Parabolic Problems with Dynamical Boundary [610087]

Parabolic Problems with Dynamical Boundary
Conditions in Perforated Media
C. Timofte
Department of Mathematics, Faculty of Physics,
University of Bucharest, P.O. Box MG-11, Bucharest-Magurele, Romania.
E-mail: [anonimizat]
Abstract
The asymptotic behavior of the solution of a parabolic dynamical boundary-value problem
in a periodically perforated domain is analyzed. The perforations, which are identical and
periodically distributed, are of size ". In the perforated domain we consider a heat equation,
with a Dirichlet condition on the exterior boundary and a dynamical boundary condition on the
surface of the holes. The limit equation, as "!0, is a heat equation with extra-terms coming
from the in°uence of the non-homogeneous dynamical boundary condition.
Key words : homogenization, energy method, dynamical boundary-value problems.
1 Introduction and Formulation of the Problem
The aim of this paper is to study the asymptotic behavior of the solution of a parabolic dynamical
boundary-value problem in a periodically perforated domain. Such problems, although not too
widely considered in the literature, are very natural in many mathematical models as partially
saturated °ows in porous media, heat transfer in a solid in contact with a moving °uid, di®usion
phenomena in porous media (see [ ?] and [ ?] and the references therein).
Let ­ be a ¯xed bounded open subset in RNand let as perforate it by holes. As a result, we
obtain an open set ­";which will be referred to as being the perforated domain ;"represents a
small parameter related to the characteristic size of the perforations. We shall deal with the case
in which the perforations (holes) are identical and periodically distributed and their size is of the
order of ". In the perforated domain we consider a heat equation, with a Dirichlet condition on the
exterior boundary and a dynamical boundary condition on the surface of the holes.
Our main motivation is to study the asymptotic behavior, as "!0;of the solution u"of the
following dynamical boundary-value problem:
@u"
@t¡¢u"=f(t; x);in ­"£(0; T); (1.1)
@u"
@n+"@u"
@t="g(t; x);on@F"£(0; T); (1.2)
u"(0; x) =u0(x);in ­"; (1.3)
1

u"(0; x) =v0(x);on@F"; (1.4)
u"= 0;on@­£(0; T): (1.5)
Here, f2L2(0; T; L2(­)); g2L2(0; T; H1
0(­)); u02L2(­); v02L2(@F");[0; T] is the time interval
of interest and @F"is the boundary of the holes.
As we shall see in Section 4, there exists an extension»u"ofu"into all ­ £(0; T) such that»u"!u
strongly in L2(0; T;L2(­)) and uis the unique solution of the following system (the macromodel):
(jY¤j
jYj+j@Fj
jYj)@u
@t¡ r(Qru) =jY¤j
jYjf+j@Fj
jYjg;in ­£(0; T); (1.6)
u= 0;on@­£(0; T); (1.7)
u(0; x) =u0(x);in ­; (1.8)
where Q= ((qij)) is the classical homogenized matrix whose entries are de¯ned by
qij=jY¤j
jYj±ij¡1
jYjZ
Y¤@´j
@yidy (1.9)
in terms of the functions ´j, solutions of the system
¡¢´j= 0;inY¤; (1.10)
@(´j¡yj)=@n= 0;on@F; (1.11)
´jisY¡periodic : (1.12)
Here, Fis the elementary hole, Yis the elementary cell of periodicity and Y¤=YnF(see
Section 2). Thus, in the limit, when "!0, we get a classical constant coe±cient heat equation,
with a Dirichlet boundary condition, with a non-homogeneous right-hand term and with a constant
(due to the periodicity) extra-term in front of the time derivative, coming from the well-balanced
contribution of the dynamical part of our boundary condition on the surface of the holes. Also, let
us note that the external force gacting on the boundary of the holes leads in the limit to a force
distributed all over the domain ­ :
The plan of the paper is the following one: in the second section we introduce some useful
notations and assumptions. In Section 3 we give the main convergence result of this paper, i. e.
the macromodel. For obtaining this macromodel, we need some preliminary results, which are given
in Section 4. The last section is devoted to the proof of some a priori estimates, independent of ",
for the solution of the micromodel and to the proof of the convergence result.
Problems closed to this one have been considered by many authors. Among others, let us
mention the papers of D. Cioranescu and P. Donato [ ?], [?], C. Conca and P. Donato [ ?], C. Conca,
J.I. D¶ ³az and C. Timofte [ ?], C. Conca, F. Murat and C. Timofte [ ?]. The homogenization of
Laplace and Poisson equations in perforated domains with holes of the same size as the period and
with homogeneous Dirichlet conditions on the surface of the holes and on the exterior boundary
of the domain was treated in [ ?]. The same problem, but with homogeneous Neumann boundary
conditions on the holes, was treated in [ ?]. For the non-homogeneous case, we can refer to [ ?].
The homogenization of the Poisson equation (or even a more general elliptic equation) with non-
homogeneous Fourier boundary conditions on surface of the holes has been treated in [ ?], [?].
2

2 Notation and Assumptions
Let ­ be a bounded connected open set in RN(N¸2), with boundary @­ of class C2and let
[0; T] be the time interval of interest.
LetY= [0; l1[£:::::£[0; lN[ be the representative cell in RNandFan open subset of Ywith
boundary @Fof class C2, such that F½Y:
We shall denote by F(";k) the translated image of "Fby the vector "kl;k2ZN,kl=
(k1l1; ::::; k NlN):
F(";k) ="(kl+F):
Also, we shall denote by F"the set of all the holes contained in ­. So
F"=[
k2KfF(";k)jF(";k)½­g:
Let ­"= ­nF":
Hence, ­"is a periodically perforated domain with holes of the same size as the period. Let us
remark that the holes do not intersect the boundary @­:
We shall use the following notations:
Y¤=YnF; (2.1)
µ=jY¤j
jYj: (2.2)
Also, we shall denote by Â"the characteristic function of the domain ­":
Let us introduce the following usual function spaces and norms:
H=L2(­);(u; v)­=Z
­uvdx; kuk2
­= (u; u)­
H=L2(0; T;H);(u; v)­;T=ZT
0(u(t); v(t))­dt;kuk2
­;T= (u; u)­;T
V=H1(­);(u; v)V= (u; v)­+ (ru;rv)­
V=L2(0; T;V);(u; v)V=ZT
0(u(t); v(t))Vdt
3 The Convergence Result
As we shall see in Section 4, for f2L2(0; T; L2(­)); g2L2(0; T; H1
0(­)); u02L2(­) and v02
L2(@F"), there exists a unique solution u"of the problem (1.1)-(1.5).
The main result of this paper is the following one:
3

Theorem 3.1 Let u"be the unique solution of the problem (1.1)-(1.5). Then, there exists an
extension»u"ofu"into all ­£(0; T)such that»u"!ustrongly in Handuis the unique solution
of the following system (the macromodel):
(jY¤j
jYj+j@Fj
jYj)@u
@t¡ r(Qru) =jY¤j
jYjf+j@Fj
jYjg;in­£(0; T); (3.1)
u= 0; on @ ­£(0; T); (3.2)
u(0; x) =u0(x);in­; (3.3)
where Q= ((qij))is the classical homogenized matrix whose entries are de¯ned as
qij=jY¤j
jYj±ij¡1
jYjZ
Y¤@´j
@yidy (3.4)
in terms of the functions ´j, solutions of the system
¡¢´j= 0; in Y¤; (3.5)
@(´j¡yj)=@n= 0; on @F; (3.6)
´jisY¡periodic: (3.7)
Thus, in the limit, when "!0;we get a classical constant coe±cient heat equation, with
a Dirichlet boundary condition, with a non-homogeneous right-hand term and with a constant
(due to the periodicity) extra-term in front of the time derivative, coming from the well-balanced
contribution of the dynamical part of our boundary condition on the surface of the holes.
Remark 3.2 The weak formulation of the problem (3.1)-(3.3) is:
Find u2L2(0; T;H1
0(­))\C([0; T];L2(­)),u(0) = u0, such that
¡jY¤j
jYjµ
u;@'
@t¶
­;T¡j@Fj
jYjµ
u;@'
@t¶
­;T+ (Qru;r')­;T=
=jY¤j
jYj(f; ')­;T+j@Fj
jYj(g; ')­;T; (3.8)
for any '2 D=C1
0((0; T)£­), or, equivalently:
¡jY¤j
jYjZT
0(u; ')­Ã0(s)ds¡jY¤j
jYj(u(0); ')­Ã(0)¡j@Fj
jYjZT
0(u; ')­Ã0(s)ds¡
¡j@Fj
jYj(u(0); ')­Ã(0) +ZT
0(Qru;r')­Ã(s)ds=
=jY¤j
jYjZT
0(f; ')­Ã(s)ds+j@Fj
jYjZT
0(g; ')­Ã(s)ds; (3.9)
for any '2H1
0(­); Ã2C1([0; T]); Ã(T) = 0 ; Ã(0)6= 0:
Remark 3.3 There is one and only one solution of the weak macromodel problem.
4

4 Preliminary Results
4.1 The Existence Result
In order to obtain the existence and the uniqueness of a solution of problem (1.1)-(15), we shall
make use of the following general result for abstract parabolic equations, due to J.L. Lions (see [ ?]
and [?]).
Theorem 4.1 LetHbe a Hilbert space, with scalar product (;) and norm j:j:We shall
identify Hwith its dual. Let Vbe another Hilbert space, with norm k:k:Suppose that V½H,
with continuous and dense embedding ;such that V½H½V0:Let0< T < 1anda:V£V!Rbe a
continuous and coercive bilinear form. For every f2L2(0; T;V0)andu02H, there exists a unique
function usuch that
u2L2(0; T;V)\C([0; T];H);du
dt2L2(0; T;V0)
and ¿du
dt(t); vÀ
+a(u(t); v) =hf(t); vi; a:e: t 2[0; T];8v2V; (4.1)
u(0) = u0: (4.2)
Here, h;idenotes the duality between V0andV. Also, one has the estimates:
kukL2(0;T;V)·C(kfkL2(0;T;V0)+ku0kH); (4.3)
°°°°@u
@t°°°°
L2(0;T;V0)·C(kukL2(0;T;V)+ 1); (4.4)
with constants Cdepending on T.
In our case, we choose
H=L2(­")£L2
@­(@­");
V=n
(u1; u2)2H1
@­(­")£H1=2
@­(@­")=u2=°(u1)o
;
a(u; v) =1
"Z
­"ru1¢ rv1dx;where u= (u1; u2); v= (v1; v2);
u0= (u0; v0):
In what follows, we shall use the standard Sobolev spaces Hr
@­(­") and Hr
@­(@­");forr¸0,
which are closed subspaces of Hr(­") and Hr(@­");respectively, and the subscript @­ means that,
respectively, traces or functions in @­";vanish on this part of the boundary of ­":Let us notice
that, in fact, we can consider the given v0as an element of L2
@­(@­"):
So, we get immediately from Theorem 4.1 the existence of a unique solution of the problem
(1.1)-(1.5),
(u"; °(u"))2L2(0; T;V)\C([0; T];H);d
dt(u"; °(u"))2L2((0; T);V0):
with the initial conditions (1.3), (1.4).
5

The weak formulation of our problem is the following one:
Find ( u"; °(u"))2L2(0; T;V)\C([0; T];H),d
dt(u"; °(u"))2L2((0; T);V0),(u"(0); °(u")(0))=( u0; v0)
such that
¿d
dt(u"; °(u"));(v; °(v))À
+1
"Z
­"ru"¢ rvdx=1
"Z
­"fvdx +Z
@F"g°(v)d¾; (4.5)
for any ( v; °(v))2V:
Note that here, in H1
@­(­");we take the scalar product1
"R
­"ru¢ rvdx:
If we suppose that we have a better regularity of the data, we can get a more regular solution
([?]). More precisely, if the initial data u02H2(­)\H1
0(­); v0=u0j@F"andf2C1([0; T]; L2(­));
g2C1([0; T]; H1
0(­));our solution satis¯es
u"2C([0; T];H2(­")\H1
@­(­"))\C1([0; T];L2(­")); u"
t2C((0; T);H1
@­(­")):
For such regular solutions, the weak formulation of the system (1 :1)¡(1:5) is the following one:
Find u"2C([0; T];H2(­")\H1
@­(­"))\C1([0; T];L2(­")),u"(0) = u0j­"such that
¡µ
u";@'
@t¶
­";T+ (ru";r')­";T¡"µ
u";@'
@t¶
@F";T= (f; ')­";T+"(g; ')@F";T; (4.6)
for any '2 D;or, in other possible form
¡ZT
0(u"; ')­"Ã0(s)ds¡(u"(0); ')­"Ã(0) +ZT
0(ru";r')­"Ã(s)ds¡
¡"ZT
0(u"; ')@F"Ã0(s)ds¡"(u"(0); ')@F"Ã(0) =
=ZT
0(f; ')­"Ã(s)ds+"ZT
0(g; ')@F"Ã(s)ds; (4.7)
for any '2H1
@­(­"); Ã2C1([0; T]); Ã(T) = 0 ; Ã(0)6= 0:
4.2 A Convergence Result
For obtaining the macromodel, we have to pass to the limit, with "!0;in some surface integrals
on the boundary of the holes. For doing this, we shall make use of a convergence result based on
a technique introduced by M. Vanninathan ([ ?]), which transforms surface integrals into volume
integrals. This method was also used for the elliptic case in [ ?] and [ ?].
For a given function h2L2(@F), following [4], let us denote
Ch=j@Fj
jY¤jM@F(h); (4.8)
where M@F(h) is the mean value of hover@F:Also, let
¹h=µCh: (4.9)
6

In particular
C1=j@Fj
jY¤j(4.10)
and
¹1=j@Fj
jYj; (4.11)
since µ=jY¤j=jYj:
Forh2L2(@F);we de¯ne the measure ¹"
hby
h¹"
h; 'i="Z
@F"h(x
")'(x)d¾;for any '2H1
0(­): (4.12)
In [3] it was proved that
¹"
h!¹hstrongly in H¡1(­); (4.13)
with ¹hgiven by (4.9). Moreover, if his constant and the boundary of Fis smooth (of class C2),
the above convergence takes place strongly in W¡1;1(­):
In the general case, when h2L2(@F), it was proved in [4] (Corollary 4.2) that, if fw"gis
a sequence such that w"2H1
0(­) and w"* w weakly in H1
0(­), then the corresponding linear
form ¹"
hde¯ned by (4.12) on H1(­") satis¯es:
D
¹"
h; w"
j­"E
!¹hZ
­wdx: (4.14)
4.3 An Extension Lemma
Since the solution u"of the problem (1.1)-(1.5) is de¯ned only in ­", we need to extend it to
the whole ­ :For ¯nding a suitable extension eu"into all ­ ;we shall use the following well-known
extension lemma ([ ?]):
Lemma 4.2 (i) Any function '2H1(Y¤)can be extended to a function»'inH1(Y), such that
°°°r»'°°°
Y·Ckr'kY¤:
(ii) Any function '"2H1(­"); '"
j@­= 0can be extended to a function»
'"inH1
0(­), such that
°°°°r»
'"°°°°
­·Ckr'"k­";
where Cis a constant independent of ".
5 A Priori Estimates. Proof of the Convergence Result
In this section we shall prove the convergence result given by Theorem 3.1, for the solution of the
problem (1.1)-(1.5) :This solution being de¯ned on ­";we need to extend it to the whole ­, in order
to be able to state a convergence result. For doing this, we shall need some accurate estimates for
the solution u", independent of ":
7

In what follows, we shall denote by Cdi®erent constants which are independent of ".
Proof of Theorem 3.1 . We shall prove ¯rst the convergence result given by Theorem 3.1 for
regular initial data and so, for regular solutions u"
n. Then, by performing a classical regularization
process of our initial data, we shall be able to prove immediately the convergence result for the
general case.
Step 1. Regular data. Let us consider ¯rst the case in which our data are regular. In fact, let
u0
n2H2(­)\H1
0(­); v0
n=u0
nj@­"such that ( u0
nj­"; v0
n)!(u0j­"; v0) inL2(­")£L2
@­(@­").
Also, let fn2C1([0; T]; L2(­)), gn2C1([0; T]; H1
0(­)) such that fn!finL2(0; T; L2(­)) and
gn!ginL2(0; T; H1
0(­)):
For such regular data, for any n;we know that there exists a unique solution ( u"
n; °(u"
n)) of the
problem (1.1)-(1.5), such that
u"
n2C([0; T];H2(­")\H1
@­(­"))\C1([0; T];L2(­"));(u"
n)t2C((0; T);H1
@­(­")):
In this step, working only with regular solutions u"
n, we shall omit to write explicitly the index n:
For the special geometry of our problem, we can use the following well-known lemma, due to
C. Conca ([ ?]):
Lemma 5.1 There exists a positive constant C, independent of ";such that
kvk­"·C("krvk­"+"1
2kvk@F"); (5.1)
for any v2H1(­"); v= 0on@­:
The next proposition gives us some classical energy estimates for such a regular solution.
Proposition 5.2 For the system (1.1)-(1.5), the following classical parabolic estimates hold:
supt2(0;T)(ku"k2
­"+"ku"k2
@F") +kru"k2
­";T·C; (5.2)
ku"
tk2
­";T+"ku"
tk2
@F";T+ supt2(0;T)ku"k2
H1
@­(­")·C: (5.3)
Proof. Let us multiply ¯rst equation (1.1) by u":We have
1
2d
dtku"k2
­"+"1
2d
dtku"k2
@F"+kru"k2
­"=Z
­"fu"dx+"Z
@F"gu"d¾: (5.4)
Using Cauchy-Schwartz and Young's inequalities, we get
d
dtku"k2
­"+"d
dtku"k2
@F"+ 2kru"k2
­"·2kfk­"ku"k­"+"kgk2
@F"+"ku"k2
@F": (5.5)
From the trace theorem
"ku"(0)k2
@F"·C(ku"(0)k2
­"+"2kru"(0)k2
­"):
So
"ku"(0)k2
@F"·C: (5.6)
Also
"kgk2
@F"·C: (5.7)
8

Now, using Poincar¶ e's and Young's inequalities, from (5.5), (5.7), we get
d
dtku"k2
­"+"d
dtku"k2
@F"+kru"k2
­"·C+"ku"k2
@F":
Integrating in time and using Gronwall's lemma, we ¯nally obtain
ku"k@F";T·C"¡1
2: (5.8)
Moreover, taking the supremum on (0 ; T);from (5.5)-(5.8) we get
supt2(0;T)(ku"k2
­"+"ku"k2
@F") +kru"k2
­";T·C: (5.9)
Let us multiply now equation (1.1) by u"
t. We have:
Zt
0Z
­"(u"
t)2+Zt
0Z
­"ru"¢ ru"
t+"Zt
0Z
@F"(u"
t)2=Zt
0Z
­"fu"
t+"Zt
0Z
@F"gu"
t: (5.10)
Using Young's inequality and evaluating the supremum over (0 ; T);we get:
ku"
tk2
­";T+"ku"
tk2
@F";T+ supt2(0;T)ku"k2
H1
@­(­")·
· ku"
0k2
H1
@­(­")+kfk2
­";T+"ZT
0Z
@F"g2: (5.11)
But
"kgk2
@F"·C: (5.12)
So, assuming the boundedness of our data u0; fandg(u0; f; g6= 0), we obtain:
ku"
tk2
­";T+"ku"
tk2
@F";T+ supt2(0;T)ku"k2
H1
@­(­")·C: (5.13)
In fact, one has
ku"kV·C: (5.14)
Using Lemma 4.2 and classical parabolic estimates, for such a regular solution, one gets
Theorem 5.3 There exists an extension eu"of the solution u"of problem (1.1)-(1.5) into ­, such
that
°°eu"°°
­+°°reu"°°
­;t+°°°°@eu"
@t°°°°
­+°°°°@reu"
@t°°°°
­;t·C; (5.15)
for any t·T:Here, Cdepends on the data and on T.
Proof. The proof follows immediately from the extension lemma and classical parabolic esti-
mates .
Let us introduce the vector »"=Â"»
ru":Recall that µis the weak- ?limit in L1(­) of Â":
Lemma 5.4 There exist a function u2 V(uwill be the unique solution of the limit system (3.1)-
(3.3)) and a function »2 H such that, at least after extraction of a subsequence, we have the
following convergences:
eu"* u weakly in Vand strongly in H; (5.16)
9

@eu"
@t*@u
@tweakly in H; (5.17)
Â"eu"* µu weakly in H; (5.18)
»"=Â"reu"* » weakly in H; (5.19)
"µ@u"
@t; '¶
@F";T!j@Fj
jYjµ@u
@t; '¶
­;T; for all ' 2 D: (5.20)
Proof of Lemma 5.4. (5.16) and (5.17) are direct consequences of the estimates given by Propo-
sition 5.2 and Theorem 5.3. (5.18) follows immediately from the fact that eu"!ustrongly in H
andÂ"* µ, weakly- ?inL1(­):Also, (5.19) follows from our a priori estimates. Indeed, we have
k»"k­;T·C;and hence, up to a sequence, there exists »2 H such that »"* » weakly in H. It
remains to prove (5.20).
Let us consider a test function '2 D. It is easy to see that choosing h= 1 and taking
w"=eu"'t, from (4.13) we get
"Z
@F"u"'td¾=­
¹"
1;eu"'tj­"®
!¹1Z
­u'tdx=j@Fj
jYjZ
­u'tdx;
which, integrating in time and using Lebesgue's convergence theorem, gives exactly (5.20).
Now, let us come back to the ¯rst step of the proof of Theorem 3.1. It remains only to obtain
the limit equation (3.1) satis¯ed by uand»:Let'2 D. We have:
Z
­"u"'tdx=Z
­Â"eu"'tdx!µZ
­u'tdx=jY¤j
jYjZ
­u'tdx; (5.21)
Z
­"ru"r'dx=Z
­Â"reu"¢ r'dx!Z
­»¢ r'dx; (5.22)
"Z
@F"g'd¾ =h¹"
1; g'i !¹1Z
­g'dx =j@Fj
jYjZ
­g'dx: (5.23)
Z
­"f'dx =Z
­Â"f'dx!µZ
­f'dx =jY¤j
jYjZ
­f'dx: (5.24)
So, all the terms in (4.6) pass to the limit, as "!0;and, therefore, we get
¡(jY¤j
jYj+j@Fj
jYj)µ
u;@'
@t¶
­;T+ (»;r')­;T=jY¤j
jYj(f; ')­;T+j@Fj
jYj(g; ')­;T; (5.25)
for any '2 D. But exactly as in the elliptical case (see [ ?]), we have
»i=qij@u
@xj(5.26)
and, ¯nally, putting together (5.20)-(5.26) and having in mind that the solution of the macromodel
is unique, the entire sequence of solutions of the microscopic model converges as necessary. So,
we get (3.1). Passing to the limit with "!0 in (4.7), we get immediately (3.3) and since (3.2) is
obviously satis¯ed, the proof of Theorem 3.1 for regular solutions is complete.
Step 2. The general case; density arguments. Let us prove now the convergence result given
by Theorem 3.1 for the general case, in which the initial data ( u0; v0)2L2(­)£L2
@­(@­") and
f2L2(0; T; L2(­)); g2L2(0; T; H1
0(­)):
10

For doing this, let us consider, as already mentioned, u0
n2H2(­)\H1
0(­); v0
n=u0
nj@­"
such that ( u0
nj­"; v0
n)!(u0j­"; v0) inL2(­")£L2
@­(@­").Also, let fn2C1([0; T]; L2(­)),
gn2C1([0; T]; H1
0(­)) such that fn!finL2(0; T; L2(­)) and gn!ginL2(0; T; H1
0(­)):
For any n;we know that
u"
n2C([0; T];H2(­")\H1
@­(­"))\C1([0; T];L2(­"));(u"
n)t2C((0; T);H1
@­(­")):
Also, we know that all the results given by Proposition 5.2, Lemma 4.2 and Lemma 5.4 hold. So,
the extension»
u"
nconverges strongly in Hor weakly in Vtoun:
The weak formulation for un, given by (3.9) is:
Find un2C([0; T];H2(­)\H1
0(­))\C1([0; T];L2(­)), un(0) = u0
nsuch that:
¡jY¤j
jYjZT
0(un; ')­Ã0(s)ds¡jY¤j
jYj(un(0); ')­Ã(0)¡
¡j@Fj
jYjZT
0(un; ')­Ã0(s)ds¡j@Fj
jYj(un(0); ')­Ã(0) +ZT
0(Qrun;r')­Ã(s)ds=
=jY¤j
jYjZT
0(fn; ')­Ã(s)ds+j@Fj
jYjZT
0(gn; ')­Ã(s)ds; (5.27)
for any '2H1
0(­); Ã2C1([0; T]); Ã(T) = 0 ; Ã(0)6= 0:
Now, by taking subsequences, if necessary, using classical energy estimates for k»
u"¡»
u"
nk, we
can pass to the limit in (5.27), with n! 1 :We get
¡jY¤j
jYjZT
0(u; ')­Ã0(s)ds¡jY¤j
jYj¡
u0; '¢
­Ã(0)¡j@Fj
jYjZT
0(u; ')­Ã0(s)ds¡
¡j@Fj
jYj¡
u0; '¢
­Ã(0) +ZT
0(Qru;r')­Ã(s)ds=
=jY¤j
jYjZT
0(f; ')­Ã(s)ds+j@Fj
jYjZT
0(g; ')­Ã(s)ds
for any '2H1
0(­); Ã2C1([0; T]); Ã(T) = 0 ; Ã(0)6= 0 and also
u(0) = u0;in ­:
So, we have exactly the weak formulation (3.9) of the homogenized problem and this ends the proof
of the theorem.
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