ON THE HOMOGENIZATION OF A TWO-CONDUCTIVITY [610101]

ON THE HOMOGENIZATION OF A TWO-CONDUCTIVITY
PROBLEM WITH FLUX JUMP
RENATA BUNOIUyAND CLAUDIA TIMOFTEz
Abstract. In this paper, we study the homogenization of a thermal diusion problem in a highly
heterogeneous medium formed by two constituents. The main characteristics of the medium are the
discontinuity of the thermal conductivity over the domain as we go from one constituent to another
and the presence of an imperfect interface between the two constituents, where both the temperature
and the
ux exhibit jumps. The limit problem, obtained via the periodic unfolding method, captures
the in
uence of the jumps in the limit temperature eld, in an additional source term, and in the
correctors, as well.
Key words. Homogenization, imperfect interface, the periodic unfolding method.
AMS subject classications. 35B27, 80M35, 80M40.
1. Introduction
Our goal in this paper is to analyze the eective thermal transfer in a periodic
composite material formed by two constituents occupying a domain
in RN(N2),
divided in two open subdomains, denoted by
"
1and
"
2, and separated by an imperfect
interface ". We assume that the phase
"
1is connected and reaches the external
xed boundary @
and that
"
2is disconnected, being the union of domains of size ",
periodically distributed in
with period of order ", where"is a positive real number
less than one. Nevertheless, if N3, our results still hold true if the domain
"
2is
connected, too. The order of magnitude of the thermal conductivity of the material
occupying the domain
"
2is"2, while the conductivity of the material occupying the
domain
"
1is supposed to be of order one. Our problem presents various sources of
singularities: the geometric one related to the interspersed periodic distribution of the
components, the material one related to the conductivities and the ones generated by
the presence of an imperfect interface between the two materials. All these singularities
are described in terms of ".
More precisely, we study the asymptotic behavior, as the small parameter "tends
to zero, of the solution u"= (u"
1;u"
2) of the following problem:
8
>>>><
>>>>:div (A"ru"
1) =fin
"
1;
div ("2A"ru"
2) =fin
"
2;
A"ru"
1
n"="h"(u"
1u"
2)G"on ";
"2A"ru"
2
n"="h"(u"
1u"
2) on ";
u"
1= 0 on@
:
The caseG"= 0, which corresponds to a continuous
ux, proportional to the jump
of the temperature eld across the imperfect interface, has attracted, in the last two
decades, the interest of a broad category of researchers. In the pioneering work [8], using
the asymptotic expansion method, the authors study the homogenization of a thermal
Both authors acknowledge the support from the Laboratoire Europ een Associ e CNRS Franco-
Roumain \Math-Mode".
yInstitut Elie Cartan de Lorraine and CNRS, UMR 7502, Universit e de Lorraine – Metz, France
([anonimizat])
zUniversity of Bucharest, Faculty of Physics, Bucharest-Magurele, P.O. Box MG-11, Romania
([anonimizat])
1

2 HOMOGENIZATION OF A TWO-CONDUCTIVITY PROBLEM WITH FLUX JUMP
problem in a two-component composite with interfacial barrier in the particular case in
which the conductivities of the two components are both of order one. For this problem,
the convergence results were rigorously justied later by using various mathematical
methods: the energy method in [21] and [33], the two-scale convergence method in [23]
and the unfolding method for periodic homogenization in [20], [39], [40] and [32], to
quote just a few of them. Also, for problems involving jumps in the solution in other
contexts, such as heat transfer in polycrystals with interfacial resistance, linear elasticity
problems or problems modeling the electrical conduction in biological tissues, see [4], [5],
[24], [25], [26], [30] and [41]. The case corresponding to the scaling of the conductivities
considered in this paper is addressed, among others, in [6], [36], [35], [34], [22], [28],
[1], [2]. After passing to the limit with respect to the small parameter ", a regularised
model of diusion is obtained, which in fact is a special case of the double-porosity
model, introduced in [37] in the frame of the heat transfer and in [9] in the context
of the
ow in porous media. For a review of such models in various types of ssured
porous media, see, for instance, [38] and the references therein.
In this paper, we consider the case G"6= 0, which corresponds to a discontinous

ux as well. We study here two representative cases for the jump function G", stated
explicitly in Section 2, relations (2.2) and (2.3), which both lead to dierent modied re-
guralized models of diusion. More precisely, in the rst case, a new global source term,
macroscopically distributed over the entire equivalent domain, appears in the right-hand
side of the homogenized equation (3.8). In the second case, the novelty brought by the
presence of the
ux jump is the emergence of the new non homogeneous Neumann cell
problem (3.25) and the presence of its solution in the corrector (3.24). We notice that
this jump plays no role in the homogenized problem (3.23). Nevertheless, in Remark 3.6
we mention a case in which the homogenized problem depends on this jump, too. This
last result is to be compared with the Neumann problem in perforated domains (see [14]
and [18]), where the same cases of G"are considered on the boundary of the holes and
a similar phenomenon occurs. More recently, this type of functions G"is encountered
in [12] for the study of a thermal problem with
ux jump, involving conductivities of
order one and a scaling of the jump in the temperature eld of order "1. We also
point out here the eect of the jump of the solution, which is recovered in the corrector
(3.10) and in the weak limit (3.15), via the solution of the local Robin problem (3.13).
This phenomenon was already noticed in the case in which G"= 0:For transmission
problems involving jump in the
ux in other contexts, such as linear elasticity, theory of
semiconductors, the study of photovoltaic systems, combustion theory or heat transfer
problems, see [7], [10], [11], [27], [29], [31].
Let us notice that if in the microscopic problem the temperature and the
ux are
continuous across "and if moreover the thermal conductivities of the two materials
are both of order one, we have a standard transmission problem, and, then, the limit
process leads to a single-diusion equation. However, if we assume that the thermal
conductivities of the two materials are both of order one, but we keep our jump condi-
tions at the interface, then the limit problem is a system of two coupled equations. The
caseG"= 0 is studied in [23] and the limit process leads to the celebrated Barenblatt
system, introduced in [9]. The case G"6= 0, which leads to a modied Barenblatt model,
is addressed in [13].
The rest of the paper is organized as follows: in Section 2, we introduce the mi-
croscopic problem and we x the notation. In Section 3, we state and prove the main
homogenization results of this paper. Corrector results are given, too. We end our pa-
per with a few concluding remarks, an appendix in which we review the denition and

R. BUNOIU AND C. TIMOFTE 3
the basic properties of the unfolding operators and their adjoints, and some references.
2. Setting of the problem
Let
be a bounded open set in RN(N>2), with a Lipschitz continuous boundary
@
and letY= (0;1)Nbe the reference cell in RN. We suppose that Y1andY2are two
non-empty disjoint connected open subsets of Ysuch thatY2YandY=Y1[Y2. We
also assume that = @Y2is Lipschitz continuous and that Y2is connected.
Throughout the paper, the small parameter "takes values in a positive real sequence
tending to zero and Cis a positive constant independent of ", whose value can change
from line to line.
For eachk2ZN, we denote Yk=k+YandYk
=k+Y, for= 1;2. We also dene
for each",Z"=n
k2ZN:"Yk
2
o
and we set
"
2=S
k2Z"
"Yk
2

and
"
1=
n
"
2. The
boundary of
"
2is denoted by "andn"is the unit outward normal to
"
2.
Our goal in this paper is to analyze the asymptotic behavior, as "!0, of the solution
u"= (u"
1;u"
2) of the following problem:
8
>>>><
>>>>:div (A"ru"
1) =fin
"
1;
div ("2A"ru"
2) =fin
"
2;
A"ru"
1
n"="h"(u"
1u"
2)G"on ";
"2A"ru"
2
n"="h"(u"
1u"
2) on ";
u"
1= 0 on@
:(2.1)
Remark 2.1. We remark that the
ux of the solution is discontinuous across ".
Indeed, we have
A"ru"
1
n""2A"ru"
2
n"=G":
The function f2L2(
) is given. Let gbe aY-periodic function that belongs to
L2(). We dene
g"(x) =gx
"
a.e. on ":
For the given function G"in (2:1), we consider the following two relevant situations
(see, also, [12], [14] and [18]):
Case 1 :G"(x) ="gx
"
;ifM(g)6= 0; (2.2)
Case 2 :G"(x) =gx
"
;ifM(g) = 0: (2.3)
Here,M(g) =1
jjZ
g(y)dydenotes the mean value of the function gon .
Moreover, we make the following assumptions on the data:
(H1)his aY{periodic function such that h2L1() and there exists h02Rwith
0<h0<h(y) a.e. on . We set
h"(x) =hx
"
a.e. on ":

4 HOMOGENIZATION OF A TWO-CONDUCTIVITY PROBLEM WITH FLUX JUMP
(H2) For;2R, with 0<, letM(;;Y ) be the set of all the matrices
A2(L1(Y))NNsuch that for any 2RN,jj2(A(y);)jj2, almost everywhere
inY. For aY-periodic symmetric matrix A2M(;;Y ), we set
A"(x) =Ax
"
a.e. in
:
In order to write the variational formulation of problem (2.1), we introduce, for
every positive "<1, the Hilbert space
H"=V"H1(
"
2):
The space V"=
v2H1(
"
1);v= 0 on@

is endowed with the norm kvkV"=
krvkL2(
"
1), for anyv2V", and the space H1(
"
2) is equipped with the standard norm.
On the space H", we consider the scalar product
(u;v)H"=Z

"
1ru1rv1dx+Z

"
2"2ru2rv2dx+"Z
"(u1u2)(v1v2)dx (2.4)
whereu= (u1;u2) andv= (v1;v2) belong to H". The norm generated by the scalar
product (2.4) is given by
kvk2
H"=krv1k2
L2(
"
1)+"2krv2k2
L2(
"
2)+"kv1v2k2
L2("): (2.5)
The variational formulation of problem (2.1) is the following one: nd u"2H"such
that
a(u";v) =l(v);8v2H"; (2.6)
where the bilinear form a:H"H"!Rand the linear form l:H"!Rare given by
a(u;v) =Z

"
1A"ru1rv1dx+"2Z

"
2A"ru2rv2dx+"Z
"h"(u1u2)(v1v2)dx
and
l(v) =Z

"
1fv1dx+Z

"
2fv2dx+Z
"G"v1dx;
respectively.
We recall in the next lemma a result from [22], which is a key argument allowing
us to prove an existence and uniqueness result and a priori estimates for the solution
of the variational problem (2.6). In the sequel, unless otherwise mentioned, by Cwe
denote a positive constant which is independent of "and whose value can change from
line to line.
Lemma 2.1. For everyvgiven in the space H", the following inequalities hold true:
kv1kL2(
"
1)CkvkH"
and
kv2kL2(
"
2)CkvkH":

R. BUNOIU AND C. TIMOFTE 5
Proof. The rst inequality is a direct consequence of the denition (2.5), together
with the Poincar e inequality applied to functions from the space V", namely
kv1kL2(
"
1)Ckrv1kL2(
"
1): (2.7)
In order to prove the second inequality, we need the following inequalities (see [22]):
kv2kL2(
"
2)C("krv2kL2(
"
2)+"1
2kv2kL2(")) (2.8)
and
"1
2kv1kL2(")C("krv1kL2(
"
1)+kv1kL2(
"
1)): (2.9)
The triangular inequality applied in (2.8), together with (2.9) and (2.7), imply
kv2kL2(
"
2)C("krv2kL2(
"
2)+"1
2kv2v1kL2(")+"1
2kv1kL2("))
C("krv1kL2(
"
1)+"krv2kL2(
"
2)+"1
2kv1v2kL2(")+kv1kL2(
"
1))
C(krv1kL2(
"
1)+"krv2kL2(
"
2)+"1
2kv1v2kL2("));
and the second inequality then follows, by using the denition (2.5). 
Theorem 2.2. For any"2(0;1), the variational problem (2:6)has a unique solution
u"2H". Moreover, there exists a constant C>0, independent of ", such that
ku"
1kL2(
"
1)C;ku"
2kL2(
"
2)C (2.10)
and
kru"
1kL2(
"
1)C; "kru"
2kL2(
"
2)C; "1=2ku"
1u"
2kL2(")C: (2.11)
Proof. In order to prove the existence and the uniqueness of the solution for problem
(2.6), we apply the Lax-Milgram theorem for the space H"endowed with the norm (2.5).
Due to the hypotheses ( H1) and (H2), we easily get that the bilinear form ais coercive
and continuous. Indeed, we have
a(v;v)Ckvk2
H";8v2H";
and
a(u;v)CkukH"kvkH";8u;v2H":
Let us prove now that the linear form lis continuous, i.e.
l(v)CkvkH";8v2H":
One obviously has
jl(v)jkfkL2(
"
1)kv1kL2(
"
1)+kfkL2(
"
1)kv2kL2(
"
2)+Z
"G"(x)v1(x)dx: (2.12)

6 HOMOGENIZATION OF A TWO-CONDUCTIVITY PROBLEM WITH FLUX JUMP
According to Proposition 3.8 in [18], we obtain the estimate of the last term in (2.12)
as follows:
(i) ifG"satises (2.2), then
Z
"G"(x)v1(x)dx=Z
""gx
"
v1(x)dx
"C
"(jM(g)j+")krv1kL2(
"
1)Ckrv1kL2(
"
1);
(ii) ifG"satises (2.3), then
Z
"G"(x)v1(x)dx=Z
"gx
"
v1(x)dx
C
"(jM(g)j+")krv1kL2(
"
1)Ckrv1kL2(
"
1);
sinceM(g) = 0.
Coming back to (2.12), we obtain:
jl(v)jC(kv1kL2(
"
1)+kv2kL2(
"
2)+krv1kL2(
"
1)):
By using Lemma 2.1 and the denition (2.5), we get the continuity of l. Thus, the
Lax-Milgram theorem applies.
In order to obtain the a priori estimates (2.10) and (2.11), we take v=u"in the
variational formulation (2.6). By using the coerciveness of aand the continuity of l, we
obtain
ku"kH"C;
which obviously imply (2.11). Estimates (2.10) are then obtained by applying Lemma
2.1. 
3. Homogenization results
Our goal in this section is to pass to the limit, with "!0, in the variational formu-
lation (2.6) of the problem (2.1). To this end, we make use of the periodic unfolding
method and the general compactness results given in the appendix of this paper.
More precisely, using the a priori estimates (2.10)-(2.11) and the general com-
pactness results from Proposition 5.5, it follows that there exist u12H1
0(
),bu12
L2(
;H1
per(Y1)),bu22L2(
;H1(Y2)) such thatM(bu1) = 0 and up to a subsequence,
for"!0, we get:
T"
1(u"
1)!u1strongly in L2(
;H1(Y1));
T"
1(ru"
1)*ru1+rybu1weakly inL2(
Y1);
T"
2(u"
2)*bu2weakly inL2(
;H1(Y2));
"T"
2(ru"
2)*rybu2weakly inL2(
Y2);
eu"
1*jY1ju1weakly inL2(
);
eu"
2*Z
Y2bu2(x;y)dyweakly inL2(
);(3.1)

R. BUNOIU AND C. TIMOFTE 7
where the space H1
per(Y1) is dened by
H1
per(Y1) =fv2H1(Y1)jvis Y-periodicg:
Remark 3.1. We notice that in (3.1) we omitted to write jYj. Indeed, since Yis the
unit cube, we have jYj= 1and, in order to simplify the presentation, in the sequel we
shall not write it.
LetWper(Y1) =fv2H1
per(Y1)jM(v) = 0g. We introduce the space
V=H1
0(
)L2(
;Wper(Y1))L2

;H1(Y2)

;
equipped with the norm
kVk2
V=krv+rybv1k2
L2(
Y1)+krybv2k2
L2(
Y2)+kvbv2k2
L2(
);
for allV= (v;bv1;bv2)2V.
In order to pass to the limit in (2.6), we need to distinguish between two cases,
depending on the form of the function G".
Case 1 :G"="gx
"
, ifM(g)6= 0.
Theorem 3.1. The unique solution u"= (u"
1;u"
2)of the variational problem (2.6) con-
verges, in the sense of (3.1), to the unique solution (u1;bu1;bu2)2V of the following
unfolded limit problem:
Z

Y1A(y)(ru1+rybu1)(r'+ry1)dxdy+Z

Y2A(y)rybu2ry2dxdy+
Z

h(y)(u1bu2)('2)dxdy=Z

Y1f(x)'(x)dxdy+Z

Y2f(x)2(x;y)dxdy+
jjM(g)Z

'(x)dx; (3.2)
for all'2H1
0(
),12L2(
;H1
per(Y1))and22L2(
;H1(Y2)).
Proof. In order to get the limit problem (3.2), we unfold the variational formulation
(2.6). Then, by using Proposition 5.2 and Lemma 5.3, we obtain
lim
"!0Z

Y1T"
1(A")T"
1(ru"
1)T"
1(rv1)dx+Z

Y2T"
2(A")T"
2("ru"
2)T"
2("rv2)dx+
Z

h(y)(T"
1(u"
1)T"
2(u"
2))(T"
1(v1)T"
2(v2))dx
=
lim
"!0Z

Y1T"
1(f)T"
1(v1)dx+Z

Y2T"
2(f)T"
2(v2)dx+1
"Z

T"
b(G")T"
b(v1)dx
:
In this unfolded problem, we choose the admissible test functions
v1='(x)+"!1(x) 1x
"
; v 2=!2(x) 2x
"
; (3.3)

8 HOMOGENIZATION OF A TWO-CONDUCTIVITY PROBLEM WITH FLUX JUMP
with';!1;!22D(
), 12H1
per(Y1), 22H1(Y2). It is not dicult to see that we have
the following convergences:
T"
1(v1)!'(x) strongly in L2(
Y1); (3.4)
T"
1(rv1)!r'(x)+ry1strongly in L2(
Y1); (3.5)
T"
2(v2)!2(x;y) strongly in L2(
Y2); (3.6)
and
T"
2("rv2)!ry2strongly in L2(
Y2); (3.7)
where  1(x;y) =!1(x) 1(y) and  2(x;y) =!2(x) 2(y).
The passage to the limit with "!0 is standard, by using convergences (3.1) and
(3.4)-(3.7). The only term which needs more attention is the one involving the function
G". For this term, we get:
1
"Z

T"
b(G")T"
b(v1)dx=Z

T"
b
gx
"
T"
b
'(x)+"!1(x) 1x
"
dx=
Z

g(y)T"
b(')(x;y)dxdy+"Z

g(y)T"
b(!1)(x;y)T"
b( 1)(x;y)dxdy!
jjM(g)Z

'(x)dx:
By the density of D(
)
H1
per(Y1) inL2(
;H1
per(Y1)) and ofD(
)
H1(Y2) in
L2(
;H1(Y2)), we obtain (3.2).
We notice that our limit problem (3.2) is similar with the one obtained in [34], the
only dierence being the right-hand side, in which an extra constant term involving the
functiongarises. Indeed, our right-hand side actually writes
Z

F(x)'(x)dx+Z

Y2f(x)2(x;y)dxdy;
with
F(x) =jY1jf(x)+jjM(g):
The existence and the uniqueness for the solution of problem (3.2) is a consequence
of the Lax-Milgram theorem. Due to the uniqueness of ( u1;bu1;bu2)2V, all the above
convergences hold true for the whole sequence. 
Theorem 3.2. The unique solution u"= (u"
1;u"
2)of the variational problem (2.6) con-
verges, in the sense of (3.1), to (u1;bu1;bu2)2V, whereu1is the unique solution of the
homogenized problem

div(Ahomru1(x)) =f(x)+jjM(g)in
;
u1= 0 on@
(3.8)

R. BUNOIU AND C. TIMOFTE 9
and
bu1(x;y) =NX
j=1@u1
@xj(x)j
1(y)in
Y1; (3.9)
bu2(x;y) =u1(x)+f(x)2(y)in
Y2: (3.10)
Here,Ahomis the constant homogenized matrix whose entries are dened, for i;j=
1;:::;N by
Ahom
ij=Z
Y1
aijNX
k=1aik@j
1
@yk!
dy: (3.11)
The vectorial function j
12H1
per(Y1)(j= 1;:::;N ) and the scalar function 22H1(Y2)
are the weak solutions of the following cell problems:
8
<
:divy(A(y)(ryj
1ej)) = 0 inY1;
(A(y)(ryj
1ej))
n= 0 on;
M(j
1) = 0(3.12)
and
divy(A(y)ry2) = 1 inY2;
A(y)ry2
n+h2= 0 on;(3.13)
wherendenotes the unit outward normal to Y2:
Moreover, the weak solution (u"
1;u"
2)of problem (2.6) veries:
eu"
1*jY1ju1weakly inL2(
) (3.14)
and
eu"
2*jY2ju1+Z
Y22(y)dy
fweakly inL2(
): (3.15)
Proof. By choosing '= 0 in the unfolded limit problem (3.2), we obtain:
Z

Y1A(y)(ru1+rybu1)ry1dxdy+Z

Y2A(y)(rybu2)ry2dxdy
Z

h(y)(u1bu2)2dxdy=Z

Y2f2dxdy: (3.16)
Then, taking  2= 0 in (3.16), we get:
Z

Y1A(y)(ru1+rybu1)ry1dxdy= 0;
which implies
divy(A(y)rybu1) = divy(A(y)ru1) in
Y1

10 HOMOGENIZATION OF A TWO-CONDUCTIVITY PROBLEM WITH FLUX JUMP
and
A(y)(ru1+rybu1)
n= 0 on
:
Classical results from the theory of homogenization then imply (3.9) and (3.12).
By choosing now  1= 0 in (3.16), we get:
Z

Y2A(y)(rybu2)ry2dxdyZ

h(y)(u1bu2)2dxdy=Z

Y2f2dxdy;
which implies
divy(A(y)rybu2) =fin
Y2
and
A(y)rybu2
n=h(y)(u1bu2) on
:
This suggests us to search the function bu2of the form
bu2(x;y) =u1(x)+f(x)2(y) in
Y2:
By replacing this form of bu2in the two previous equations, we obtain
divy(A(y)(f(x)ry2)) =f(x) in
Y2;
and
A(y)(f(x)ry2)
n=h(y)f(x)2(y) on
;
which imply that the scalar function 2is the solution of the Robin cell problem (3.13).
By choosing now  1= 2= 0 in (3.2), we obtain:
Z

Y1A(y)(ru1+rybu1)r'dxdy+Z

h(y)(u1bu2)'dxdy=
jY1jZ

f(x)'(x)dx+jjM(g)Z

'(x)dx: (3.17)
We have the equality
Z

h(y)(u1bu2)'dxdy=Z

h(y)f(x)2(y)'(x)dxdy=

Z
h(y)2(y)dyZ

f(x)'(x)dx=Z
A(y)ry2(y)
ndyZ

f(x)'(x)dx=
Z
Y2divy(A(y)ry2(y))dyZ

f(x)'(x)dx=

R. BUNOIU AND C. TIMOFTE 11
Z
Y2(1)dyZ

f(x)'(x)dx=jY2jZ

f(x)'(x)dx;
and then relation (3.17) becomes:
Z

Y1A(y)(ru1+rybu1)r'dxdy=Z

f(x)'(x)dx+jjM(g)Z

'(x)dx:
We integrate this last equality by parts with respect to xand, by using (3.9) and (3.12),
we are led to the homogenized problem (3.8).

Remark 3.2. Due to the right scaling "in front of the function g"given at the interface
", we obtain at the limit a new source term distributed all over the domain
. Our
initial problem (2.1) can be also studied for a nonzero function gwith mean-valueM(g)
equal to zero. In this situation, there is no contribution of gin the right-hand side of
the homogenized equation and, thus, the limit problem is the same as in the case with
nogat all in the microscopic problem.
Remark 3.3. The solution u1(x)of problem (3.8) represents the contribution coming
from the rst material distributed in
"
1and the solution bu2(x;y), verifying relation
(3.10), is an additional contribution coming from the second material distributed in

"
2. This shows that the diusion within the domain
Y2has more than a local
character. The limit problems keep information from the two dierent materials, but on
two dierent scales, and this is an important particularity of such models.(see e.g. [3])
We remark that the homogenized matrix Ahomand, so, the solution u1, are independent
of the function h, while the limit bu2depends on h, via the function 2.
Remark 3.4. All the above results remain true for the case in which the set Y2is not
connected, but consists on a nite number of connected components, as in [20].
We can also state the convergence of the energy and corrector results for the solution
u"= (u"
1;u"
2) of problem (2.1). They are obtained in a classical way, by adapting to our
case the proof of Proposition 4.7 in [20]. We have the following result:
Theorem 3.3. Under the assumptions of Theorem 3.1, if u"= (u"
1;u"
2)is the unique
solution of problem (2.1), then
lim
"!0 Z

"
1A"ru"
1ru"
1dx+Z

"
2"2A"ru"
2ru"
2dx!
=
Z

Y1A(y)(ru1+rybu1)(ru1+rybu1)dxdy+
Z

Y2A(y)rybu2rybu2dxdy; (3.18)
lim
"!0 Z
"
1jru"
1j2dx+Z
"
2jru"
2j2dx!
= 0; (3.19)
T"
1(ru"
1)!ru1+rybu1strongly in L2(
Y1) (3.20)

12 HOMOGENIZATION OF A TWO-CONDUCTIVITY PROBLEM WITH FLUX JUMP
and
T"
2("ru"
2)!rybu2strongly in L2(
Y2): (3.21)
Moreover, the following corrector result holds true:

ru"
1ru1+NX
j=1U"
1@u1
@xj
U"
1
ryj
1

L2(
"
1)!0
and
kru"
2ru1f(x)U"
2(ry2)kL2(
"
2)!0:
Let us analyse now the second relevant situation for the jump function G".
Case 2 :G"(x) =gx
"
, ifM(g) = 0.
Theorem 3.4. The unique solution u"= (u"
1;u"
2)of the variational problem (2.6) con-
verges, in the sense of (3.1), to the unique solution (u1;bu1;bu2)2V of the following
unfolded limit problem:
Z

Y1A(y)(ru1+rybu1)(r'+ry1)dxdy+Z

Y2A(y)rybu2ry2dxdy+
Z

h(y)(u1bu2)('2)dxdy=Z

Y1f(x)'(x)dxdy+
Z

Y2f(x)2(x;y)dxdy+Z

g(y)1(x;y)dxdy; (3.22)
for all'2H1
0(
),12L2(
;H1
per(Y1)),22L2(
;H1(Y2)).
Proof. To obtain the problem (3.22), we pass to the limit in the unfolded form of
the variational formulation (2.6) with the test functions (3 :3), which satisfy (3.4)-(3.7).
The only dierence with respect to the proof of Theorem 3.1 is the passage to the limit
in the term involving the function G". More precisely, we have now:
1
"Z

T"
b(G")T"
b(v1)dx=1
"Z

T"
b
gx
"
T"
b
'(x)+"!1(x) 1x
"
dx=
1
"Z

g(y)T"
b(')(x;y)dxdy+Z

g(y)T"
b(!1)(x;y)T"
b( 1)(x;y)dxdy=
1
"jjM(g)Z

'(x)dx+Z

g(y)!1(x) 1(y)dxdy:
Then, from the fact that M(g) = 0, by using the density of D(
)
H1
per(Y1) in
L2(
;H1
per(Y1)) and ofD(
)
H1(Y2) inL2(
;H1(Y2)), we are led to the unfolded
limit problem (3.22).

R. BUNOIU AND C. TIMOFTE 13
Due to the uniqueness of ( u1;bu1;bu2)2V, which can be proven by the Lax-Milgram
theorem, all the above convergences hold true for the whole sequence and our theorem
is proven.

Remark 3.5. Let us notice that the termZ

g(y)1(x;y)dxdyin (3.22) constitutes
the main dierence with respect to the unfolded equation (3.2), where the term involving
gis a nonzero constant, appearing explicitly in the right-hand side of the homogenized
equation (3.8). This is not the case here, since this term involves now explicitly both
variablesxandy. Our task now is to understand the contribution in the homogenized
problem of this non standard term generated by the discontinuity of the
ux in the
microscopic problem. As we shall see in Theorem 3.5, apart from the cell problems
(3.12) and (3.13), an additional non homogeneous Neumann cell problem needs to be
introduced.
Theorem 3.5. The unique solution u"= (u"
1;u"
2)of the variational problem (2.6) con-
verges, in the sense of (3.1), to (u1;bu1;bu2)2V, whereu1is the unique solution of the
homogenized problem

div(Ahomru1(x)) =f(x)in
;
u1= 0 on@
(3.23)
and
bu1(x;y) =NX
j=1@u1
@xj(x)j
1(y)+(y); (3.24)
bu2(x;y) =u1(x)+f(x)2(y):
Here,Ahomis the homogenized matrix whose entries are given by (3.11) and the func-
tionsj
1and2are dened by (3.12) and (3.13). The Y-periodic function is the
unique solution of the following non homogeneous Neumann cell problem:
8
<
:divy(A(y)ry) = 0 inY1;
A(y)ry
n=g(y)on;
M() = 0:(3.25)
Convergences (3.14) and (3.15) still hold true, with u1solution of (3.23).
Proof. By taking'= 0 in the unfolded limit problem (3.22), we obtain:
Z

Y1A(y)(ru1+rybu1)ry1dxdy+Z

Y2A(y)rybu2ry2dxdy
Z

h(y)(u1bu2)2dxdy=
Z

Y2f(x)2(x;y)dxdy+Z

g(y)1(x;y)dxdy: (3.26)
By choosing  1= 0 in (3.26), we have
Z

Y2A(y)rybu2ry2dxdyZ

h(y)(u1bu2)2dxdy=Z

Y2f(x)2dxdy:

14 HOMOGENIZATION OF A TWO-CONDUCTIVITY PROBLEM WITH FLUX JUMP
By taking now suitable test functions  2, we obtain
divy(A(y)rybu2) =fin
Y2
and
A(y)rybu2
n=h(y)(u1bu2) on
: (3.27)
We then nd the functions bu2and2exactly like in the proof of Theorem 3.2.
Now, let us take  2= 0 in (3.26). We obtain
Z

Y1A(y)(ru1+rybu1)ry1dxdy=Z

g(y)1dxdy:
By taking suitable test functions  1, we obtain
divy(A(y)rybu1) = divy(A(y)ru1) in
Y1; (3.28)
A(y)(rxu1+rybu1)
n=g(y) on
: (3.29)
We remark that (3.27) and (3.29) imply that we also have a discontinuity type
condition:
A(y)(rxu1+rybu1)
nA(y)rybu2
n=h(y)(u1bu2)g(y) on
:
The presence of the function gin relation (3.29) suggests us to search bu1in the
following non standard form:
bu1(x;y) =NX
j=1@u1
@xj(x)j
1(y)+(y); (3.30)
where the functions j
1are dened by (3.12) and the function remains to be found.
To this end, we replace bu1given by (3.30) in (3.28)-(3.29). We obtain:
8
<
:divy(A(y)(rxu1ry1+ry(y)) = divy(A(y)rxu1) in
Y1;
A(y)(rxu1rxu1ry1+ry)
n=g(y) on
;
M() = 0:(3.31)
By using (3.12), we deduce that the scalar function is the unique Y-periodic solution
of the cell problem
8
<
:divy(A(y)ry) = 0 inY1;
A(y)ry
n=g(y) on ;
M() = 0:(3.32)
We observe that (3.32) is a non homogeneous Neumann problem. The compatibility
condition
Z
(y)dy= 0:
is satised, thanks to the hypothesis (2.3) imposed on the function g.

R. BUNOIU AND C. TIMOFTE 15
By choosing now  1= 2= 0 in (3.22), we get:
Z

Y1A(y)(ru1+rybu1)r'dxdy+Z

h(y)(u1bu2)'dxdy=
Z

Y1f(x)'(x)dxdy: (3.33)
Also, since
u1(x)bu2(x;y) =f2(x)2(y) in
Y2;
we have, as in the proof of Theorem 3.2, the equality
Z

h(y)(u1bu2)'dxdy=jY2jZ

f(x)'(x)dx
and relation (3.33) then becomes:
Z

Y1A(y)(ru1+rybu1)r'dxdy=Z

f(x)'(x)dx:
We integrate this last equality by parts with respect to xand, using (3.30) and the
denition (3.11) of the matrix Ahom, we obtain
divx(Ahomru1) =f+divxZ
Y1A(y)r(y)dy
in
;
which leads immediately to the homogenized problem (3.23). We notice that this prob-
lem does not involve the function g. Nevertheless, the in
uence of the
ux jump g
appears in the corrector function bu1, via the cell problem (3.25).

Remark 3.6. The above results can be generalized to the case in which A"is a sequence
of matrices inM(;;
)such that
T"
(A")!Astrongly in L1(
Y);
for some matrix A=A(x;y)inM(;;
Y). The heterogeneity of the medium de-
scribed by such a matrix generates dierent eects in our limit problems (3.2) and
(3.22), respectively. In both situations, due to the fact that the correctors j
1depend
also on the variable x, the new homogenized matrix Ahom
x is no longer constant, but it
depends on x. A more interesting eect occurs in the second case. As proven in Theo-
rem 3.5, if the matrix Adepends only on the variable y, the function is independent
ofxand there is no contribution of the term containing gin the decoupled form of the
limit problem. So, the limit equation is the same as that corresponding to the case with
no jump of the
ux in the microscopic problem. Now, the dependence of Aonxpre-
vents this phenomenon to occur, and, as a consequence, the function ggives an explicit
contribution in the homogenized problem, which becomes
divx(Ahom
xru) =f+divxZ
Y1A(x;y)r(x;y)dy
in
:

16 HOMOGENIZATION OF A TWO-CONDUCTIVITY PROBLEM WITH FLUX JUMP
A similar eect was observed in the homogenization of the Neumann problem in perfo-
rated domains (see [14]).
In this second case too, a corrector result similar to the one stated in Theorem 3.3
holds true. The main dierence now is that the function , solution of the cell problem
(3.25), appears in the correctors of the solution u"= (u"
1;u"
2) of problem (2.1), as well.
Theorem 3.6. Under the assumptions of Theorem 3.4, if u"= (u"
1;u"
2)is the unique
solution of problem (2.1), then (3.18)-(3.21) hold true. Moreover, we have the following
corrector result:

ru"
1ru1+NX
j=1U"
1@u1
@xj
U"
1
ryj
1
U"
1(ry)

L2(
"
1)!0
and
kru"
2ru1f(x)U"
2(ry2)kL2(
"
2)!0:
Remark 3.7. Let us point out that similar corrector results can be stated in the case
in which the matrix Adepends both on xandy, as in Remark 3.6.
4. Conclusions
Using the periodic unfolding method, the eective thermal transfer in a periodic
composite material formed by two constituents, with dierent thermal properties, was
analyzed. The main features of the considered composite material were the discontinuity
of the thermal conductivity over the domain as we go from one constituent to another
and the presence of an imperfect interface between the two constituents, where both
the temperature and the
ux exhibit jumps. The limit problem captures the in
uence
of the jumps in the limit temperature eld, in an additional source term, and in the
correctors, as well.
5. Appendix
We brie
y recall here the denitions and the main properties of the unfolding op-
eratorsT"
1andT"
2, introduced, for a two-component domain, by P. Donato et al. in
[20] (see, also, [14], [15], [16] and [19]) and of the boundary unfolding operator T"
b,
introduced in [16] and [17]. The main particularity of these operators is that they map
functions dened on the oscillating domains
"
1,
"
2and, respectively, ", into functions
dened on the xed domains
Y1,
Y2and
, respectively.
Forx2RN, we denote by [ x]Yits integer part k2ZN, such thatx[x]Y2Yand we
setfxgY=x[x]Yforx2RN. So, for every x2RN, we havex="x
"
Y+x
"
Y

. For
dening the above mentioned periodic unfolding operators, we consider the following
sets (see [20]):
bZ"=
k2ZNj"Yk

;b
"= int[
k2bZ"
"Yk
;"=
nb
";
b
"
=[
k2bZ"
"Yk


;"
=
"
nb
"
;b"=@b
"
2:

R. BUNOIU AND C. TIMOFTE 17
Definition 5.1. For any Lebesgue measurable function 'on
"
,2f1;2g, we dene
the periodic unfolding operators by the formula
T"
(')(x;y) =8
<
:'
"hx
"i
Y+"y
for a.e. (x;y)2b
"Y
0 for a.e. (x;y)2"Y
If'is a function dened in
, for simplicity, we write T"
(')instead ofT"
('j
").
For any function 'which is Lebesgue-measurable on ", the periodic boundary un-
folding operatorT"
bis dened by
T"
b(')(x;y) =8
<
:'
"hx
"i
Y+"y
for a.e. (x;y)2b
"
0 for a.e. (x;y)2"
Remark 5.1. We notice that if '2H1(
"
), thenT"
b(') =T"
(')jb
".
We give now a few useful properties of these operators (see, e.g., [14], [19] and [20]).
Proposition 5.2. Forp2[1;1)and2f1;2g, the operatorsT"
are linear and con-
tinuous from Lp(
"
)toLp(
Y)and
(i) if'and are two Lebesgue measurable functions on
"
, one has
T"
(' ) =T"
(')T"
( );
(ii) for every '2L1(
"
), one has
1
jYjZ

YT"
(')(x;y)dxdy=Z
b
"'(x)dx=Z

"'(x)dxZ
"'(x)dx;
(iii) iff'"g"Lp(
)is a sequence such that '"!'strongly in Lp(
), then
T"
('")!'strongly in Lp(
Y);
(iv) if'2Lp(Y)isY-periodic and '"(x) ='(x="), then
T"
('")!'strongly in Lp(
Y);
(v) if'2W1;p(
"
), thenry(T"
(')) ="T"
(r')andT"
(')belongs to
L2

;W1;p(Y)

.
Moreover, for every '2L1("), one has
Z
b"'(x)dx=1
"jYjZ

T"
b(')(x;y)dxdy:
The following result was proven, for our geometry, in [20].
Lemma 5.3. Ifu"= (u"
1;u"
2)is a sequence in H"and'2D(
), then, for"small enough
and2f1;2gwe have
"Z
"h"(u"
1u"
2)'dx=Z

h(y)(T"
1(u"
1)T"
2(u"
2))T"
(')dxdy:
We remind some general compactness results obtained in [22] for bounded sequences
inH".

18 HOMOGENIZATION OF A TWO-CONDUCTIVITY PROBLEM WITH FLUX JUMP
Lemma 5.4. Letu"= (u"
1;u"
2)be a bounded sequence in H". Then, there exists a
constantC>0, independent of ", such that
kT"
1(u"
1)kL2(
Y1)C;
kT"
2(u"
2)kL2(
Y2)C;
kT"
1(ru"
1)kL2(
Y1)C;
"kT"
2(ru"
2)kL2(
Y2)C;
kT"
2(u"
1)T"
1(u"
2)kL2(
)C:
Proposition 5.5. Letu"= (u"
1;u"
2)be a bounded sequence in H". Then, up to a
subsequence, still denoted by ", there exist u12H1
0(
),bu12L2

;H1
per(Y1)

andbu22
L2

;H1(Y2)

such that
T"
1(u"
1)!u1strongly in L2

;H1(Y1)

;
T"
1(ru"
1)*ru1+rybu1weakly inL2(
Y1);
T"
2(u"
2)*bu2weakly inL2(
;H1(Y2));
"T"
2(ru"
2)*rybu2weakly inL2(
Y2);
eu"
1*jY1j
jYju1weakly inL2(
);
eu"
2*1
jYjZ
Y2bu2(x;y)dyweakly inL2(
);
whereM(bu1) = 0 for almost every x2
and~
denotes the extension by zero of a func-
tion to the whole of the domain
.
Finally, we give for 2f1;2gthe denition of the adjoints Uof the unfolding
operators and we state some useful properties for them (see [14] and [20]).
Definition 5.6. Forp2[1;1), the averaging operators U"
:Lp(
Y)!Lp(
"
), are
given by
U"
()(x) =8
<
:1
jYjZ
Y
"hx
"i
Y+"z;nx
"o
Y
dzfor a.e.x2b
"
;
0 for a.e.x2"
:
It is not dicult to see that these averaging operators are almost left-inverses of the
corresponding unfolding operators T"
, i.e., for any '2Lp(
"
), we have
U"
(T"
('))(x) =8
<
:'(x) for a.e.x2b
"
;
0 for a.e. x2"
:
Proposition 5.7. Forp2[1;1), the operatorsU"
are linear and continuous from
Lp(
Y)toLp(
"
)and
(i)kU"
()kLp(
")!0for every2Lp(
);
(ii) if'"2Lp(
"
), then the following statements are equivalent:
 T"
('")!b'strongly in Lp(
Y)andZ
"
j'"jpdx!0;
 k'"U"
(b')kLp(
")!0.

R. BUNOIU AND C. TIMOFTE 19
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