Homogenization of a degenerate parabolic
problem in a highly heterogeneous medium with
highly anisotropic fibers
A. Boughammoura
University of Monastir, Tunisia
CIMPA 2009
We consider the homogenization of a heat transfer problem in a
periodic medium consisting of a set of highly anisotropic fibers
surrounded by insulating layers, the whole is being embedded in a
third material having a conductivity of order 1. The conductivity
along the fibers is of order 1 but the conductivities in the
transverse directions and in the insulating layers are very small, and
related to the scales µ and λ respectively. We assume that µ (resp.
λ) tends to zero with a rate µ = µ(ε) (resp. λ = λ(ε)), where ε is
the size of the basic periodicity cell. The heat capacities ci of the
i-th component are positive, but may vanish at some subsets such
that the problem can be degenerate (parabolic-elliptic). We show
2
that the critical values of the problem are γ = limε→0 εµ and
2
δ = limε→0 ελ , and we identify the homogenized limit depending
on whether γ and δ are zero, strictly positive, finite or infinite.
THE PHYSICAL and GEOMETRY ASSUMPTION
Now, we introduce a reference periodic medium as follows.
e and Y the unit cube in RN−1 and RN
We denote by Y
e × I, I =]0, 1[. We assume that Y
e is
respectively : Y = Y
e =Y
e1 ∪ Y
e13 ∪ Y
e2 ∪ Y
e23 ∪ Y
e3 where Y
e1 , Y
e2 , Y
e3 are
partitioned as Y
e
e
e
e3 = ∅
three connected open subsets such that Y1 ∩ Y2 = ∅, ∂ Y ∩ Y
e
e
e
and where Yα3 , α = 1, 2 is the interface between Yα and Y3 ; thus
e3 separates Y
e1 and Y
e2 (see Figure 1).
Y
e1
Y
e2
Y
e3
Y
e in 2D
Figure: A typical basic cell Y
For i = 1, 2, 3 we denote χi the characteristic function of
Yi := Yei × I and θ1 , θ2 , θ3 their respective Lebesgue measures are
ei the
supposed to be of the same magnitude order. Let E
N−1
N−1
ei , i.e., E
ei := Y
ei + Z
Z
-translates of Y
and e
Γα3 , α = 1, 2 the
e
e
e2 is
surface separating Eα and E3 . We shall assume that only E
ε
e
e
connected. We introduce the contracted sets Yi := εYi ,
e ε := εE
ei , i = 1, 2, 3 and e
E
Γεα3 := εe
Γα3 , α = 1, 2, where ε is a
i
e ε is filled with a
small positive parameter. We assume that E
1
material having a finite conductivity in the direction xN of the
fibers and a conductivity tending to zero with a rate µ(ε) in the
eε
transverse directions xα , α = 1, 2, · · · , N − 1. The phase E
3
consists of a material with a conductivity that tends to zero with a
e ε is filled by a material with a finite
rate λ(ε), and the matrix E
2
conductivity.
e be a regular bounded domain in RN−1 . We denote by
Now, let Ω
ε
e
e
e ε , and S
e ε := Ω
e ∩e
e × I be the
Ωi := Ω ∩ E
Γα3 . Finally, let Ω := Ω
α3
i
e and a height 1 and
cylinder having a base Ω
ε
ε
e
Ωi := Ωi × I, i = 1, 2, 3.
Henceforth, x = (e
x , xN ) and y = (e
y , yN ) denote points of RN
and Y respectively and by ye and e
x we denote the transverse
vectors (y1 , · · · , yN−1 ) and (x1 , · · · , xN−1 ) respectively. We use
the notation ∂xi for the partial derivative with respect to xi . Let
T > 0 be given. We define, then, the corresponding space-time
domains Q = (0, T ) × Ω and Qiε = (0, T ) × Ωεi , i = 1, 2, 3.
For k = 1, 2, 3, let ck , ak ∈ L∞ (RN ). Here ck is the heat capacity
of the k-th component. These functions are Y -periodic with
respect to y with a period Y and verify the following assumptions
(A1) : 0 ≤ ck (y ) a.e y
(A2) : a0 ≤ ak (y ) a.e y where a0 > 0, independent
of y .
(A3) : a1 is independent of the vertical coordinate,
i.e., : a1 (y ) := a1 (e
y ). The corresponding ε-periodic coefficients are
defined by
x
ckε (x) = ck ( ),
ε
x
akε (x) = ak ( ), x ∈ Ωεk , k = 1, 2, 3.
ε
(1)
Let :
Aε1 (x)
=
a1ε (x)
µ(ε)IN−1
0
0
1
where IN is the matrix of identity.
, Aε2 (x) = a2ε (x)IN , Aε3 (x) = a3ε (x)IN ,
Then, the conductivity tensors of the material filling the sets Eiε
are respectively Aε1 , Aε2 , λ(ε)Aε3 and the global conductivity and
the heat capacity of the medium are respectively
Aε (x) = χε1 (x)Aε1 (x) + χε2 (x)Aε2 (x) + λχε3 (x)Aε3 (x)
and
c ε (x) = χε1 (x)c1ε (x) + χε2 (x)c2ε (x) + χε3 (x)c3ε (x).
where
x
χεk (x) = χk ( ), k = 1, 2, 3.
ε
Let us assume that the lateral and bottom boundaries of Ω are
maintained at a fixed temperature (homogeneous Dirichlet
condition), while the top boundary is insulated (homogeneous
Neumann condition), and that the initial distribution of the
temperature on Ω is given for every ε as
u0ε (x) =
3
X
ε
χεi (x)u0i
(x).
i=1
Then, the evolution of the temperature u ε (t, x) is governed by the
following initial boundary value problem, being in fact, a sequence
of problems Pε indexed by ε :











ε
∂
ε
ε
ε
ε
∂t (c (x)u (t, x))= div(A (x)∇u (t, x))+f (t, x),
∂
ε
x , 1) =
∂xN u (t, e
(t, x) ∈ R∗+×Ω
e t>0
0, e
x ∈ Ω,



u ε (0, x) = u0ε (x), x ∈ Ω






 ε
e × [0, 1[, t > 0.
u (t, e
x , xN ) = 0, (e
x , xN ) ∈ ∂ Ω
where f ε (t, x) = f (t, x), f ∈ L2 (Q) represents a given
time-dependent heat source. The coefficients c ε and aε being
discontinuous, the above equation has to be interpreted in the
sense of distributions on Ω.
To have a weak formulation of the problem which allows us to
handle less regular data and especially give a precise meaning to
the initial and the boundary conditions, we shall use the convenient
mathematical model for the problem Pε , built in [Mabrouk, 2005]
using the functional framework, developed by [Showalter,19997] for
degenerate parabolic equations. Let us recall the precise meaning
of the initial condition.
1 (Ω) as the closed subspace of H 1 (Ω):
We define HLB
n
o
1
HLB
(Ω) = u ∈ H 1 (Ω) : u = 0 on ∂ Ω̃ × [0, 1[
The subscript L (resp. B) stands for the lateral (resp. bottom)
boundary.
1 (Ω), V = L2 (0, T ; V ) and
Let V = HLB
0
1
V = (HLB (Ω))0 , V 0 = L2 (0, T ; V 0 ) be their dual spaces. For every
ε > 0, let B ε , Aε : V → V 0 , the continuous operators are defined
by the two continuous bilinear forms on V × V :
Z
ε
ε
(u, v ) ∈ V × V 7→ hB u, v iV 0 ,V = b (u, v ) =
c ε (x)u(x)v (x)dx
ZΩ
ε
ε
Aε (x)∇u(x)∇v (x)dx
(u, v ) ∈ V × V 7→ hA u, v iV 0 ,V = a (u, v ) =
Ω
Let Vbε be the completion of V with the semi-scalar product,
0
defined by the form b ε and let Vbε be its dual. Then, we have
1
1
0
Vbε = {u : (c ε ) 2 u ∈ L2 (Ω)} and Vbε = {(c ε ) 2 u, u ∈ L2 (Ω)}. The
0
operator B ε admits a continuous extension from Vbε into Vbε
denoted also by B ε . Given f ε ∈ L2 (0, T ; L2 (Ω)) or more generally
0
f ε in V 0 and w0ε in Vbε , we are now able to give a weak
formulation of the above initial-boundary value problem as the
following abstract Cauchy problem :
Find u ∈ V :
d ε
B u + Aε u = f ε ∈ V 0 ,
dt
0
B ε u(0) = w0ε ∈ Vbε
Here, Aε and B ε are the realization of Aε and B ε as operators
from V to V 0 , that is precisely
(Aε u(t), B ε u(t)) = (Aε (u(t)), B ε (u(t))) for a.e. t ∈ (0, T ). Let
us underline that, in the abstract formulation above, we implicitly
d ε
require that
B u belongs to V 0 . This allows us to give a precise
dt
meaning to the initial condition B ε u(0).
0
Indeed, the scalar products in Vbε , Vbε are defined by
Z
c ε (x)u(x)v (x)dx
(B ε u, B ε v )V 0 ε = hB ε u, v iV 0 ε ,V ε = (u, v )Vbε =
b
b
b
Ω
0
Thus, given u0ε in Vbε and w0ε in Vbε related by w0ε = c ε u0ε , we can
express the initial condition by one of the two equivalent equalities
0
1
1
(B ε u ε )(0) = B ε u ε (0) = w0ε ∈ Vbε ⇐⇒ (c ε ) 2 u ε (0) = (c ε ) 2 u0ε ∈ L2 (Ω)
d ε
We define the Hilbert space W2ε (0, T ) =: {u ∈ V : dt
B ∈ V 0 },
then, the abstract Cauchy problem can, thereby, be written more
explicitly as :
PCε :

Find u in W2ε (0, T ) s.t.


d ε
B u(t) + Aε u(t) = f ε (t) ∈ V 0

dt
 ε
0
B u(0) = w0ε in Vbε
for a.e. t ∈ (0, T ),
The initial condition is meaningful since u is in W2ε (0, T ). Some
equivalent variational formulations of the problem PCε are given in
the Proposition 1.2 of [?]. For the present study, we need only the
following useful variational formulation. The function u is a
solution of the problem PCε , if and only if, u is in V and for all v
in H 1 (0, T ; V ) with v (T ) = 0, we have
Z
Z
Z
ε ε 0
ε
ε
− c u v dxdt + µ a1 (x)∇ex u ∇ex vdxdt + a1ε (x)∂xNu ε ∂xNvdxdt
Q1ε
Q
Qε
Z 1
a2ε (x)∇x u ε ∇x vdxdt + λ a3ε (x)∇x u ε ∇x vdxdt
Z
+
Qε
Q3ε
Z2
=
Z
fvdxdt +
Q
Ω
c ε u0ε v (0, x)dx.
Throughout this work, we shall assume that :
u0ε (x) = u0 (x) ∈ L2 (Ω).
Thus u0ε belongs to Vbε and the initial condition is meaningful for
any ε > 0.
Our objective is to study the behavior of the sequence {u ε } as
ε → 0. This will be achieved below. We show that the limit
2
2
depends on the values γ = limε↓0 εµ and δ = limε↓0 ελ . We
consider, in Section 3, the case δ < ∞ which will be processed
according to whether γ is zero, strictly positive, finite or infinite.
The critical case 0 < δ < ∞, which gives the most interesting
results, will be presented with sufficient details. The case δ = ∞ is
shortly considered in Section 4, since the results of the convergence
are exactly the same as in [Mabrouk, Boughammoura,2004, 2007],
to which we refer for the details that remain unchanged. Our
further analysis will be based on the method of the two-scale
convergence [Allaire, 19992, Nguetseng,1989].
The problems of homogenization in composite media with fibers
was considered by many authors in several contexts, see [Bouchité
& Bellieud,1998,2002; Cailleria & Dinari, 1987; Cherednichenko,
Smyshlyaev & Zhikov,2006; Panasenko, 1991; Sili, 2004] and
further references therein. Most of the previous works dealt with
the case of the fiber-reinforced composite material. Motivated by
the study of the effects of the combination of high heterogeneity
and high anisotropy in the overall behavior of composite media
(see [Mabrouk & Boughammoura, 2004, 2007]), we consider here,
a special class of fiber structure exhibiting a non-standard
homogenized problem. Especially, the present work concerns a
degenerate elliptic-parabolic problem in a highly heterogeneous
medium involving highly anisotropic fibers with high contrasts
between the conductivity along the fibers and the conductivities in
the transverse directions, for which, in the critical case, the
homogenized problem is an integro-differential one, displaying
non-locality along the fibers.
Fewer are the papers dealing with highly anisotropic ”coated”
fibers. As far as we know, the closest work in this context seems to
be the recent paper [Cherednichenko, Smyshlyaev & Zhikov,2006],
where the authors studied the elliptic stationary case in a
composite, with highly anisotropic fibers with the so called ”double
porosity” type scaling ( i.e. µ = ε2 ), expressing the high contrast
between the conductivity along the fibers and the conductivity in
the transverse directions. Thus, our work goes beyond that of
[Cherednichenko, Smyshlyaev & Zhikov,2006] at least in three
directions:
1. We consider an evolution ( and degenerate ) problem.
2. We investigate a more complex and a more interesting
transversal geometry.
3. We analyse a fiber-reinforced composite taking into account the
combined effects of fiber coatings together with the highly
anisotropy of the material in the fibers.
Our results reflect these facts and a much more precise comparison
with those results is presented below in remark 3.3.
Lemma
1 (Ω), we
There exists a constant C such that, for every v ∈ HLB
have
||v ||2L2 (Ω) ≤ C ||∂xN v ||2L2 (Ωε ) + ||∇v ||2L2 (Ωε ) + ε2 ||∇v ||2L2 (Ωε )
1
A priori estimates
2
3
A priori estimates
ε2
Let δ = lim . Then
ε↓0 λ
A priori estimates
ε2
Let δ = lim . Then
ε↓0 λ
• Case 1: δ < +∞, f ε = f . There exists a constant C such that :
ku ε kL2 (Q) , ku ε kL∞ (0,T ,V ε ) ≤ C
b
µ k∇ex u ε k2L2 (Q ε ) , k∂xN u ε kL2 (Q ε ) ≤ C
1
1
ε
k∇u kL2 (Q ε ) ≤ C
2
λ k∇u ε k2L2 (Q ε )
3
≤C
A priori estimates
ε2
Let δ = lim . Then
ε↓0 λ
• Case
2:
√ δ = +∞. We
√ introduce the scaled sequences :
λ ε
λ ε
vε =
u and z ε =
v . Then
ε
ε
A priori estimates
ε2
Let δ = lim . Then
ε↓0 λ
• Case
2:
√ δ = +∞. We
√ introduce the scaled sequences :
λ ε
λ ε
vε =
u and z ε =
v . Then
ε
ε
√
• Case 2.1: f ε = χε1 + χε2 + ελ χε3 f . Then the above estimates
are true and we have
ε
ku ε kL2 (Ωε ∪Ωε ) ≤ C , ku ε kL2 (Ωε ) ≤ C + C √
1
2
3
λ
µ k∇ex v ε k2L2 (Q ε ) , k∂xN v ε kL2 (Q ε ) ≤ C
1
1
ε
k∇v kL2 (Q ε ) ≤ C
2
λ k∇v ε k2L2 (Q ε )
3
≤C
ε
ε
kv ε kL2 (Ωε ∪Ωε ) ≤ C + C √ , kz ε kL2 (Ωε ) ≤ C √
1
2
3
λ
λ
A priori estimates
ε2
Let δ = lim . Then
ε↓0 λ
• Case
2:
√ δ = +∞. We
√ introduce the scaled sequences :
λ
λ ε
vε =
u ε and z ε =
v . Then
ε
ε
ε
• Case 2.2: f = f .
µ||∇ex v ε ||2L2 (Q ε ) , k∂xN v ε kL2 (Q ε ) , ||∇x v ε ||2L2 (Q ε ) , λ||∇x v ε ||2L2 (Q ε ) ≤ C
1
2
1
||v ε ||2L2 (Q ε ∪Q ε ) ≤ C , ||v ε ||2L2 (Q ε ) ≤ C + C
1
2
3
3
ε2
λ
ε2
µ||∇ex z ε ||2L2 (Q ε ) , k∂xN z ε kL2 (Q ε ) , ||∇x z ε ||2L2 (Q ε ) , λ||∇x z ε ||2L2 (Q ε ) ≤ C
1
2
3
1
λ
||z ε ||2L2 (Q ε ∪Q ε ) ≤ C
1
2
ε2
, ||z ε ||2L2 (Q ε ) ≤ C
3
λ
Convergence
Definition of the two-scale convergence
Let p > 1 and p 0 its conjugate.
A sequence u ε in Lp (Q) two-scale converges to a function
2s,p
2s
u 0 ∈ Lp (Q × Y ), and we denote this u ε −→ u 0 (u ε −→ u 0 if p=2
), if, for any φ(t, x, y ) ∈ D (Q, C] (Y )),
Z
Z Z
x
ε
u 0 (t, x, y )φ(t, x, y )dtdxdy . (2s)
lim
u (t, x)φ(t, x, )dtdx =
ε→0 Q
ε
Q Y
Convergence (2s) can be extended to test functions
0
0
φ ∈ Lp (Q, C] (Y )) or φ ∈ C(Q, Lp ] (Y )).
Lemma
ε2
. Assume that δ < +∞, γ < +∞ and f ε = f .
ε→0 µ
1 (Ω) , v ∈ L2 Q; H 1 Y
e1 /R
There exists u2 ∈ L2 0, T ; HLB
1
]
Q
(v2 , v3 ) ∈ 3i=2 L2 Q; H]1 (Yi ) /R such that we have the
Let γ := lim
2s
following two-scale (denoted −→) convergences :
2s
1. u ε (t, x) −→χ1 (y )v1 (t, x, ye) + χ2 (y )u2 (t, x) + χ3 (y )v3 (t, x, y )
2s
2. (χε1 u ε , εχε1 ∇ex u ε ) −→ χ1 (y )v1 (t, x, ye), χ1 (y )∇ye v1 (t, x, ye)
2s
3. (χε2 u ε , χε2 ∇x u ε ) −→(χ2 (y )u2 (t, x), χ2 (y )[∇x u2 (t, x) + ∇y v2 (t, x, y )])
2s
4. (χε3 u ε , εχε3 ∇x u ε ) −→(χ3 (y )v3 (t, x, y ), χ3 (y )∇y v3 (t, x, y ))
Moreover, there exists a unique function w3 ∈ L2 Q; H]1 (Y3 )
such that

 v3 (t, x, y ) = u2 (t, x) + w3 (t, x, y ) in Y3
e13 × I
w3 (t, x, y ) = v1 (t, x, ye) − u2 (t, x) on Y13 := Y

e23 × I
w3 (t, x, y ) = 0 on Y23 := Y
and u ε converges weakly in L2 (Q) to the function
Z
Z
U(t, x) = (1 − θ1 ) u2 (t, x) +
v1 (t, x, ye)d ye +
e1
Y
Y3
w3 (t, x, y )dy
Homogenized Problems in the case 0 < δ < ∞
• Case 1 : γ = 0
we shall introduce the following auxiliary functions : Let wk2 ,
k = 1, ..., N, be the unique solutions of the cellular problems
#
"



2

−divy a2 (y ) ∇y wk + ek = 0 in Y2



(cell.2 )
a2 (y ) ∇y wk2 + ek .n(y ) = 0 on Y23



∂w 2


Y − periodic,
 y 7→ wk2 (y ), a2 (y ) k .n(y )
∂y
∂Y2 ∩∂Y
where (e1 , e2 , · · · , eN ) is the canonical basis of RN . Then, we
define the homogenized matrix A2,hom by its entries
Z
=
A2,hom
∇y wk2 + ek ∇y wl2 + el dy
kl
Y2
Homogenized Problems in the case 0 < δ < ∞
• Case 1 : γ = 0
Let η1 ,η2 and η3 be the solutions of the following cellular problems

∂η1


(t,
y
)−
div
a
(y
)∇
η
= δ(t) in Y3
c
(y
)

3
y
3
y
1


∂t





 η1 (t, y ) = 0 on Y13 , η1 (t, y ) = 0 on Y23
(cell.3,1 )




c3 (y )η1 (0, y ) = 0 in Y3






 y 7→ η , Y − periodic
1
Homogenized Problems in the case 0 < δ < ∞
• Case 1 : γ = 0

∂η2


c3 (y )
(t, y )− divy a3 (y )∇y η2 = 0 in Y3



∂t





 η2 (t, y ) = 0 on Y13 , η2 (t, y ) = 0 on Y23
(cell.3,2 )




c3 (y )η2 (0, y ) = c3 (y ) in Y3






 y 7→ η , Y − periodic
2
Homogenized Problems in the case 0 < δ < ∞
• Case 1 : γ = 0

∂η3


c3 (y )
(t, y )− divy a3 (y )∇y η3 = 0 in Y3



∂t





 η3 (t, y ) = −δ(t) on Y13 , η3 (t, y ) = 0 on Y23
(cell.3,3 )




c3 (y )η3 (0, y ) = 1 in Y3






 y 7→ η , Y − periodic
1
where δ(t) is the Dirac measure at t = 0. Then we define the
following exchange coefficients
Z
∂ηq
1
α
Kq (t) =
a3 (y )
dS(y ), α = 1, 2, q = 1, 2, 3.
δ Yα3
∂n3
Homogenized Problems in the case 0 < δ < ∞
• Case 1 : γ = 0
Theorem 1.
1
(uα , v2 , w3 ) ∈ L2 (0, T ; HLB
(Ω))×L2 (Q; H]1 (Y2 )/R)×L2 (Q; H]1 (Y3 ))
are the unique solutions of the following homogenized coupled
problems
:

∂u
1

hom 2
1
10

e
c
−
a
∂
u
+
(u
−
u
)∗
K
−
K

1
1
1
2
1
x
3
2
N
 ∂t


1 − K1 u in Q

=
θ
f
−
f
∗K

1
2 0 
1


∂u
2

2,hom
2
20

e
−
K
c
−
div
A
∇u
+
(u
−
u
)∗
K

1
2
2
2
x
3
2

 ∂t
= θ2 f − f ∗K12 − K22 u0 in Q
Z
Z


hom

e
e
e
a1 (e
y )d ye
cα uα (0, x) = cα u0 (x) in Ω, cα =
cα (y )dy , a1 =


e1

Y
Yα



e × [0, 1[

uα (t, x) = 0 on (0, T ) × ∂ Ω



∂u
∂u


 w3 = − 2 ∗ η1 + η1 ∗ f + η2 u0 + η3 ∗ (u1 − u2 ) , v2 = 2 wk2
∂t
∂xk
0
0
where the prime in K21 , K22 indicates derivation with respect to t.
Homogenized Problems in the case 0 < δ < ∞
• Case 2 : 0 < γ < ∞
Theorem 2.
1
e1 )/R)×L2 (Q; H 1 (Y2 )/R)
(u2 , v1 , v2 , w3 ) ∈ L2 (0, T ; HLB
(Ω))×L2 (Q; H]1 (Y
]
are the unique solutions of the following two-scale homogenized
problems :

1Z
∂w3
∂u2

2,hom

− divx A
∇u2 +
ce2
a3 (y )
(t, x, y )dS(y ) =


∂t
δ Y23
∂n3




θ f in Q


Z
 2
ce2 u2 (0, x) = ce2 u0 (x) in Ω, ce2 =
c2 (y )dy
(∗)

Y2



e × [0, 1[

u2 (t, x) = 0 on (0, T ) × ∂ Ω



∂u


 v2 = 2 wk2
∂xk
Homogenized Problems in the case 0 < δ < ∞
• Case 2 : 0 < γ < ∞
Theorem 2.

∂v1 1

e1

y )∂x2N v1 = f in Y
y )∇ye v1 − a1 (e
− divye a1 (e
hc1 iI


∂t
γ 
∂v1
γ
∂w3
(∗∗)
e13
a1 (e
y)
=
a3 (y )
on (0, T ) × Y


∂n
δ
∂n

3
3
I


e1
hc1 iI (e
y )v1 = hc1 iI (e
y )u0 ye ∈ Y
where h.iI denotes the integration with respect to yN over I.
Homogenized Problems in the case 0 < δ < ∞
• Case 2 : 0 < γ < ∞
Theorem 2.

∂u2
∂w3
1


c3 (y )
(t, x) +
(t, x, y ) − divy a3 (y )∇y w3 (t, x, y )


∂t
∂t
δ


 = f (t, x) in Y
3
(∗∗∗)
 w3 (t, x, y ) = v1 (t, x, ye) − u2 (t, x) on (0, T ) × Y13



w (t, x, y ) = 0 on (0, T ) × Y23


 3
c3 (y )w3 (0, x, y ) = c3 (y ) (u0 (x) − u2 (0, x)) y ∈ Y3
Remark :
Contrary to the previous case, here the problems (∗), (∗∗) and
(∗ ∗ ∗) involve both macroscopic and microscopic variables (x, y ), a
unique macroscopic function u2 and three microscopic functions
v1 , v2 , w3 . The functions v1 , w3 are ”strongly”
coupled
via the
γ
∂w3
∂v1
e13
=
a3 (y )
on (0, T ) × Y
boundary conditions a1 (e
y)
∂n3
δ
∂n3 I
and w3 (t, x, y ) = v1 (t, x, ye) − u2 (t, x) on (0, T ) × Y13 , therefore,
the uncoupling between v1 and w3 could not be possible. As a
consequence, we can not eliminate the microscopic variable.
Moreover, it should be noted that the third term in the first
equation in (∗), the second equation in (∗∗) and (∗ ∗ ∗)
respectively might be interpreted as the combined effects of fiber
coatings together with the highly anisotropy of the fibers in the
overall behavior of the composite.
Remark :
These results should be compared with those of [Panasenko, 1991]
and [Cherednichenko et al., 2006] as follows :
a) The structure of the homogenized operator in (∗), which can be
viewed as the effective properties of the matrix, is the same as in
[Panasenko, 1991] (see Theorem 6).
b) In the stationary case ( ck = 0 a.e. for k ∈ {1, 2, 3}), we obtain
exactly the statement obtained by [Cherednichenko et al. 2006]
(see Theorem 2.2).
Homogenized Problems in the case δ = 0 and δ = ∞
Theorem 3.
The case δ = 0
• When γ = 0, the functions
1
(u, v2 ) ∈ L2 (0, T ; HLB
(Ω)) × L2 (Q; H]1 (Y2 )/R)
are the unique solutions of the following homogenized problems :

∂u
2,hom
hom 2


e
−
a
∂
u
−
div
A
∇u
=f in Q
c
x

1
xN

∂t
Z
Z



hom
 e
a1 (e
y )d ye
c u(0, x) = e
c u0 (x) in Ω, e
c=
c(y )dy , a1 =
e1
Y
Y

e × [0, 1[

u(t, x) = 0 on (0, T ) × ∂ Ω




∂u

 v2 =
w2
∂xk k
Homogenized Problems in the case δ = 0 and δ = ∞
Theorem 3.
The case δ = 0
• When 0 < γ < ∞, the functions
1
e1 )/R)×L2 (Q; H 1 (Y2 )/R)
(u2 , v1 , v2 ) ∈ L2 (0, T ; HLB
(Ω))×L2 (Q; H]1 (Y
]
are the unique solutions of the following two-scale homogenized
problems :

∂u2
2,hom


e
−
div
A
∇u
c
2 = θ2 f in Q
2
x


Z
∂t



 ce2 u2 (0, x) = ce2 u0 (x) in Ω, ce2 =
c2 (y )dy
Y2

e × [0, 1[

u2 (t, x) = 0 on (0, T ) × ∂ Ω




∂u

 v2 = 2 wk2
∂xk
Homogenized Problems in the case δ = 0 and δ = ∞
Theorem 3.
The case δ = 0
• When 0 < γ < ∞, the functions
1
e1 )/R)×L2 (Q; H 1 (Y2 )/R)
(u2 , v1 , v2 ) ∈ L2 (0, T ; HLB
(Ω))×L2 (Q; H]1 (Y
]
are the unique solutions of the following two-scale homogenized
problems :

∂v1 1


e1
− divye a1 (e
y )∇ye v1 − a1 (e
y )∂x2N v1 = f in Y
hc i


 1 I ∂t
γ
∂v1
e13
a1 (e
y)
= 0 on (0, T ) × Y


∂n

1


e1
hc1 i (e
y )v1 = hc1 i (e
y )u0 ye ∈ Y
I
I
Homogenized Problems in the case δ = 0 and δ = ∞
Theorem 3.
Homogenized Problems in the case δ = 0 and δ = ∞
Theorem 3.
The case δ = ∞
• When γ = 0,
1
(uα , v2 , w3 ) ∈ L2 (0, T ; HLB
(Ω))×L2 (Q; H]1 (Y2 )/R)×L2 (Q; H]1 (Y3 ))
are the unique solutions of the following homogenized coupled
problems :

∂u1


− a1hom ∂x2N u1 (t, x) = θ1 f in Q
ce1



∂t


∂u2

2,hom

e
c
−
div
A
∇u
= θ2 f in Q

2
x
2

Z
Z
 ∂t
hom
e
e
e
cα uα (0, x) = cα u0 (x) in Ω, cα =
cα (y )dy , a1 =
a1 (e
y )d ye

e1

Yα
Y



e × [0, 1[

uα (t, x) = 0 on (0, T ) × ∂ Ω



∂u2 2


w
 w3 = u0 , in ∈ Y3 , v2 =
∂xk k
Homogenized Problems in the case δ = 0 and δ = ∞
Theorem 3.
The case δ = ∞
• When 0 < γ < ∞,
1
e1 )/R)×L2 (Q; H 1 (Y2 )/R)
(u2 , v1 , v2 , w3 ) ∈ L2 (0, T ; HLB
(Ω))×L2 (Q; H]1 (Y
]
are the unique solutions of the following two-scale homogenized
problems :

∂u

ce2 2 − divx A2,hom ∇u2 = θ2 f in Q


∂t
Z



 ce2 u2 (0, x) = ce2 u0 (x) in Ω, ce2 =
c2 (y )dy
Y2

e × [0, 1[,

w3 = u0 , y ∈ Y3 , u2 (t, x) = 0 on (0, T ) × ∂ Ω




∂u

 v2 = 2 wk2 .
∂xk
Homogenized Problems in the case δ = 0 and δ = ∞
Theorem 3.
The case δ = ∞
• When 0 < γ < ∞,
1
e1 )/R)×L2 (Q; H 1 (Y2 )/R)
(u2 , v1 , v2 , w3 ) ∈ L2 (0, T ; HLB
(Ω))×L2 (Q; H]1 (Y
]
are the unique solutions of the following two-scale homogenized
problems :

∂v1 1


e1
− divye a1 (e
y )∇ye v1 − a1 (e
y )∂x2N v1 = f in Y
hc i


 1 I ∂t
γ
∂v1
e13
a1 (e
y)
= 0 on (0, T ) × Y


∂n

1


e1
hc1 i (e
y )v1 = hc1 i (e
y )u0 ye ∈ Y
I
I