i.math+εY⊆Ω/comma.math∀/i.math=1/comma.math/n.math/bracerightbig
/comma.math
where/e.math/i.mathare the elements of the canonical basis of R/n.math.
We define
Πε=int/parenleftBigg/uniondisplay
/k.math∈Iε(ε/k.math+εY2)/parenrightBigg
and
Ωε= Ω\Πε/comma.math
and we set θ=/vextendsingle/vextendsingleY\Y2/vextendsingle/vextendsingle.
Let us consider a family of inhomogeneous media occu-
pying the region Ω, parameterized by εand represented
by/n.math×/n.mathmatricesAε(/x.math)of real-valued coefficients defined
onΩ. We shall deal with periodic structures, defined by
Aε(/x.math) =A(/x.math
ε). HereA=A(/y.math)is a smooth matrix-valued
function on R/n.mathwhich isY-periodic. We use the symbol #
to denote periodicity properties. We shall assume that


A∈L∞
#(Ω)/n.math×/n.math/comma.math
Ais a symmetric matrix,
For some
0<γ<λ/comma.math γ|ξ|2≤A(/y.math)ξ·ξ≤λ|ξ|2∀ξ/comma.math /y.math∈R/n.math/comma.math
and we shall denote the matrix AbyA1inY1and byA2
inY2, respectively.If we denote by (0/comma.mathT)the time interval of interest, we
shall analyze the asymptotic behavior of the solutions of
the following nonlinear system:


−div(Aε
1∇/u.mathε) +β(/u.mathε) =/f.mathinΩε×(0/comma.mathT)/comma.math
−div(Aε
2∇/v.mathε) =/f.math/comma.mathinΠε×(0/comma.mathT)/comma.math
Aε
1∇/u.mathε·ν=Aε
2∇/v.mathε·νonSε×(0/comma.mathT)/comma.math
Aε
1∇/u.mathε·ν+αε∂
∂/t.math(/u.mathε−/v.mathε) =ε/g.math(/v.mathε−/u.mathε)
onSε×(0/comma.mathT)/comma.math
/u.mathε= 0 on∂Ω×(0/comma.mathT)/comma.math
/u.mathε(0/comma.math/x.math)−/v.mathε(0/comma.math/x.math) =/c.math0(/x.math)/comma.mathonSε/period.math(1)
Here,νis the exterior unit normal to Ωε,/f.math∈
L2(0/comma.mathT;L2(Ω)),/c.math0∈H1
0(Ω)andα > 0. The function
βis continuous, monotonously non-decreasing and such
thatβ(0) = 0 and the function /g.mathis continuously differen-
tiable, monotonously non-decreasing and with /g.math(0) = 0 .
We shall suppose that there exist a positive constant C
and an exponent /q.math, with 0≤/q.math</n.math/ (/n.math−2), such that
|β(/v.math)|≤C(1 +|/v.math|/q.math)/comma.math/vextendsingle/vextendsingle/vextendsingle/vextendsingle/d.math/g.math
/d.math/v.math/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(1 +|/v.math|/q.math)/period.math(2)
As particular examples of such functions we can con-
sider, for instance, the following important practical ones:
β(/v.math) =δ/v.math
1 +γ/v.math/comma.math δ/comma.mathγ > 0(Langmuir kinetics), β(/v.math) =
|/v.math|/p.math−1/v.math/comma.math0</p.math< 1(Freundlich kinetics), /g.math(/v.math) =/a.math/v.mathor
/g.math(/v.math) =/a.math/v.math3, with/a.math>0.
Well posedness of problem (1) in suitable function spaces
and proper energy estimates have been obtained in [1], [4]
and [13].
Since the period of the structure is small compared to
the dimension of Ω, in other words, since the nonhomo-
geneities are small compared to the global dimension of
the structure, an asymptotic analysis becomes necessary.
Two scales are important for a suitable description of the
given structure: one which is comparable with the dimen-
sion of the period, called the microscopic scale and de-
noted by/y.math=/x.math/ε, and one which is of the same order of
magnitude as the global dimension of our system, called
themacroscopic scale and denoted by /x.math.
The main goal of the homogenization method is to pass
from the microscopic scale to the macroscopic one; more
precisely, using the homogenization method, we try to de-
scribe the macroscopic properties of our nonhomogeneous
system in terms of the properties of its microscopic struc-
ture. Intuitively, the nonhomogeneous real system, having
a very complicated microstructure, is replaced by a ficti-
tious homogeneous one, whose global characteristics rep-
resent a good approximation of the initial system. Hence,
the homogenization method provides a general framework
556

Claudia Timofte
for obtaining these macroscale properties, eliminating the
difficulties related to the explicit determination of a solu-
tion of the problem at the microscale and offering a less
detailed description, but one which is applicable to much
more complex systems.
Also, from the point of view of numerical computation, the
homogenized equations will be easier to solve. This is due
to the fact that the homogenized equations which describe
the effective behavior of the system are defined on a fixed
domain Ωand they have constant coefficients: the effec-
tive or homogenized coefficients (see Sec. 2). The orig-
inal equations have rapidly oscillating coefficients, they
are defined on a complicated domain and satisfy nonlin-
ear boundary conditions. The dependence on the real
microstructure is given through the homogenized coeffi-
cients.
Hence, our goal is to obtain the asymptotic behavior, when
ε→0, of the solution of problem (1). Using Tartar’s
method of oscillating test functions (see [14]), coupled with
monotonicity methods and results from the theory of semi-
linear problems (see [4] and [9]), we can prove that the
asymptotic behavior of the solution of our problem is gov-
erned by a new nonlinear system, similar to the famous
Barenblatt’s model (see [3] and [12]), with additional terms
capturing the effect of the interfacial barrier, of the dynam-
ical boundary condition and of the nonlinear sources. Our
results constitute a generalization of those obtained in [3]
and [12], by considering nonlinear sources, nonlinear dy-
namical transmission conditions and different techniques
in the proofs. Similar problems have been considered, us-
ing different techniques, in [2] and [13], for studying elec-
trical conduction in biological tissues.
In conclusion, at a macroscopic level, despite of the dis-
crete cellular structure, the composite material can be rep-
resented by a continuous model, describing the averaged
properties of the complex structured composite material.
The resulting macroscopic model describes the composite
material as the superimposition of two interpenetrating
continuous media, coexisting at every point of the domain.
Also, note that the above model is a degenerate parabolic
system, as the time derivatives involve the unknown /v.math−/u.math,
instead of the unknowns /u.mathand/v.mathoccurring in the second-
order conduction term.
Similar problems to that described herein were consid-
ered by D. Cioranescu and P. Donato [5], D. Cioranescu,
P. Donato and H. I. Ene [6], C. Conca and P. Donato [10],
C. Conca, J. I. Díaz and C. Timofte [9], C. Conca, F. Murat
and C. Timofte [11], C. Timofte [16], [17] and [18], H. Ene
and D. Polisevski [12], C. Timofte [15], for the determinis-
tic case, W. Wang and J. Duan [20] for the stochastic one.
The plan of the paper is as follows: in the second section
we give the main convergence result of this paper. Thelast section is devoted to the proof of this convergence
result.
2. The main result
Using well-known extension results (see, for in-
stance, [8], [5] and [13]) and suitable test functions, we
can take the limit in the variational formulation of prob-
lem (1) and obtain the effective behavior of the solution of
our microscopic model. Therefore, the main result of this
paper can be formulated as follows:
Theorem 2.1.
One can construct two extensions Pε/u.mathεandPε/v.mathεof the
solutions/u.mathεand/v.mathεof problem (1)such thatPε/u.mathε/arrowrighttophalf
/u.math/comma.math Pε/v.mathε/arrowrighttophalf/v.math, weakly in L2(0/comma.mathT;H1
0(Ω)), where


α|Γ|∂
∂/t.math(/u.math−/v.math)−div/parenleftBig
A1∇/u.math/parenrightBig
+θβ(/u.math)−|Γ|/g.math(/v.math−/u.math)
=θ/f.mathinΩ×(0/comma.mathT)/comma.math
α|Γ|∂
∂/t.math(/v.math−/u.math)−div/parenleftBig
A2∇/v.math/parenrightBig
+|Γ|/g.math(/v.math−/u.math)
= (1−θ)/f.mathinΩ×(0/comma.mathT)/comma.math
/u.math(0/comma.math/x.math)−/v.math(0/comma.math/x.math) =/c.math0(/x.math)onΩ/period.math
(3)
Here,A1andA2are the homogenized matrices, defined
by:
A1
/i.math/j.math=/integraldisplay
Y1/parenleftbigg
/a.math/i.math/j.math+/a.math/i.math/k.math∂χ1/j.math
∂/y.math/k.math/parenrightbigg
/d.math/y.math/comma.math
A2
/i.math/j.math=/integraldisplay
Y2/parenleftbigg
/a.math/i.math/j.math+/a.math/i.math/k.math∂χ2/j.math
∂/y.math/k.math/parenrightbigg
/d.math/y.math/comma.math
in terms of the functions χ1/k.math∈H1
/p.math/e.math/r.math(Y1)/R/comma.math χ2/k.math∈
H1
/p.math/e.math/r.math(Y2)/R,/k.math= 1/comma.math/period.math/period.math/period.math/comma.math/n.math, weak solutions of the cell prob-
lems


−∇/y.math·/parenleftbigg
A1(/y.math)∇/y.mathχ1/k.math/parenrightbigg
=∇/y.mathA1(/y.math)/e.math/k.math/comma.math /y.math∈Y1/comma.math
/parenleftbigg
A1(/y.math)∇/y.mathχ1/k.math/parenrightbigg
·ν=−A1(/y.math)/e.math/k.math·ν/comma.math /y.math∈Γ/comma.math


−∇/y.math·/parenleftbigg
A2(/y.math)∇/y.mathχ2/k.math/parenrightbigg
=∇/y.mathA2(/y.math)/e.math/k.math/comma.math /y.math∈Y2/comma.math
/parenleftbigg
A2(/y.math)∇/y.mathχ2/k.math/parenrightbigg
·ν=−A2(/y.math)/e.math/k.math·ν/comma.math /y.math∈Γ/period.math
Thus, in the limit, we obtain a system similar to the so-
called Barenblatt model . Alternatively, such a system is
similar to the so-called bidomain model , appearing in the
context of electrical activity of the heart.
557

Multiscale analysis in nonlinear thermal diffusion problems in composite structures
3. Proof of the main result
From [1] and [13], we know that there exists a unique weak
solution (/u.mathε/comma.math/v.mathε)of (1), with/u.mathε∈L2(0/comma.mathT;H1(Ωε)),/u.mathε= 0
on∂Ωand/v.mathε∈L2(0/comma.mathT;H1(Πε)).
Using our assumptions on the data and the Cauchy-Schwartz, Poincaré’s, Young’s and Gronwall’s inequalities,
we can obtain suitable energy estimates, independent of
ε, for our solution (see [2], [13], [9], [15] and [20]). More
precisely, if we multiply the first equation in (1) by /u.mathε,
the second one by /v.mathεand we integrate formally by parts,
we obtain, for 0</t.math<T ,
/integraldisplay/t.math
0/integraldisplay
ΩεAε
1∇/u.mathε·∇/u.mathε/d.math/x.math/d.mathτ+/integraldisplay/t.math
0/integraldisplay
ΠεAε
2∇/v.mathε·∇/v.mathε/d.math/x.math/d.mathτ+/integraldisplay/t.math
0/integraldisplay
Ωεβ(/u.mathε)/u.mathε/d.math/x.math/d.mathτ+εα
2/integraldisplay
Sε(/v.mathε−/u.mathε)2/d.mathσ
+ε/integraldisplay/t.math
0/integraldisplay
Sε/g.math(/v.mathε−/u.mathε)(/v.mathε−/u.mathε)/d.mathσ/d.mathτ =εα
2/integraldisplay
Sε/parenleftbig
/c.math0/parenrightbig2/d.mathσ+/integraldisplay/t.math
0/integraldisplay
Ωε/f.math/u.mathε/d.math/x.math/d.mathτ+/integraldisplay/t.math
0/integraldisplay
Πε/f.math/v.mathε/d.math/x.math/d.mathτ/period.math (4)
Therefore, using the conditions imposed for βand/g.mathand our hypotheses on the data, we have:
/integraldisplay/t.math
0/integraldisplay
ΩεAε
1∇/u.mathε·∇/u.mathε/d.math/x.math/d.mathτ+/integraldisplay/t.math
0/integraldisplay
ΠεAε
2∇/v.mathε·∇/v.mathε/d.math/x.math/d.mathτ+ε/integraldisplay
Sε(/v.mathε−/u.mathε)2/d.mathσ≤C/comma.math (5)
whereCis independent of ε.
The above computations can be made perfectly rigorous by taking the limit in the analogous stability estimates for a
suitable regularized or discretized system (see [13]).
As a matter of fact, one can obtain the needed a priori estimates by writing the variational formulation of problem (1) in
a compact form. To this end, let us introduce the following space:
/V.kaliε=/braceleftbig
/w.mathε= (/u.mathε/comma.math/v.mathε)∈(H1(Ωε)∩H1
0(Ω))×H1(Πε)|/v.mathε−/u.mathε∈L2(Sε)/bracerightbig
and let us take /w.math= (/u.math/comma.math/v.math)∈/V.kaliεand the test function /w.math= (/u.math/comma.math/v.math)∈/V.kaliε. If we introduce the bilinear forms
/a.mathε(/w.math/comma.math/w.math) =/integraldisplay
ΩεAε
1∇/u.math·∇/u.math/d.math/x.math+/integraldisplay
ΠεAε
2∇/v.math·∇/v.math/d.math/x.math/comma.math
/b.mathε(/w.math/comma.math/w.math) =ε/integraldisplay
Sε(/v.math−/u.math)(/v.math−/u.math)/d.mathσ/comma.math
and
Fε(/w.math/comma.math/w.math) =ε/integraldisplay
Sε/g.math(/v.math−/u.math)(/v.math−/u.math)/d.mathσ+/integraldisplay
Ωεβ(/u.math)/u.math/d.math/x.math/comma.math
and we adopt the point of view to consider /u.mathεand/v.mathεas time dependent functions with values in a space dependent
functional space (see [13]), we can write the variational formulation of problem (1) in the following compact form:
/d.math
/d.math/t.math/b.mathε(/w.mathε/comma.math/w.math) +/a.mathε(/w.mathε/comma.math/w.math) +Fε(/w.mathε/comma.math/w.math) =/integraldisplay
Ωε/f.math/u.math/d.math/x.math+/integraldisplay
Πε/f.math/v.math/d.math/x.math/comma.math
/b.mathε(/w.mathε(0)/comma.math/w.math) =/b.mathε(/w.mathε
0/comma.math/w.math)/comma.math
where/w.mathε= (/u.mathε/comma.math/v.mathε)/comma.math/w.math∈/V.kaliε, with Λε(/w.math)<+∞and/w.mathε
0= (/u.mathε
0/comma.math/v.mathε
0)∈/V.kaliε/comma.math(/u.mathε
0−/v.mathε
0)|Sε=/c.math0,Λε(/w.mathε
0))<+∞and
/a.mathε(/w.mathε
0) = min{/a.mathε(/w.mathε)|(/u.mathε−/v.mathε)|Sε=/c.math0}.
Here,
Λε(/w.math) =ε/integraldisplay
Sεφ(/v.math−/u.math)/d.mathσ/comma.math
558

Claudia Timofte
φ(/x.math) =/integraldisplay/x.math
0/g.math(ξ)/d.mathξ
and
/a.mathε(/w.math) =/a.mathε(/w.math/comma.math/w.math)/period.math
If we denote
/b.mathε(/w.math) =/b.mathε(/w.math/comma.math/w.math)/comma.math
one can prove (see, for details, [13]) that there exists a unique solution /w.mathε= (/u.mathε/comma.math/v.mathε)∈C0([0/comma.mathT];/V.kaliε)of the above problem
and, moreover, we have the following estimates:
sup
/t.math∈[0/comma.mathT](/b.mathε(/w.mathε) + Λε(/w.mathε) +/a.mathε(/w.mathε))≤C(/b.mathε(/w.mathε
0) + Λε(/w.mathε
0) +/a.mathε(/w.mathε
0))/comma.math
/integraldisplayT
0/b.mathε(∂/t.math(/v.mathε−/u.mathε))/d.math/t.math≤C(/b.mathε(/w.mathε
0) + Λε(/w.mathε
0) +/a.mathε(/w.mathε
0))/comma.math
for a constant Cindependent of εand the initial data.
Let us take now Φ1∈C∞
0(Ω) =/D.kali(Ω)/comma.mathΦ2∈C∞
0(Ω)andψ∈C∞
0((0/comma.mathT)) =/D.kali((0/comma.mathT)). We have:
/integraldisplayT
0/integraldisplay
ΩεAε
1∇/u.mathε·∇Φ1ψ/d.math/x.math/d.math/t.math +/integraldisplayT
0/integraldisplay
ΠεAε
2∇/v.mathε·∇Φ2ψ/d.math/x.math/d.math/t.math +/integraldisplayT
0/integraldisplay
Ωεβ(/u.mathε)Φ1ψ/d.math/x.math/d.math/t.math
+αε/integraldisplayT
0/integraldisplay
Sε(/u.mathε−/v.mathε)(Φ2−Φ1)/d.mathψ
/d.math/t.math/d.mathσ/d.math/t.math+ε/integraldisplayT
0/integraldisplay
Sε/g.math(/v.mathε−/u.mathε)(Φ2−Φ1)ψ/d.mathσ/d.math/t.math
=/integraldisplayT
0/integraldisplay
Ωε/f.mathΦ1ψ/d.math/x.math/d.math/t.math +/integraldisplayT
0/integraldisplay
Πε/f.mathΦ2ψ/d.math/x.math/d.math/t.math/period.math (6)
As already mentioned, we know that there exists a unique weak solution (/u.mathε/comma.math/v.mathε)of (6). Denoting by Pε/u.mathεandPε/v.mathε
the extensions of /u.mathεand, respectively, /v.mathεto the whole of Ω(see [8] and [13]), we can see that Pε/u.mathεandPε/v.mathεare
bounded in L2(0/comma.mathT;H1
0(Ω))(see, for details, [9], [13], [15] and [20]). So, by passing to a subsequence, we have Pε/u.mathε/arrowrighttophalf
/u.mathweakly inL2(0/comma.mathT;H1
0(Ω)). Also,Pε/u.mathε/arrowrighttophalf/u.mathweakly inL2(Ω×(0/comma.mathT))and strongly in L1
/l.math/o.math/c.math(0/comma.mathT;L1(Ω))(see [2], Remark
2.1, Lemma 7.1 and Lemma 7.4). In a similar manner, we obtain that Pε/v.mathε/arrowrighttophalf /v.mathweakly inL2(0/comma.mathT;H1
0(Ω)), weakly in
L2(Ω×(0/comma.mathT))and strongly in L1
/l.math/o.math/c.math(0/comma.mathT;L1(Ω)). Moreover, /u.math/comma.math /v.math∈C0([0/comma.mathT];H1
0(Ω))and∂
∂/t.math(/u.math−/v.math)∈L2(0/comma.mathT;L2(Ω))
(see [13]).
It is well-known by now how to pass to the limit, with ε→0, in the linear terms of (6) defined on ΩεandΠε(see, for
instance [9], [12], [15] and [20]). Also, recall that θis the weak- ⋆limit inL∞(Ω)of the characteristic function χεof the
domain Ωε. Thus, we obtain:
/integraldisplayT
0/integraldisplay
ΩεAε
1∇/u.mathε·∇Φ1ψ/d.math/x.math/d.math/t.math→/integraldisplayT
0/integraldisplay
ΩA1∇/u.math·∇Φ1ψ/d.math/x.math/d.math/t.math (7)
and/integraldisplayT
0/integraldisplay
ΠεAε
2∇/v.mathε·∇Φ2ψ/d.math/x.math/d.math/t.math→/integraldisplayT
0/integraldisplay
ΩA2∇/v.math·∇Φ2ψ/d.math/x.math/d.math/t.math/comma.math (8)
for any Φ1/comma.mathΦ2∈/D.kali(Ω)/comma.math ψ∈/D.kali((0/comma.mathT)).
Let us see now how we can pass to the limit in the nonlinear terms in (6).
For the third term in the left-hand side of (6), let us notice
that, exactly like in [9], one can prove that for any /z.mathε/arrowrighttophalf/z.math
weakly inH1
0(Ω), we getβ(/z.mathε)→β(/z.math)strongly inL/q.math(Ω),where/q.math=2/n.math
/q.math(/n.math−2) +/n.math. Therefore, since for /t.mathfixed,
Pε/u.mathε(·/comma.math/t.math)converges weakly in H1
0(Ω)(see [13]), it follows
559

Multiscale analysis in nonlinear thermal diffusion problems in composite structures
that/integraldisplay
Ωεβ(/u.mathε)Φ1/d.math/x.math→/integraldisplay
Ωβ(/u.math)θΦ1/d.math/x.math/comma.math
for any Φ1∈/D.kali(Ω). Using our hypotheses on the data, we
see that, for any /t.math,/bardblβ(/u.mathε)/bardbl2
L2(Ωε)≤C,/bardblβ(Pε/u.mathε)/bardbl2
L2(Ω)≤C,
withCdepending on Tand the data, but independent of
ε. Here, byCwe have denoted different constants, which
can vary from line to line.
Using Lebesgue’s convergence theorem, we have
/integraldisplayT
0/integraldisplay
Ωεβ(/u.mathε)Φ1ψ/d.math/x.math/d.math/t.math→/integraldisplayT
0/integraldisplay
Ωβ(/u.math)θΦ1ψ/d.math/x.math/d.math/t.math/comma.math (9)
for any Φ1∈/D.kali(Ω)and for any ψ∈/D.kali((0/comma.mathT)).
To take the limit in the other terms in (6), we recall a
result from [5].
Let us introduce, for any /h.math∈L/s.math/prime(Γ),1≤/s.math/prime≤∞, the
linear formµε
/h.mathonW1/comma.math/s.math
0(Ω)defined by
/angbracketleftµε
/h.math/comma.mathψ/angbracketright=ε/integraldisplay
Sε/h.math(/x.math
ε)ψ/d.mathσ∀ψ∈W1/comma.math/s.math
0(Ω)/comma.math
with1//s.math+ 1//s.math/prime= 1. It is proven in [5] that
µε
/h.math→µ/h.mathstrongly in (W1/comma.math/s.math
0(Ω))/prime/comma.math (10)
where
/angbracketleftµ/h.math/comma.mathψ/angbracketright=µ/h.math/integraldisplay
Ωψ/d.math/x.math/comma.math
with
µ/h.math=/integraldisplay
Γ/h.math(/y.math)/d.mathσ/period.math
If/h.math∈L∞(Γ)or even if/h.mathis constant, we have (see [6])
µε
/h.math→µ/h.mathstrongly inW−1/comma.math∞(Ω)/period.math (11)
Also, for obtaining the limiting behavior of our homoge-
nization problem, let us recall another result from [9]. Let
Hbe a continuously differentiable function, monotonously
non-decreasing and such that H(/v.math) = 0 if and only if
/v.math= 0. We shall suppose that there exist a positive con-
stantCand an exponent /q.math, with 0≤/q.math</n.math/ (/n.math−2), such
that|H|≤C(1+|/v.math|/q.math). If we denote by /q.math=2/n.math
/q.math(/n.math−2) +/n.math,
one can prove (see [9]) that for any /z.mathε/arrowrighttophalf /z.mathweakly in
H1
0(Ω), we obtain
H(/z.mathε)/arrowrighttophalfH(/z.math)weakly inW1/comma.math/q.math
0(Ω)/period.math (12)Now, using (12) written for /g.mathand the convergence (11)
written for/h.math= 1, we obtain
ε/integraldisplay
Sε/g.math(/v.mathε−/u.mathε)(Φ2−Φ1)/d.mathσ→|Γ|/integraldisplay
Ω/g.math(/v.math−/u.math)(Φ2−Φ1)/d.math/x.math/comma.math
for any Φ1/comma.mathΦ2∈ /D.kali(Ω). Integrating in time and using
Lebesgue’s convergence theorem, it is not difficult to see
that
ε/integraldisplayT
0/integraldisplay
Sε/g.math(/v.mathε−/u.mathε)(Φ2−Φ1)ψ/d.mathσ/d.math/t.math
→|Γ|/integraldisplayT
0/integraldisplay
Ω/g.math(/v.math−/u.math)(Φ2−Φ1)ψ/d.math/x.math/d.math/t.math/period.math (13)
Also, we have
αε/integraldisplayT
0/integraldisplay
Sε(/u.mathε−/v.mathε)(Φ2−Φ1)/d.mathψ
/d.math/t.math/d.mathσ/d.math/t.math
→α|Γ|/integraldisplayT
0/integraldisplay
Ω(/u.math−/v.math)(Φ2−Φ1)/d.mathψ
/d.math/t.math/d.math/x.math/d.math/t.math/period.math (14)
Since it is obvious how to pass to the limit in the other
linear terms in (6), putting together (7)-(9) and (13)-(14),
we can pass to the limit in all the terms in (6) and, using
standard density arguments, we obtain exactly the varia-
tional formulation of the limit problem (3). Also, we can
easily pass to the limit, with ε→0, in the initial condition
and we have
/u.math(0/comma.math/x.math)−/v.math(0/comma.math/x.math) =/c.math0(/x.math)/comma.math /x.math∈Ω/period.math
As/u.mathand/v.mathare uniquely determined (see [13]), the whole
sequencesPε/u.mathεandPε/v.mathεconverge to /u.mathand, respectively,
/v.math, and this completes the proof of Theorem 2.1. 
Acknowledgements
This work was supported by the CNCSIS Grant Ideas 992,
under contract 31/2007.
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