Proc. Indian Acad. Sci. (Math. Sci.) Vol. 112, No. 1, February 2002, pp. 195–207.
© Printed in India
Homogenization of a parabolic equation in perforated domain
with Neumann boundary condition
A K NANDAKUMARAN and M RAJESH∗
Department of Mathematics, Indian Institute of Science, Bangalore 560 012, India
Corresponding author
E-mail: nands@math.iisc.ernet.in; rajesh@math.iisc.ernet.in
∗
Abstract. In this article, we study the homogenization of the family of parabolic
equations over periodically perforated domains
∂t b
x
, uε − diva
, uε , ∇uε = f (x, t) in ε × (0, T ),
ε
ε
x
a
, uε , ∇uε .νε = 0 on ∂Sε × (0, T ),
ε
uε = 0 on ∂ × (0, T ),
uε (x, 0) = u0 (x) in ε .
x
Here, ε =  \ Sε is a periodically perforated domain. We obtain the homogenized equation and corrector results. The homogenization of the equations on a fixed
domain was studied by the authors [15]. The homogenization for a fixed domain and
b xε , uε ≡ b(uε ) has been done by Jian [11].
Keywords.
Homogenization; perforated domain; two-scale convergence; correctors.
1. Introduction
Let  be a bounded domain in Rn with smooth boundary ∂. Let T > 0 be a constant
and let ε > 0 be a small parameter which eventually tends to zero. Let Y = (0, 1)n and
S(closed) ⊂ Y . We define a periodically perforated domain ε as follows: First define
Iε = {k ∈ Zn : ε(k + S) ⊂ } and,
[
ε(k + S).
Sε =
(1.2)
ε = \ Sε .
(1.3)
(1.1)
k∈Iε
Set
We assume that, (i) ∂ε is smooth and, (ii) ε is connected. Note that in the definition
of the holes Sε we have taken them to have no intersection with the external boundary
∂. This is only done to avoid technicalities and the results of our paper will remain valid
without this assumption (cf. [2] for a refinement of our arguments).
195
196
A K Nandakumaran and M Rajesh
We consider the following family of equations with Neumann conditions on the boundary
of the holes
x
x
, uε − diva
, uε , ∇uε = f (x, t) in ε × (0, T ),
∂t b
ε
x ε
, uε , ∇uε .νε = 0 on ∂Sε × (0, T ),
a
ε
uε = 0 on ∂ × (0, T ),
(1.4)
uε (x, 0) = u0 (x) in ε ,
where u0 is a given function on  and f is a given function on  × (0, T ). For given
ε, the Cauchy problem (1.4) will also be denoted by (Pε ). It is known that under suitable
assumptions on a and b (cf. assumptions (A1)–(A4) below), that the problem (Pε ) has a
solution uε . Our aim in this paper is to study the homogenization of the equations (Pε )
as ε → 0 i.e. to study the limiting behaviour of uε as ε → 0 and obtain the limiting
equation satisfied by the limit. The theory of homogenization is a well-developed area
of research [5,12] and we are not going into the details of this theory. The case when b
is linear (i.e. b(y, s) = s) has been studied quite widely [4, 6–8, 10,16]. When b is not
linear, the homogenization of the equation in a fixed domain was studied by Jian [11] for
b(y, s) ≡ b(s) and in the general case, by the authors [15].
Assuming that ueε are a priori bounded in L∞ ( × (0, T )) (where e denotes the
extension of uε by zero in the holes), we show that ueε converges, for a subsequence, to a
solution of the homogenized problem
∂t b∗ (u) − div A(u, ∇u) = m∗ f (x, t) in  × (0, T ),
u = 0 on ∂ × (0, T ),
u(x, 0) = 0 in ,
(1.5)
where b∗ and A have been identified (cf. eqs (2.13)–(2.15)); m∗ denotes the volume fraction
of Y \ S in Y , i.e. m∗ = |Y \ S|, the Lebesgue measure of Y \ S. We also improve the
weak convergence of ueε by obtaining correctors. Such results are very useful in obtaining
numerical approximations and they form an important aspect of homogenization.
The layout of the paper is as follows: In §2, we give the weak formulation for the problem
(Pε ). Then we state our main results viz. Theorems 2.4 and 2.6. In §3, we sketch some
preliminary results and recall facts about the two-scale convergence method. Finally, in
§4, we prove our main results in the framework of two-scale convergence.
2. Assumptions and main results
.
For p > 1, p∗ will denote the conjugate exponent p/(p − 1). Define the space Vε = {v ∈
.
W 1,p (ε ) : v = 0 on ∂}; Vε∗ is its dual and Eε = Lp (0, T ; Vε ). We define uε ∈ Eε to
be a weak solution of (Pε ) if it satisfies:
x
x
∗
b
(2.1)
, uε ∈ L∞ (0, T ; L1 (ε )), ∂t b
, uε ∈ Lp (0, T ; Vε∗ ),
ε
ε
that is
Z
T
0
D
∂t b
x
ε
Z
E
, uε , ξ(x, t) dt +
ε
T
0
Z
D x
x
E
, uε − b
, u0
b
ε
ε
ε
∂t ξ dx dt = 0
(2.2)
Homogenization of a parabolic equation
197
for all ξ ∈ Eε ∩ W 1,1 (0, T ; L∞ (ε )) with ξ(T ) = 0; and,
Z TZ
Z TD x
E
x
a
, uε , ξ(x, t) dt +
, uε , ∇uε .∇ξ(x, t) dx dt
∂t b
ε
ε
ε
ε
0
0
Z TZ
f (x, t)ξ(x, t) dx dt
(2.3)
=
0
ε
for all ξ ∈ Eε . Here h . , . iε denotes the duality bracket with respect to Eε∗ , Eε .
Note: For any ξ ∈ Lp (0, T ; W0 ()), its restriction to ε can be used as a test function
in (2.3) for the problem (Pε ).
1,p
We make the following assumptions on a and b:
(A1) The function b(y, s) is continuous in y and s, Y -periodic in y and non-decreasing
in s and b(y, 0) = 0.
(A2) There exists a constant θ > 0 such that for every δ and R with 0 < δ < R, there
exists C(δ, R) > 0 such that
|b(y, s1 ) − b(y, s2 )| > C(δ, R)|s1 − s2 |θ
(2.4)
for all y ∈ Y and s1 , s2 ∈ [−R, R] with δ < |s1 |.
Remark 2.1. The prototype for b is a function of the form c(y)|s|k sgn(s) for some positive
real number k and continuous and Y -periodic function, c(.), which is positive on Y .
(A3) The mapping (y, µ, λ) 7→ a(y, µ, λ) defined from Rn × R × Rn to Rn is
measurable and Y -periodic in y and continuous in (µ, λ). Further, it is assumed
that there exists positive constants α, r such that
(2.5)
a(y, µ, λ) . λ ≥ α|λ|p ,
(2.6)
(a(y, µ, λ1 ) − a(y, µ, λ2 ))(λ1 − λ2 ) > 0, ∀ λ1 6= λ2
|a(y, µ, λ)| ≤ α −1 (1 + |µ|p−1 + |λ|p−1 ),
(2.7)
|a(y, µ1 , λ) − a(y, µ2 , λ)| ≤ α −1 |µ1 − µ2 |r (1 + |µ1 |p−1−r
+|µ2 |p−1−r + |λ|p−1−r ).
(2.8)
(A4) We assume that the data, f ∈ L∞ ( × T ). Under the above assumptions, it is
known that (Pε ) admits a solution uε (cf. [3]).
Remark 2.2. From hereon, we make the following simplifying assumption to reduce
the technicalities in the proof of the homogenization, viz. that a(y, µ, λ) is of the form
a(y, λ). The same results can be carried over for general a (cf. see [11,15] for suitable
modifications).
(A5) The strong monotonicity condition;
(a(y, λ1 ) − a(y, λ2 )) . (λ1 − λ2 ) ≥ α|λ1 − λ2 |p .
(2.9)
198
A K Nandakumaran and M Rajesh
Remark 2.3. We use the condition (A5) only for obtaining the corrector result. It will not
be required for identifying the homogenized problem.
We now state our main theorems.
Theorem 2.4. Let uε be a family of solutions of (Pε ). Assume that there is a constant
C > 0, such that
sup kuε kL∞ (ε ×(0,T )) ≤ C.
ε
(2.10)
Then there exists a subsequence of ε, still denoted by ε, such that for all q with 0 < q < ∞,
we have
ueε → u strongly in Lq (T )
(2.11)
and u solves,
∂t b∗ (u) − div A(∇u) = m∗ f in T ,
u = 0 on ∂ × (0, T ),
u(x, 0) = u0 in ,
(2.12)
where b∗ and A are defined below by (2.13)–(2.15), m∗ = |Y \ S| and T denotes the set
 × (0, T ).
The functions b∗ and A are defined through,
Z
b∗ (s) =
b(y, s) dy,
Y\ S
Z
a(y, λ + ∇8λ (y)) dy,
A(λ) =
Y\ S
where λ ∈ Rn and 8λ ∈ Wper (Y ) solves the periodic boundary value problem
Z
a(y, λ + ∇8λ (y)).∇φ(y) dy = 0
(2.13)
(2.14)
1,p
Y \S
(2.15)
1,p
for all φ ∈ Wper (Y ). See [9] for further details about eq. (2.15) and for the properties of A.
Remark 2.5. The assumption (2.10) is true in special cases (see [13]) and it is reasonable
on physical grounds (see [11]).
We see that by extending uε by zero in the holes we do not get functions in
gε can be shown to be a bounded sequence in Lp (T ))
However, ∇u
and has a weak limit. We will improve the weak convergence through a corrector result.
1,p
For this, we consider the solution U1 (x, t, y) ∈ Lp ( × (0, T ); Wper (Y )) of
Z TZ Z
a(y, ∇x u(x, t) + ∇y U1 (x, t, y)).∇y 8(x, t, y) dy dx dt = 0
1,p
Lp (0, T ; W0 ()).
0
 Y\ S
for all 8 ∈ Lp ( × (0, T ); Wper (Y )) (we will see that such a U1 exists). Let χ be the
characteristic function of Y \ S defined on Y and extended Y -periodically to Rn . Then,
χ (x/ε) is the characteristic function of ε . The corrector result is as follows:
1,p
Homogenization of a parabolic equation
199
Theorem 2.6. Let the functions uε , u, U1 be as above. Assume further that (A5) holds
and u, U1 are sufficiently smooth, i.e. that they belong to C 1 ( × (0, T )) and C( ×
1 (Y )) respectively. Then
(0, T ); Cper
x
→ 0 strongly in Lp (T ) and,
ueε − u − εU1 x, t,
ε
gε − χ x ∇x u + ∇y U1 x, t, x
→ 0 strongly in Lp (T ).
∇u
ε
ε
gε is m∗ ∇u(x, t) +
2.7. Note that, in particular, the Lp weak limit of ∇u
RRemark
∂U1
∂S ∂ν (x, t, y) dσ (y).
3. Some preliminary results
As we closely follow the treatment in [15], many results will only be sketched and we refer
the reader to [15] for more details as and when necessary.
We first obtain a priori bounds under the assumption (2.10). From now on, C will denote
a generic positive constant which is independent of ε.
Lemma 3.1.
Let uε be a family of solutions of (Pε ) and assume that (2.10) holds. Then,
sup k∇uε kLp (ε ×(0,T )) ≤ C,
ε
x
, ∇uε ∗
≤ C,
sup a
ε
Lp ( ×(0,T ))
ε
x ε , uε ≤ C.
sup ∂t b
ε
Eε∗
ε
Proof. Define the function B(., .) : Rn × R → R by
Z s
B(y, s) = b(y, s)s −
b(y, τ ) dτ.
(3.1)
(3.2)
(3.3)
(3.4)
0
As in [15] we deduce that
Z TZ
Z
x
x
, uε (x, T ) dx +
, ∇uε .∇uε dx dt
B
a
ε
ε
ε
ε
0
Z TZ
Z
x
, u0 dx +
B
f uε dx dt
=
ε
ε
ε
0
and from this we obtain
Z TZ
Z
x
x
B
a
, uε (x, T ) dx +
, ∇uε .∇uε dx dt ≤ C
ε
ε

ε
0
(3.5)
by (2.10) and the assumptions on b and a. Then, (3.1) follows from (3.5) and (2.5), as B is
non-negative, while (3.2) follows from (3.1) and (2.7). The estimate (3.3) may be obtained
from (3.1), (3.2) and (2.3). Thus the lemma.
We now show that ueε → u a.e.  × (0, T ), for a subsequence and for some u. This is the
crucial part of the present analysis. This result is to be seen in the context of the estimate
200
A K Nandakumaran and M Rajesh
(3.1) where there is no time gradient estimate. Hence, we cannot use any ready-made
compact imbedding theorem to get the strong convergence of uε to u in some Lr (T ).
However, we are able to achieve this by adapting a technique found in [3]. This is given
in the following lemma.
Lemma 3.2. Let uε be as above. Then, the sequence {ueε }ε>0 is relatively compact in
Lθ (T ), where θ is as in (A2). As a result, there is a subsequence of uε such that,
ueε → u a. e. in T .
(3.6)
Proof. Analogous to Step 1, Lemma 3.3 in [15], we can derive the following estimate
Z T −h Z x
x
b
, uε (t + h) − b
, uε (t) (uε (t + h)
h−1
ε
ε
ε
0
− uε (t)) dx dt ≤ C
for some constant C which is independent of ε and h. Thus, as we have assumed in (A1)
that b(y, 0) = 0, we get
Z T −h Z x
x
, ueε (t + h) − b
, ueε (t) (ueε (t + h)
b
h−1
ε
ε

0
− ueε (t)) dx dt ≤ C.
Proceeding as in [15] we obtain the desired result.
From Lemma 3.2 above, the continuity of b and the assumption (2.10) we derive the
following corollary.
COROLLARY 3.3
We have, b xε , ueε − b( xε , u) → 0 strongly in Lq (T ) ∀ q, 0 < q < ∞.
Proof. By the a priori bound (2.10), it is enough to consider the function b on Y ×[−M, M]
for a large M > 0. As b is continuous, it is uniformly continuous on Y × [−M, M].
Therefore, given h0 > 0, there exists a δ > 0 such that,
|b(y, s) − b(y 0 , s 0 )| < h0 ,
whenever |y − y 0 | + |s − s 0 | < δ.
Now, since ueε → u a.e in T , by Egoroff’s theorem, given h1 > 0, there exists E ⊂ T
such that its Lebesgue measure m(E) < h1 and ueε converges uniformly to u on (T \ E),
which we denote by E 0 . Therefore, we can find ε1 > 0 such that
kueε − uk∞,E 0 < δ ∀ε < ε1 .
Therefore, for ε < ε1 we have
Z x q
x
, ueε − b
, u dx dt
b
ε
ε
T
Z x q
x
=
, ueε − b
, u dx dt
b
ε
ε
E0
Z x q
x
, ueε − b
, u dx dt
+
b
ε
ε
E
q
≤ h0 m(T ) + 2q sup(|b|q ) m(E)
q
≤ h0 m(T ) + 2q sup(|b|q ) h1 .
(3.7)
Homogenization of a parabolic equation
201
This completes the proof as h0 and h1 can be chosen arbitrarily small.
As a consequence, we have the following corollary.
COROLLARY 3.4
We have the following convergences:
x
, ueε * b(u) weakly in Lq (T ),
b
x xε
b
, ueε * b∗ (u) weakly in Lq (T )
χ
ε
ε
for q > 1. Further, b(u) = b∗ (u).
Proof. We note that
x x
x x
, ueε = b
, ueε − b
,u + b
,u
b
ε
ε
ε
ε
→ 0 + b(u)
by Corollary 3.3 and the averaging principle for periodic functions mentioned below.
Similarly,
x x
x x x x x
b
, ueε = χ
b
, ueε − b
,u + χ
b
,u
χ
ε
ε
ε
ε
ε
ε
ε
→ 0 + b∗ (u).
From χ xε b
corollary.
x
eε
ε,u
= b
x
eε
ε,u
, we readily obtain the last of the conclusions in the
The homogenization will be done in the framework of two-scale convergence. So to
conclude this section, we recall an averaging principle which is at the basis of this method,
the definition of two-scale convergence and a few related results which we will use in the
sequel. For an exposition of this method and its main features, see [14,1,15].
q
For f (x, y) ∈ Lloc (Rn ; Cper (Y )), the oscillatory function f x, xε converges
R
q
n
weakly in Lloc (R ) to Y f (x, y) dy, for all q > 1. A similar result holds for
n
f (x, y) ∈ L∞
loc (R ; Cper (Y )), with weak convergence replaced by weak-* convergence.
DEFINITION 3.5
Let 1 < q < ∞. A sequence of functions vε ∈ Lq (T ) is said to two-scale converge to a
function v ∈ Lq (T × Y ) if
Z Z
Z
x
ε→0
dx dt →
vε ψ x, t,
v(x, t, y) ψ(x, t, y) dy dx dt
ε
T
T Y
∗
2−s
for all ψ ∈ Lq (T ; Cper (Y )). We write vε → v.
Theorem 3.6.
Let 1 < q < ∞ and let vε be a bounded sequence in Lq (T ). Then there
2−s
exists a function v ∈ Lq (T × Y ) such that, up to a subsequence, vε → v(x, t, y).
202
A K Nandakumaran and M Rajesh
The following theorem [1] is useful in obtaining the limit of the product of two two-scale
convergent sequences. Let 1 < q < ∞.
Let q > 1. Let vε be a sequence in Lq (T ) and wε be a sequence in
Theorem 3.7.
q∗
2−s
2−s
L (T ) such that vε → v and wε → w. Further, assume that the sequence wε satisfies
Z
Z Z
∗
ε→0
q∗
|wε | (x, t) dx dt →
|w(x, t, y)|q dy dx dt.
(3.8)
T
Then,
T
Z
ε→0
T
vε wε dx dt →
Z
Y
Z
T
Y
v(x, t, y)w(x, t, y) dy dx dt.
(3.9)
Remark 3.8. A sequence wε which two-scale converges and satisfies (3.8) is said to be
strongly two-scale convergent. Examples of strongly two-scale convergent sequences are
q
{ψ(x, t, xε )}ε for any ψ ∈ Lper (Y ; C(T )).
We now state a result for perforated domains the proof of which may be found in [1].
Let q > 1. Let vε ∈ Lq (ε × (0, T )) and let ∇vε ∈ Lq (ε × (0, T )) be
Theorem 3.9.
such that
sup kvε kLq (ε ×(0,T )) ≤ C,
ε
sup k∇vε kLq (ε ×(0,T )) ≤ C.
(3.10)
ε
Then there exist v ∈ Lq (0, T ; W 1,q ()) and V1 ∈ Lq (T ; Wper (Y )) such that, up to a
subsequence
1,q
2−s
veε → v(x, t),
gε 2−s
→ χ(y)(∇x v(x, t) + ∇y V1 (x, t, y)).
∇v
(3.11)
4. Proofs of the main Theorems
We first obtain the two-scale homogenized equation. From the estimates (3.1) and (3.2)
∗
1,p
and Theorems 3.8 and 3.5, there exists U1 ∈ Lp (T ; Wper (Y )) and a0 ∈ Lp (T × Y )
such that, up to a subsequence,
gε 2−s
→ χ(y) (∇x u(x, t) + ∇y U1 (x, t, y)),
(4.1)
∇u
2−s
x x
gε → a0 (x, t, y).
a
, ∇u
(4.2)
χ
ε
ε
Theorem 4.1. The limits u, a0 satisfy the following two-scale homogenized problem
Z T
Z Z
h∂t b∗ (u), φi dt +
a0 (x, t, y).(∇x φ + ∇y 8(x, t, y)) dy dx dt
T Y \ S
0
Z
f φ dx dt
(4.3)
=
T
for all φ ∈
C0∞ (T )
∞ (Y )).
and 8 ∈ C0∞ (T ; Cper
Homogenization of a parabolic equation
203
Proof. We pass to the limit in (2.3) in the presence of appropriate test functions. Let
∞ (Y )). Set,
φ ∈ C0∞ (T ) and 8 ∈ C0∞ (T ; Cper
x
φε = φ(x, t) + ε8 x, t,
.
ε
We may use these as test functions in (2.3) as the restrictions of φε to ε are in Eε and we
note
Z TD Z TZ
x x
E
x
gε
χ
∂t b
, uε , φε dt +
a
, ∇u
ε
ε
ε
ε

0
0
Z TZ
x x × ∇x φ +∇y 8 x, t,
dx dt + o(1) =
f (x, t)φ(x, t) dx dt +o(1).
χ
ε
ε

0
(4.4)
Now we pass to the limit in the above identity. We may handle the first term as follows.
Observing that φε (. , T ) = 0 = φε (. , 0), we have from (2.2),
Z TD Z TZ
x
E
x
b
∂t b
, uε , φε dt = −
, uε ∂t φε dx dt
ε
ε
ε
0
0 ε
Z TZ
x x
χ
b
, ueε ∂t φ dx dt + o(1)
=−
ε
ε
0 
Z TZ
→−
b∗ (u) ∂t φ dx dt
Z
=
0

0
T
h∂t b∗ (u) , φi dx dt,
where the convergence in the above follows from Corollary 3.4.
For passing to the limit in the other two terms in (4.4) we observe that
Z TZ
x
x a
, ∇uε . ∇x φ + ∇y 8 x, t,
dx dt
ε
ε
ε
0
Z TZ
x x
gε . ∇x φ + ∇y 8 x, t, x
a
, ∇u
dx dt
χ
=
ε
ε
ε

0
Z TZ Z
→
a0 (x, t, y).(∇x φ + ∇y 8(x, t, y)) dy dx dt,
0
 Y
where the convergence in the last step follows
from Theorem 3.6. For this, one observes
from Remark 3.7 that ∇x φ + ∇y 8 x, t, xε is strongly two-scale convergent to ∇x φ +
gε is zero in the holes, we have
∇y 8(x, t, y). Further, observe that since χ xε a xε , ∇u
a0 (x, t, y) to be zero in S and the integral over Y can be replaced by that over Y \ S.
Finally, note that
Z TZ
Z TZ Z
x f (x, t)φ(x, t) dx dt →
χ
χ (y)f (x, t)φ(x, t) dy dx dt
ε

 Y
0
0
Z TZ
=
m∗ f φ dx dt.
0

Thus, from the above we deduce that the limit of the equations (2.3) with φε as test functions
is nothing but (4.3). This is referred to as the two-scale homogenized problem.
204
A K Nandakumaran and M Rajesh
We note from the two-scale homogenized problem (4.3), by setting 8 ≡ 0
Remark 4.2.
that
Z
∗
∂t b (u) − div
Y\ S
a0 (x, t, y)dy
= m∗ f in  × (0, T )
and by setting φ ≡ 0 we get
Z TZ Z
a0 (x, t, y).∇y 8(x, t, y) dy dx dt = 0
0
 Y \S
(4.5)
(4.6)
∞ (Y )). In view of (4.5) and (4.6), it is enough to show that
for all 8 ∈ C0∞ (T ; Cper
a0 (x, t, y) = a(y, ∇x u(x, t) + ∇y U1 (x, t, y)) to conclude Theorem 2.4.
Proof of Theorem 2.4. Note that (2.11) follows from the convergence (3.6) and the
assumption (2.10). We now justify (2.12), for which it is enough to show that a0 (x, t, y)
equals a(y, ∇x u(x, t) + ∇y U1 (x, t, y)), by the preceding remark.
1,p
Let ξ ∈ Lp (0, T ; W0 ()) ∪ W 1,1 (0, T ; L∞ ()) with ξ(. , T ) = 0 = ξ(. , 0). This
may be used as a test function in (2.2) and we have
Z TZ
Z TD x
E
x
b
, uε , ξ(x, t) dt +
, uε ∂t ξ dx dt = 0.
∂t b
ε
ε
ε
ε
0
0
This can be rewritten as
Z TZ
Z TD
x
x x
E
, uε , ξ dt +
b
, ueε ∂t ξ dx dt = 0.
χ
Rε∗ ∂t b
ε
ε
ε
0
0 
(4.7)
x
∗
∗
ε , uε kLp (0,T ;W −1,p () ) ≤ C by (3.3) where Rε is the restric∗
∗
1,p
Lp (0, T ; W0 ()) to Eε . Thus for some w ∈ Lp (0, T ; W −1,p ())
Observe that supε kRε∗ ∂t b
tion operator from
and for a subsequence
x
∗
∗
, uε * w weakly * in Lp (0, T ; W −1,p ()).
Rε∗ ∂t b
ε
Thus, passing to the limit in (4.7) we get
Z TZ
Z T
hw, ξ i +
b∗ (u) ∂t ξ dx dt = 0
0
0

b∗ (u)
for all ξ as above. So, ∂t
= w in the sense of distributions.
We now show that a0 (x, t, y) = a(y, ∇x u(x, t) + ∇y U1 (x, t, y)). Let φ, 8 be as in the
∞ (Y ))n . Set
proof of Theorem 4.1. Let λ > 0 and ψ ∈ C0∞ (T ; Cper
x
x
.
+ λψ x, t,
.
ηε (x, t) = ∇x φ(x, t) + ∇y 8 x, t,
ε
ε
By (2.6) we note that
Z TZ x
x
, ∇uε − a
, ηε .(∇uε − ηε ) dx dt ≥ 0.
a
ε
ε
ε
0
(4.8)
Homogenization of a parabolic equation
205
We have
Z TD Z TZ
Z TZ
x
E
x
, ∇uε .∇uε = −
, uε , uε dt +
a
f uε dx dt
∂t b
ε
ε
ε
ε
0
0
0
Z TZ
Z T
∗
h∂t b (u), ui +
m∗ f u dx dt
→−
0
Z
T
=
 Y\ S
0

0
Z Z
a0 (x, t, y).∇x u dy dx dt,
where the last step follows from (4.5).
Also, since ηε and a( xε , ηε ) are strongly two-scale convergent, Theorem 3.6 gives
Z
T
→
a
ε
0
T
Z
Z
x
ε
Z Z
 Y\ S
0
Z
, ηε .∇uε dx dt =
Z
T

0
a
x
ε
gε dx dt
, ηε .∇u
a(y, ∇x φ + ∇y 8 + λψ).(∇x u + ∇y U1 ) dy dx dt.
Similarly,
Z
T
→
a
ε
0
T
Z
Z
x
ε
Z Z
 Y\ S
0
Z
, ηε .ηε dx dt =
T
Z

0
χ
x x
a
, ηε .ηε dx dt
ε
ε
a(y, ∇x φ + ∇y 8 + λψ).(∇x φ + ∇y 8 + λψ) dy dx dt
and,
Z
0
T
Z
ε
a
x
Z
, ∇uε .ηε dx dt =
ε
Z Z
Z
T
→
 Y\ S
0
Z
T
=
0
0
T
Z

χ
x x
gε .ηε dx dt
a
, ∇u
ε
ε
a0 (x, t, y).(∇x φ + ∇y 8 + λψ) dy dx dt
Z Z
 Y\ S
a0 (x, t, y).(∇x φ + λψ) dy dx dt.
The equality in the last step follows from (4.6). Thus
Z TZ Z
Z TZ x
x
a0 (x, t, y)
, ∇uε − a
, ηε .(∇uε − ηε ) =
a
0 ≤ lim
ε→0 0
ε
ε
ε
 Y\ S
0
Z TZ Z
× (∇x (u − φ) − λψ) −
a(y, ∇x φ + ∇y 8 + λψ)
0
 Y \S
× (∇x (u − φ) + ∇y (U1 − 8) − λψ).
Letting φ → u and 8 → U1 in the above and using the continuity of a we get
Z TZ Z
(a(y, ∇x u + ∇y U1 + λψ) − a0 (x, t, y)).λψ dy dx dt ≥ 0
0
 Y \S
206
A K Nandakumaran and M Rajesh
for all λ and ψ. Dividing the above inequality by λ and letting λ → 0 we get,
Z
T
Z Z
 Y \S
0
(a(y, ∇x u + ∇y U1 ) − a0 (x, t, y)).ψ dy dx dt ≥ 0
for all ψ. From this we conclude that a0 (x, t, y) = a(y, ∇x u + ∇y U1 ).
Proof of Theorem 2.6. Set
x
.
.
ηε (x, t) = ∇x u(x, t) + ∇y U1 x, t,
ε
From (2.9) it follows that
Z
T
0
Z
x
x
, ∇uε −a
, ηε .(∇uε − ηε ) dx, dt
a
ε
ε
ε
Z TZ
|∇uε − ηε |p dx dt,
≥α
0
ε
that is,
Z
Z
x
gε − a x , ηε . ∇u
gε − χ x ηε dx dt
, ∇u
a
ε
ε
ε
ε
0
Z TZ x h
i
x p
∇x u(x, t) + ∇y U1 x, t,
≥α
dx dt.
∇ ueε − χ
ε
ε

0
T
As we have assumed that u, U1 are smooth, we may proceed as in the proof of Theorem
2.4 to obtain
Z TZ x x p
α lim sup
∇x u(x, t) + ∇y U1 x, t,
dx dt
∇ ueε − χ
ε
ε

0
ε→0
Z TZ x
x g
gε − χ x ηε dx dt
, ∇uε − a
, ηε . ∇u
a
≤ lim
ε→0 0
ε
ε
ε
ε
= 0.
This completes the proof.
Remark 4.3. A weaker form of the corrector result with no smoothness assumptions can
also be obtained. See [15] for the corresponding formulation.
Acknowledgement
The second author would like to thank the National Board of Higher Mathematics, India
for financial support. He would also like to thank the organizers for their invitation to
give a talk at the Workshop on ‘Spectral and Inverse Spectral Theories for Schrödinger
Operators’ held at Goa, where he presented this work.
Homogenization of a parabolic equation
207
References
[1] Allaire G, Homogenization and two-scale convergence, SIAM J. Math. Anal. 23 (1992)
1482–1518.
[2] Allaire G, Murat M and Nandakumar A K, Appendix of ‘Homogenization of the Neumann
problem with nonisolated holes’, Asymptotic Anal. 7(2) (1993) 81–95
[3] Alt H W and Luckhaus S, Quasilinear elliptic-parabolic differential equations, Math. Z.
183 (1983) 311–341
[4] Avellaneda M and Lin F H, Compactness methods in the theory of homogenization,
Comm. Pure Appl. Math. 40 (1987) 803–847
[5] Bensoussan B, Lions J L and Papanicolaou G, Asymptotic Analysis of Periodic Structures
(Amsterdam: North Holland) (1978)
[6] Brahim-Omsmane S, Francfort G A and Murat F, Correctors for the homogenization of
wave and heat equations, J. Math. Pures Appl. 71 (1992) 197–231
[7] Clark G W and Showalter R E, Two-scale convergence of a model for flow in a partially
fissured medium, Electron. J. Differ. Equ. 1999(2) (1999) 1–20
[8] Douglas Jr. J, Peszyńska M and Showalter R E, Single phase flow in partially fissured
media, Trans. Porous Media 28 (1995) 285–306
[9] Fusco N and Moscariello G, On the homogenization of quasilinear divergence structure
operators, Ann. Mat. Pura Appl. 146(4) (1987) 1–13
[10] Hornung U, Applications of the homogenization method to flow and transport in porous
media, Notes of Tshingua Summer School on Math. Modelling of Flow and Transport in
Porous Media (ed.) Xiao Shutie (Singapore: World Scientific) (1992) pp. 167–222
[11] Jian H, On the homogenization of degenerate parabolic equations, Acta Math. Appl.
Sinica 16(1) (2000) 100–110
[12] Jikov V V, Kozlov S M and Oleinik O A, Homogeniztion of Differential Operators and
Integral Functionals (Berlin: Springer-Verlag) (1994)
[13] Ladyzenskaya O A, Solonnikov V and Ural’tzeva N N, Linear and quasilinear equations
of parabolic type, Am. Math. Soc. Transl. Mono.? 23, (1968)
[14] Nguetseng G, A general convergence result of a functional related to the theory of homogenization, SIAM J. Math. Anal. 20 (1989) 608–623
[15] Nandakumaran A K and Rajesh M, Homogenization of a nonlinear degenerate parabolic
differential equation, Electron. J. Differ. Equ. 2001(17) (2001) 1–19
[16] Oleinik O A, Kozlov S M and Zhikov V V, On G-convergence of parabolic operators,
Ouspekhi Math. Naut. 36(1) (1981) 11–58