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. 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