Electronic Journal of Differential Equations, Vol. 2000(2000), No. 15, pp. 1–11.
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
ftp ejde.math.swt.edu ftp ejde.math.unt.edu (login: ftp)
A diffusion equation for composite materials ∗
Mohamed El Hajji
Abstract
In this article, we study the asymptotic behavior of solutions to the
diffusion equation with non-homogeneous Neumann boundary conditions.
This equation models a composite material that occupies a perforated
domain, in RN , with small holes whose sizes are measured by a number rε .
We examine the case when rε < εN/(N−2) with zero-average data around
the holes, and the case when limε→0 rε /ε = 0 with nonzero-average data.
1
Introduction
As a model for a composite material occupying a perforated domain in RN ,
the diffusion equation with non-homogeneous boundary conditions has been the
object of many studies. In particular, we are interested in the properties of the
solution to the equation
− div (Aε ∇uε ) = f
in Ωε ,
ε
(A (x)∇uε ) · n = hε on ∂Sε ,
uε = 0 on ∂Ω,
(1)
where Ωε is the perforated domain obtained by extracting a set of holes Sε from
Ω, f and hε are given functions, and Aε is an operator in the space
M (α, β; Ω) =
2
A ∈ [L∞ (RN )]N : (A(x)λ, λ) ≥ α|λ|2 ,
|A(x)λ| ≤ β|λ| ∀λ ∈ RN , p.p · x ∈ Ω
which is defined for all real numbers α and β.
When hε ∈ L2 (∂Sε ) and the domain has holes of size rε , solutions to (1) have
been studied by D. Cioranescu and P. Donato [3] for rε = ε and Aε (x) = A( xε )
with A ∈ M (α, β; Ω), and by C. Conca and P. Donato [5] for A = I and rε ε.
Using the concept of H 0 -convergence introduced by M. Briane et al. [2], P.
Donato and M. El Hajji [6] showed convergence of solutions in not-necessarily
∗ 1991 Mathematics Subject Classifications: 31C40, 31C45, 60J50, 31C35, 31B35.
Key words and phrases: Diffusion equation, composite material,
asymptotic behavior, H 0 -convergence.
c
2000
Southwest Texas State University and University of North Texas.
Submitted October 14, 1999. Published February 22, 2000.
1
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Diffusion equation for composite materials
EJDE–2000/15
periodic domains. The H 0 -convergence is proven by showing strong convergence
in H −1 (Ω) of the distribution, concentrated on the boundary of Sε , given by
hνhε , ϕiH −1 (Ω), H01 (Ω) = hhε , ϕiH −1/2 (∂Sε ), H 1/2 (∂Sε ) ,
∀ϕ ∈ H01 (Ω).
(2)
This method allows the study the asymptotic behavior of solutions to (1) for
hε ∈ L2 (∂Sε ) with rε > εN/N −2 (see [6]), and for perforated domains with
double periodicity with hε ∈ H −1/2 (∂Sε ) and rε /ε → 0 as ε → 0 (see T. Levy
[8]).
In this article, we study the perforated domains with rε < εN/N −2 , and
perforated domains such that rε /ε → 0 as ε → 0. In these situation the distribution given by (2) does not converge strongly in H −1 (Ω), and so the method
described above can not be applied. In spite of this, we describe the asymptotic
behavior of solutions to (1) using oscillating test functions. This method was
introduced by L. Tartar [10] and has been used by many authors.
2
Statement of the main result
Let Ω be a bounded open set of RN , Y = [0, l1 [×.. × [0, lN [ be a representative
cell, and S be an open set of Y with smooth boundary ∂S such that S ⊂ Y . Let
ε and rε be terms of positive sequences such that rε ≤ ε. Let c denote positive
constants independent of ε. We denote by τ (rε S) the set of translations of rε S
of the form εk1 + rε S with k ∈ ZN . Let kl = (k1 l1 , .., kN lN ) represent the holes
in RN .
We assume that the holes τ (rε S) do not intersect the boundary ∂Ω. If Sε is
the set of the holes enclosed in Ω, it follows that there exists a finite set K ∈ ZN
such that
[
Sε =
rε (kl + S).
k∈K
We set
Ωε = Ω \ Sε ,
(3)
and denote by χΩε the characteristic function of Ωε . Let Vε denote the Hilbert
space
Vε = v ∈ H 1 (Ωε ), v|∂Ω = 0
equipped with the H 1 -norm. Let A(y) = (aij (y))ij be a matrix such that
A ∈ L ∞ RN
N 2
,
A is Y-periodic, and there there exist α > 0 such that
N
X
aij (y) λi λj ≥ α |λ|2 ,
a.e. y in RN ,
∀ λ ∈ RN .
(4)
i,j=1
We note that for every ε > 0,
x
Aε (x) = A( )
ε
a.e. x in RN .
(5)
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Mohamed El Hajji
3
In this paper, we study the system
− div(Aε ∇uε ) = 0 in Ωε ,
(Aε ∇uε ).n = hε on ∂Sε ,
uε = 0,
(6)
on ∂Ω,
where Ωε is given by (3), Aε is given by (5), and hε is given by
hε (x) = h(
x
),
rε
(7)
where h ∈ L2 (∂S) is Y -periodic function. Set
Z
1
h dσ .
Ih =
|Y | ∂S
(8)
6 0, and the case where
We examine the case where rε < εN/(N −2) with Ih =
limε→o rε /ε = 0 with Ih = 0. The following result describes the asymptotic
behavior of the solution to (6) in the two cases.
Theorem 1 Let uε be the solution of (6). Suppose that one of the following
hypotheses is satisfied

rε
 lim
=0
if N > 2,
N/(N
ε→0
ε−2 −2)
(9)
Ih 6= 0 and
−1
 lim ε (ln(ε/rε )) = 0 if N = 2,
ε→0
or
rε
= 0.
(10)
ε
Then, for every ε > 0, there exists an extension operator Pε defined from Vε to
H01 (Ω) satisfying
Pε ∈ L Vε , H01 (Ω) ,
(11)
Ih = 0
such that
(
and
lim
ε→0
(Pε v)|Ωε = v ∀v ∈ Vε ,
(12)
k∇ (Pε v)k(L2 (Ω))N ≤ C k∇vk(L2 (Ωε ))N , ∀v ∈ Vε .
(13)
rε −N/2
)
Pε uε * u
ε
weakly in H01 (Ω),
for N > 2
and
Pε [(
rε −1/2
ε
)
(log )−1/2 uε ] * u
ε
rε
weakly in H01 (Ω),
where u is the solution of the problem
− div(A0 .∇u) = 0 in Ω,
u = 0 on ∂Ω,
for N = 2,
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Diffusion equation for composite materials
EJDE–2000/15
and the matrix A0 = (a0ij )ij has entries
a0ij =
1
|Y |
Z
Y
(aji −
N
X
aki
k=1
∂χj
dy),
∂yk
(14)
in Y
(15)
and χj is a Y -periodic function that satisfies
− div(At ∇(yj − χj )) = 0
Remark. One can replace the first equation of system (6) by
− div(Aε ∇uε ) = fε
in Ωε ,
with
( rεε )−N/2 fε * f
( rεε )−1 (ln(ε/rε ))−1/2 fε
weakly in L2 (Ω)
*f
if N > 2,
2
weakly in L (Ω)
if N = 2.
Then u will be the solution of
− div(A0 .∇u) = f in Ω,
u = 0 on ∂Ω.
This approach has been used in [5] for the case A = I, in [3] when A = I and
rε = ε, and in [6] for the case where rε > εN/(N −2) using the H 0 -convergence
and some arguments given by S. Kaizu in [7].
3
Proof of the main result
Observe first that Sε is admissible in Ω, in the sense of the H 0 -convergence
([6, 4, 5, 10]). Then there exists an extension operator Pε satisfying (14).
On the other hand, the matrix A0 can be defined by
Z
1
t
0
t
A λ = MY ( A∇wλ ) =
A∇w − λ dy, ∀λ ∈ RN ,
|Y | Y
where for every λ ∈ R, wλ is the solution of the problem
− div(t A∇wλ ) = 0
with wλ − λy
For x ∈ RN , let
in Y,
(16)
Y -periodic.
x
wλε (x) = εwλ ( ).
ε
To simplify notation, let
(rε /ε)−N/2
δε =
(rε /ε)−1 (ln(ε/rε ))−1/2
(17)
if N > 2,
if N = 2.
(18)
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Mohamed El Hajji
5
Taking uε as a test function in the variational formulation of (6), and using the
classical techniques of a priori estimates, one can easily show the existence of
two constants c and c’ independent of ε such that
c0 ≤ k∇(δ ε uε )kL2 (Ωε ) ≤ c.
Hence, from (13), up to a subsequence,
Pε (δ ε uε ) * u
weakly in H01 (Ω).
(19)
Set now ξ ε = Aε ∇[Pε (δ ε uε )]. Then using (19), (11)-(13) and (4)-(5), one shows
that ξ ε is bounded in L2 (Ω), and so up to a subsequence
ξε * ξ
weakly in L2 (Ω).
(20)
Case where (9) is satisfied. Let φ ∈ D(Ω). Then from the variational
formulation of (6), one has
Z
Z
χΩε ξ ε .∇φ dx = δ ε
hε φ dσ .
(21)
Ω
∂Sε
If N (ε) denotes the number of the holes included in Ω, one has then
Z
XZ
x
ε
ε
hε φ dσ| ≤ kφkL∞ (Ω) δ
|h( )| dσ(x)
|δ
r
ε
∂Sε
k∈K ∂(rε (S+k))
Z
|h| dσ
≤ cδ ε N (ε)rεN −1
(22)
∂S
cδ ε
≤
rεN −1
|∂S|1/2 khkL2 (∂S) .
εN
From (18), one can write
rN −1
δε ε N
ε
(
=
N −2
( rεεN )1/2
[ε−2 (ln(ε/rε ))−1 ]1/2
and so in virtue of (9),
lim δ ε
ε→0
hence
lim δ
ε→0
ε
if N > 2,
if N = 2,
rε N −1
= 0,
εN
(23)
Z
∂Sε
hε φ dσ = 0 .
On the other hand, it is easy to show that
χΩε → 1 strongly in Lp (Ω)
hence
Z
Ω
ε
χε ξ .∇φ dx →
∀p ∈ [1, ∞[,
Z
Ω
ξ.∇φ dx.
(24)
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Diffusion equation for composite materials
One can deduce that, as ε → 0 in (21),
Z
ξ∇φ dx = 0,
EJDE–2000/15
∀φ ∈ D(Ω).
Ω
Consequently
− div ξ = 0
in Ω.
(25)
It remains to identify the function ξ. Let wλε be the function defined by (16)(17). Then
wλε * λx
weakly in H 1 (Ω), and so Lp (Ω) strong ∀p < 2∗
where 1/2∗ = 1/2 − 1/N, with N ≥ 2. Let φ ∈ D(Ω), by choosing φwλε as a test
function in the variational formulation of (6), one has
Z
Z
ξ ε ∇(φwλε ) dx = δ ε
hε φwλε dx .
(26)
Ωε
∂Sε
To pass to the limit as ε → 0 in (26), we set
Z
ξ ε ∇(φwλε ) dx = J1ε + J2ε ,
(27)
Ωε
Z
where
J1ε
=
ξ
Ωε
ε
∇φ.wλε
dx
J2ε
and
Z
=
Ωε
ξ ε ∇wλε .φ dx.
Using the results given by G. Stampacchia in [9] (see also [1]), one can deduce
that wλ ∈ L∞ (Y ), so
χΩε wλε → λx
strongly in L2 (Ω).
This with convergence (20), gives
Z
Z
J1ε =
χΩε wλε ξ ε ∇φ dx →
λxξ∇φ dx
Ω
as ε → 0 .
Ω
(28)
(29)
Now, we may write
J2ε =
One the one hand,
Z
Z
ξ ε ∇wλε φ dx =
Ω
t
Ω
= −
Z
Z
Ω
ξ ε ∇wλε φ dx −
Z
Sε
ξ ε ∇wλε φ dx .
Aε ∇wλε ∇[Pε (δ ε uε )φ] dx −
Ω
t
Aε ∇wλε ∇φPε [(δ ε uε ] dx
Z
Ω
t
(30)
Aε ∇wλε ∇φPε [(δ ε uε ] dx
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Mohamed El Hajji
7
because, from the definition of wλε ,
Z
t ε
A ∇wλε ∇[Pε (δ ε uε )φ] dx = 0 .
Ω
From the definition of A0 , t Aε ∇wλε * A0 λ weakly in L2 (Ω). From (19), up to
a subsequence,
Pε (δ ε uε ) → u strongly inL2 (Ω),
which implies
Z
t
Ω
ε
A
∇wλε ∇φPε (δ ε uε ) dx
Z
Hence
Ω
On the other hand,
Z
|
Sε
ξ
ε
∇wλε φ dx
Z
→
Z
→−
Ω
Ω
A0 λu∇φ dx .
A0 λ∇φu dx .
ξ ε ∇wλε φ dx| ≤ ckξ ε kL2 (Ω) k∇wλε kL2 (Sε ) .
Since kξ ε kL2 (Ω) is bounded,
Z
ξ ε ∇wλε φ dx| ≤ ck∇wλε kL2 (Sε ) .
|
Sε
Note that
k∇wλε k2L2 (Sε )
Z
=
|(∇wλε )(x)|2 dx
XZ
|(∇wλε )(x)|2 dx
Sε
=
k∈K
rε (S+k)
XZ
x
|(∇y wλ )( )|2 dy
ε
k∈K rε (S+k)
Z
|(∇y wλ )(y)|2 dy
= N (ε)εN
=
rε
ε
Z
≤ c
rε
ε
S
S
|∇wλ |2 dy .
Since rε /ε → 0 and wλ ∈ H 1 (Y ), it follows that
Z
|∇wλ |2 dy → 0 .
rε S/ε
Using (33), one has
Z
Sε
ξ ε ∇wλε φ dx → 0
as ε → 0.
(31)
(32)
(33)
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Diffusion equation for composite materials
EJDE–2000/15
This, with (30) and convergence (31) imply that
Z
J2ε → − A0 λ∇φu dx.
(34)
Ω
Next we pass to the limit in the right hand of (26). With the same argument
as in (22),
Z
Z
|δ ε
hε φwλε dσ| ≤ cδ ε N (ε)rεN −1
|hwλ | dσ
∂Sε
∂S
rN −1
cδ ε ε N kwλ kL2 (∂S) |∂S|1/2 khkL2 (∂S) .
ε
≤
Since we have shown that
lim δ ε
ε→0
rεN −1
= 0,
εN
from (9) one deduces that
δ
ε
Z
∂Sε
hε φwλε dσ → 0.
Finally, by passing to the limit as ε → 0 in (26), and using (32) and (34) one
obtains
Z
Z
λxξ∇φ dx −
A0 λ∇φu dx = 0,
Ω
Ω
hence, from (25) it follows that
Z
Z
ξλφ dx =
A0 λ∇uφ dx∀φ ∈ D(Ω), ∀λ ∈ RN ,
Ω
Ω
i.e., ξ = A0 ∇u.
Case where (10) is satisfied. Let φ ∈ D(Ω). Then from the variational
formulation of (6),
Z
Z
χΩε ξ ε .∇φ dx = δ ε
hε φ dσ.
(35)
Ω
∂Sε
The arguments used the proof of (24) can be applied here to obtain
Z
Z
ε
χΩε ξ .∇φ dx →
ξ.∇φ dx .
Ω
Ω
To pass to the limit in the right-hand side of (35), we introduce N as the
solution to
− div N = 0 in S,
N.n = −h on ∂S .
EJDE–2000/15
Mohamed El Hajji
9
The existence of N is assured by the hypothesis Ih = 0. Set Nε (x) = N (
for x in (εY \ rε S)k . Then
Z
Z
∂Sε
hence
δε|
Z
∂Sε
x − εk
),
rε
hε φ dσ =
Sε
∇φ.Nε dx,
hε φ dσ| ≤ δ ε k∇φkL2 (Sε ) kNε kL2 (Sε ) .
Note that kNε kL2 (Sε ) ≤ c( rεε )N/2 kN kL2 (S) , so
δε|
Z
∂Sε
hε φ dσ| ≤ cδ ε (
Since
rε
δ ( )N/2 =
ε
ε
rε N/2
)
k∇φkL1 (Sε ) .
ε
1
(ln(ε/rε ))−1/2
(36)
if N > 2
if N = 2,
it follows from (36), when N = 2, that
Z
lim δ ε |
hε φ dσ| = 0 .
ε→0
For N > 2, one has
ε
δ |
∂Sε
Z
∂Sε
hε φ dσ| ≤ ck∇φkL2 (Sε ) .
Since χΩε → 1 strongly in Lp (Ω), for all p ∈ [1, ∞[ and φ ∈ D(Ω), one deduces
that
Z
(1 − χε )|∇φ|2 dx → 0 .
Ω
Hence, by passing to the limit as ε → 0 in (35), one obtains
Z
ξ.∇φ dx = 0 ,
Ω
then − div ξ = 0 in Ω.
Let wλε be the function defined by (16)-(17) and φ ∈ D(Ω). As in the previous
case, by using φwλε as a test function in the variational formulation of (6), one
has
Z
Z
ξ ε .∇(φwλε ) dx = δ ε
hε φwλε dσ .
Ωε
∂Sε
From (27), (29) and (34), one has
Z
Z
Z
ξ ε .∇(φwλε ) dx →
λx.∇φ dx −
A0 λ.∇φu dx.
Ωε
Ω
Ω
(37)
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Diffusion equation for composite materials
Now we show that
δε|
Z
∂Sε
EJDE–2000/15
hε φwλε dσ| → 0.
(38)
One has
δε |
Z
hε φwλε dσ|
Z
ε
δ |
∇(φwλε ).Nε dx|
Sε
Z
Z
ε
ε
ε
δ |
∇φ.wλ .Nε dσ| + δ |
∂Sε
=
≤
≤
≤
≤
≤
Sε
Sε
φ.∇wλε .Nε dσ|
δ ε kNε kL2 (Sε ) k∇φwλε kL2 (Sε ) + δ ε kNε kL2 (Sε ) kφ∇wλε kL2 (Sε )
rε
cδ ε ( )N/2 k∇φwλε kL2 (Sε ) + kφ∇wλε kL2 (Sε )
ε
rε
cδ ε ( )N/2 k∇φkL∞ (Ω) kwλε kL2 (Sε ) + kφkL∞ (Ω) k∇wλε kL2 (Sε ) .
ε
ε
ε rε N/2
cδ ( )
kwλ kL2 (Sε ) + k∇wλε kL2 (Sε ) .
ε
Note that
k∇wλε k2L2 (Sε )
Z
=
Sε
|∇wλε |2
≤ N (ε)εN
Since wλ ∈ H 1 (S) and rε /ε → 0,
Z
lim
ε→0
rε
ε
dx =
k∈K
Z
rε
ε
S
XZ
S
rε (S+k)
|∇wλ |2 dx ≤ c
|∇wλε |2 dx
Z
r−ε
ε S
|∇wλ |2 dx .
|∇wλ |2 dx = 0 .
Hence k∇wλε kL2 (Sε ) → 0. On the other hand, one has kwλε kL2 (Sε ) ≤ c. Finally,
as
rε
lim δ ε ( )N/2 = 0 ,
ε→0
ε
one deduces (38). This and (37) completes the proof, using the same arguments
as in the previous case.
Acknowledgments The author would like to thank Professor Patrizia Donato for her help on this work.
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Mohamed El Hajji
11
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Mohamed El Hajji
Université de Rouen, UFR des Sciences
UPRES-A 60 85 (Labo de Math.)
76821 Mont Saint Aignan, France
e-mail: Mohamed.Elhajji@univ-rouen.fr