ESTIMATES FOR ELLIPTIC HOMOGENIZATION PROBLEMS IN NONSMOOTH DOMAINS. Zhongwei Shen

Transcription

1 W,p ESTIMATES FOR ELLIPTIC HOMOGENIZATION PROBLEMS IN NONSMOOTH DOMAINS Zhongwei Shen Abstract. Let L = div`a` x, > be a family of second order elliptic operators with real, symmetric coefficients on a bounded Lipschitz domain Ω in R n, subject to the Dirichlet boundary condition. Assuming that A(x) is periodic and belongs to VMO, we show that there exists δ > independent of such that Riesz transforms (L ) /2 are uniformly bounded on L p (Ω), where < p < 3+δ if n 3, and < p < 4+δ if n = 2. The ranges of p s are sharp. In the case of C domains, we establish the uniform L p boundedness of (L ) /2 for < p < and n 2. As a consequence, we obtain the uniform W,p estimates for the elliptic homogenization problem L u = divf in Ω, u = on Ω.. Introduction This paper continues the study in [S] of the L p boundedness of Riesz transforms associated with second order elliptic operators. Here we consider a family of second order elliptic operators of divergence form in a nonsmooth domain Ω, (.) L = div ( A ( x) ), >, arising in the theory of homogenization, subject to the Dirichlet boundary condition. We assume that A(x) = ( a ij (x) ) is a n n real symmetric matrix and satisfies the following elliptic and periodic conditions: (.2) (.3) µ ξ 2 a ij (x)ξ i ξ j µ ξ 2 for any x, ξ R n, A(x + y) = A(x) for any x R n, y Z n, where µ >. We will also assume that the coefficient matrix A V MO(R n ); i.e., (.4) lim t ω(t) =, 2 Mathematics Subject Classification. Primary 35J5, 35J25; Secondary 42B2. Key words and phrases. Riesz transform; Homogenization; Lipschitz domain. The author is supported in part by the NSF (DMS-5257). Typeset by AMS-TEX

2 2 ZHONGWEI SHEN where (.5) ω(t) = sup x R n B(x, r) B(x,r) A(y) A(z)dz B(x, r) B(x,r) dy. <r<t Under these assumptions we establish the uniform L p boundedness of Riesz transforms (L ) /2 on Lipschitz or C domains. Theorem. below is the main result of the paper. We point out that the ranges of p s in Theorem. as well as in Corollary.2 are sharp even for operators with constant coefficients [JK2]. Theorem.. Let Ω be a bounded Lipschitz domain in R n, n 2. Suppose that the real symmetric matrix A(x) satisfies conditions (.2), (.3) and (.4). Then there exists a constant δ > depending only on µ, n, the Lipschitz character of Ω, and function ω(t) such that (.6) (L ) /2 f p C f p for any f L p (Ω), for < p < 3 + δ if n 3, and < p < 4 + δ if n = 2, where C depends only on µ, n, p, the Lipschitz character of Ω, and ω(t). If Ω is a C domain, estimate (.6) holds for any < p <. As a direct consequence of Theorem., we obtain the uniform W,p estimate for the elliptic homogenization problem (.7) L u = divf in Ω, u = on Ω. Corollary.2. Under the same assumptions as in Theorem., there exists δ >, depending only on n, µ, the Lipschitz character of Ω, and ω(t), such that if f L p (Ω) L 2 (Ω) with p 2 < 6 + δ for n 3, and p 2 < 4 + δ for n = 2, the unique solution to the Dirichlet problem (.7) in W,2 (Ω) satisfies (.8) u p C f p, where C depends only on n, µ, p, the Lipschitz character of Ω, and ω(t). If Ω is a C domain, estimate (.8) holds for any < p <. If the coefficients a jk (x) are periodic and Hölder continuous, it was shown by Avellaneda and Lin [AL4] that (L ) /2 are bounded on L p (R n ) for any < p <. A similar theorem, based on the results in [AL,AL3], was obtained by Alexopoulos [A]. We should point out that the results in [AL4] were established for systems of elliptic operators with periodic coefficients, and extended to bounded domains with C,α boundaries. In [CP] Caffarelli and Peral obtained the uniform interior W,p (2 < p < ) estimates under the assumption that A(x) are continuous and periodic. By Theorem A in [S] this implies the boundedness of (L ) /2 on L p (R n ) for operators with continuous and periodic coefficients. We mention that the boundedness of (L ) /2 on L p (R n ) has been established in [ERS] for second-order periodic elliptic operators in divergence form with complex continuous coefficients. For related results on W,p estimates and Riesz transforms for second order elliptic operators without the periodicity assumption, we refer the reader to [Au,AC,ACDH,AT,AT2,AQ, B, BW,CD] and their references. Our starting point for the proof of Theorem. is the following.

3 HOMOGENIZATION PROBLEMS 3 Theorem.3. Let Ω be a bounded Lipschitz domain in R n, n 2. Let A(x) be a real symmetric n n matrix with bounded measurable entries satisfying (.2). Let p > 2. Suppose that there exist constants C >, α 2 > α > and r independent of > such that for any ball B(x, r) with the property that < r < r and either x Ω or B(x, α 2 r) Ω, and for any weak solution of L u = in Ω B(x, α 2 r) and u = on B(x, α 2 r) Ω (if x Ω), one has u L p (Ω B(x, r)) and (.9) ( r n Ω B(x,r) ) / p ( u p dx C r n B(x,α r) u 2 dx) /2. Then there exists δ > depending only on µ, n, α, α 2, C and the Lipschitz character of Ω such that (.6) holds for < p < p + δ, where C depends only on µ, n, p, α, α 2, C and the Lipschitz character of Ω. Theorem.3 follows directly from Theorem B in [S], which states that given any second order elliptic operator L of divergence form with real, symmetric, bounded, measurable coefficients and any p > 2, the boundedness of the Riesz transforms (L) /2 on L p (Ω) is equivalent to the scale-invariant interior and boundary W,p estimates (.9) for weak solutions of Lu =. For the interior estimates, one may extend either the approximation method in [CP] or the compactness method developed in [AL,AL2] to the case of VMO coefficients. As expected for nonsmooth domains, the main difficulty lies in the uniform boundary W,p estimates. For the second order systems of elliptic operators with periodic and Hölder continuous coefficients, Avellaneda and Lin [AL] established a uniform boundary L estimate for the gradients of weak solutions of L u = on C,α domains. Such L gradient estimate, however, fails in general for C domains, even in the case of constant coefficients. Suppose Ω and (.) Ω B(, r ) = { (x, x n ) R n : x n > ψ(x ) } B(, r ). Let r = {(x, ψ(x )) : x < r}. Our main novelty in the proof of Theorem. is to reduce the weak reverse Hölder inequality (.9) to the following decay estimate, (.) tr r 2r C t p+α, 2r for any < t < and < r < cr, where α >, L u = in Ω B(, 2r) and u = on 2r. Note that estimate (.) only involves u, not u. This, together with the observation that (.) holds for solutions of elliptic equations with constant coefficients, makes it accessible via a variant of the three-step compactness argument of Avellaneda and Lin in [AL,AL2].

4 4 ZHONGWEI SHEN The paper is organized as follows. In section 2 we establish the uniform interior W,p estimate for operators {L } with VMO coefficients for any p > 2. The reduction from (.9) to (.) as well as the proof of (.) for p = 3 if n 3, and for p = 4 if n = 2, is given in section 3. A similar approach gives (.9) on C domains for any 2 < p <. This is outlined in section 4. Finally in section 5 we give the proof of Theorem. and Corollary.2. Throughout the rest of this paper, we assume that A(x) is a real symmetric matrix satisfying conditions (.2)-(.4). We will denote / x j by j. The standard summation convention will also be used. 2. Interior W,p estimates Let B(x, r) denote the ball centered at x with radius r. For α >, we use the notation αb = B(x, αr). In this section we establish the following theorem. Theorem 2.. Let u W,2 (3B) be a weak solution of L u = in 3B. Then u L p (B) and (2.) { /p u dx} p C B B { } /2 u 2 dx 2B 2B for any p > 2, where C depends only on µ, n, p and function ω(t). Theorem 2. was proved in [CP] for the case of continuous coefficients. The proof in [CP] extends easily to the case that A V MO(R n ). We remark that if the coefficients are Hölder continuous, estimate (2.) follows from the L estimates on u established in [AL, Lemma 6]. Here we present a proof of Theorem 2., using the three-step compactness method of Avellaneda and Lin. Let χ j be the unique function such that L (χ j ) = i a ij on R n, χ j is periodic with respect to Z n, and (2.2) [,] n χ j dx =. The function χ = (χ, χ 2,..., χ n ) is called the corrector for L. ρ >, Observe that for any (2.3) L ρ {x + ρχ(x/ρ)} = in R n. It is also not hard to see that (2.4) χ L (R n ) C(n, µ). Let L = i b ij j be a second order elliptic operator with real constant coefficients. Assume that coefficients satisfy b ij = b ji and the ellipticity condition (.2). Let B θ =

5 HOMOGENIZATION PROBLEMS 5 B(, θ). Suppose that L u = in B. Then there exists C depending only on µ and n such that (2.5) sup u (x) u () < x, ( u ) Bθ > C θ 2 u L (B ), x <θ for any θ (, /2), where ( u ) Bθ denotes the average of u over B θ. For any fixed η (, ), we choose θ (, /2) so that C θ θ η, where C is given by (2.5). The following was proved in [AL, Lemma 4] by a compactness argument. Lemma 2.2. There exists depending only on µ, n and η such that (2.6) sup u (x) u () < x + χ(x/), ( u ) Bθ > θ +η u L (B ), x <θ for any < < and any weak solution u of L u = in B. By an iteration argument, Lemma 2.2, together with (2.3) and (2.4), leads to the following. Lemma 2.3. Let η, θ and be the same constants as in Lemma 2.2. There exists a constant C, depending only on µ, n and η, such that for any weak solution u of L u = in B with θ l+ < θ l, (2.7) sup x <θ l u (x) u () d l < x + χ(x/), G l > θ l(+η) u L (B ), where d l R and G l Rn are constants with the property (2.8) d l + G l C u L (B ). Proof. See [AL, Lemma 6]. We are in a position to give the proof of Theorem 2.. Proof of Theorem 2.. Fix B = B(x, r ). By translation and dilation, one may assume that x = and r =. Also, by the local W,p estimates for solutions of second order elliptic equations with VMO coefficients, one may further assume that < < θ, where θ and are given by Lemma 2.2. Let v(x) = u (x). Then L v = in B(, 2/). It follows that for any 2 < p <, (2.9) v Lp (B(, 4 )) C v L 2 (B(, 2 )). Thus, (2.) { n B(, 4 ) u p dx} /p C { n B(, 2 ) u 2 dx} /2.

6 6 ZHONGWEI SHEN By Caccioppoli s inequality, this implies that (2.) { n B(, 4 ) Note that by Lemma 2.3, we have u p dx} /p C sup x < u (x) u () (2.2) sup x < In view of (2.) and (2.2), we have proved that (2.3) B(, 4 ) u (x) u (). C u L (B(,)). u p dx C n u p L (B(,)). It follows by translation that for any y B(, /2), (2.4) B(y, 4 ) u p dx C n u p L (B(, 3 2 )). By covering B(, /2) with a finite number of balls of radius /4, we may deduce from (2.4) that (2.5) u p dx C u p C u 2 B L ( p 3 2 B) L 2 (2B). Since u β is also a solution for any β R, we may replace u in the right side of (2.5) by u β. This, together with the Poincaré inequality, gives (2.6) { } /p 2 B u p dx C 2 B { /2 u dx} 2. 2B 2B By a simple covering argument it is not hard to see that estimate (2.6) is equivalent to (2.). This completes the proof of Theorem Boundary W,p Estimates Let ψ : R n R be a Lipschitz function such that ψ() =. For r >, let (3.) (3.2) r = { (x, ψ(x )) R n : x < r }, D r = { (x, x n ) R n : x < r and ψ(x ) < x n < ψ(x ) + r }. Let p = 3 for n 3, and p = 4 for n = 2. This section is devoted to the proof of the following theorem.

7 HOMOGENIZATION PROBLEMS 7 Theorem 3.. There exists a constant C >, depending only on µ, n, ψ and function ω(t), such that if u W,2 (D 6r ) is a weak solution of L u = in D 6r and u = on 6r, then u L p (D r ) and (3.3) { } / p { } /2 r n u p dx C D r r n u 2 dx. D 6r Let d(x) = d(x, x n ) = x n ψ(x ). To prove (3.3), we first show that it suffices to estimate the L p norm of u /d on D 2r. Lemma 3.2. Suppose that L u = in D 2r and u = on 2r. Then for any 2 < p <, p (3.4) u p dx C u (x) D r d(x) dx, where C > depends only on µ, n, p, ψ and function ω(t). Proof. Let ρ(x) = dist(x, D 4r ). Then ρ(x) d(x) for any x D 3r. Choose c = c(n, ψ ) (, /4) so small that B(x, 2cρ(x)) D 2r for any x D r. By interior estimate (2.), we have (3.5) B(x,cρ(x)) D 2r u (y) p dy C B(x,2cρ(x)) u (y) ρ(y) for any x D r. Next we multiply both sides of (3.5) by ρ(x) n and integrate the resulting inequality over D r. This gives { } u (y) p dx x Dr D r x y <cρ(x) ρ(x) n dy (3.6) C u (y) { p } dx ρ(y) ρ(x) n dy. D 2r Finally we observe that if x y < cρ(x), then x D r x y <2cρ(x) (3.7) ρ(y) ρ(x) + x y ( + c)ρ(x). p dy Similarly, ρ(x) ρ(y) + x y ρ(y) + cρ(x) and thus ( c)ρ(x) ρ(y). ρ(y) ρ(x). It follows that for any y D r, dx (3.8) ρ(x) n c ρ(y) n dx c, x D r x y <ρ(x) x D r x y <cρ(y) Hence and for any y D 2r, (3.9) x D r x y <2cρ(x) dx ρ(x) n C ρ(y) n x D r x y Cρ(y) dy C.

8 8 ZHONGWEI SHEN The desired estimate (3.4) now follows from (3.6), (3.8) and (3.9). By the well known De Giorgi-Nash regularity estimates and the Poincaré inequality, (3.) { } /p { r n u p dx C D r r n u 2 dx D 2r { Cr r n u 2 dx D 2r for any 2 < p <. In view of (3.4) and (3.), we see that the estimate (3.3) would follow from r (3.) u (x, ψ(x p ) + s) s dx ds C 4r r p. x <r x <4r Lemma 3.3. Suppose that there exist positive constants, α and C depending only on µ, n, ψ and function ω(t) such that for < t r <, t (3.2) x <r C } /2 } /2 ( ) p+α t 2r r x <2r whenever L u = in D 2r and u = on 2r. Then estimate (3.3) holds. Proof. By the boundary W,p estimates on Lipschitz domains for operators with VMO coefficients (see Theorem C and its proof in [S]), estimate (3.3) holds for ( /4). Assume that < < /4. Since w(x) = u (x) is a weak solution of L w =, the same W,p estimates also give us (3.3) r x < r u (x, ψ(x ) + s) s C 2r (r) p p x < 2r dx ds. By covering r with surface balls of radius r/, we may deduce from (3.3) that r u (x, ψ(x p ) + s) x <r s dx ds (3.4) C 2r (r) p x <2r C 4r r p, x <4r

9 HOMOGENIZATION PROBLEMS 9 where we have used the assumption (3.2) in the last step. Finally we let f(x, s) = s u (x, ψ(x ) + s) and write r f(x, s) p dx ds (3.5) = x <r r x <r + j j= 2 j r 2 j r x <r r + 2 j r x <r f(x, s) p dx ds, where 2 j 2 j. The first term in the right side of (3.5) is handled by (3.4), while the estimate of the last term is trivial. To control the term involving the summation over j, we use the assumption (3.2). This gives estimate (3.), from which inequality (3.3) follows. It remains to establish estimate (3.2). By dilation we may assume that r =. We first observe that (3.2) holds for solutions of the elliptic equations with constant coefficients. Indeed suppose that L u = i b ij j u = in D 3/2 and u = on 3/2, where b ij are real constants satisfying b ij = b ji and (.2). By the L 2 regularity estimates on Lipschitz domains [JK], we have (3.6) ( u ) (x, ψ(x )) 2 dx C u 2 dx, x < D 3/2 where ( u ) (x, ψ(x )) = sup{ u (x, x n ) : ψ(x ) < x n < ψ(x n ) + r}. This, together with the observation that u (x, s+ψ(x )) s( u ) (x, ψ(x )) and the boundary Hölder estimates, gives (3.7) t x < u (x, ψ(x ) + s) 3 dx ds Ct 3+β 3 2 x < 3 2 u (x, ψ(x ) + s) 3 dx ds, for any < t <, where β > and C > depend only on µ, n and ψ. Note that if n = 2, one has a stronger boundary Hölder estimate, (3.8) u (x, ψ(x ) + s) Cs +β 2 u dy D 3/2 for some β > depending only on µ and ψ. Together with (3.6), this shows that for n = 2, (3.9) t x < for any < t <. u (x, ψ(x ) + s) 4 dx ds Ct 4+β 3 2 x < 3 2 u (x, ψ(x ) + s) 4 dx ds We now fix < α < β. Choose t (, /4) so that 2Ct β α, where β and C are given by (3.7) and (3.9).

10 ZHONGWEI SHEN Lemma 3.4. There exists > depending only on µ, n and ψ such that for any <, (3.2) t x < t p+α 2 x <2 whenever L u = in D 2 and u = on 2., Proof. We prove estimate (3.2) by contradiction. Suppose that there exist {L k }, { k } and {u k } such that k as k, ) (3.2) L k k u k = i (a k ( x ) ij j u k = k in {(x, x n ) : x < 2 and ψ k (x ) < x n < ψ k (x ) + 2}, u k = on {(x, ψ k (x )) : x < 2}, and (3.22) 2 t x <2 x < u k (x, ψ k (x ) + s) p dx ds =, u k (x, ψ(x ) + s) p dx ds > t p+α, where the coefficients a k ij (x) of Lk are real symmetric and satisfy (.2)-(.3), ψ k M and ψ k () =. Let (3.23) b k ij = a k il(y) { δ jl l χ k j (y) } dy, [,] n where χ k = (χ k,..., χ k n) are correctors for L k. Since b k ij subsequence, we may assume that are bounded, by passing to a (3.24) b ij = lim k bk ij exists for i, j n. It is known that the constant matrix (b ij ) is symmetric and satisfies (.2) [BLP]. Note that the sequence {ψ k } is equi-continuous on {x : x 2}. Thus, without loss of generality, we may assume that ψ k converges uniformly to ψ on {x : x 2}. Clearly ψ M and ψ () =. By the classical regularity estimates, {u k } is uniformly Hölder continuous on {(x, x n ) : x r and ψ k (x ) x n ψ k (x ) + r} for any < r < 2. It follows that the sequence {u k (x, ψ k (x ) + s)} is equi-continuous on Q r = {(x, s) : x r and s r} for any < r < 2. Hence, by passing to a subsequence, we may

11 HOMOGENIZATION PROBLEMS assume that u k (x, ψ k (x ) + s) converges uniformly to u (x, ψ (x ) + s) on Q 3/2. We may also assume that u k (x, ψ k (x ) + s) converges weakly to u (x, ψ (x ) + s) in W,2 (Q 3/2 ). By (3.22) and the uniform convergence of u k, we have (3.25) 3 2 t x < 3 2 x < u (x, ψ (x ) + s) p dx ds, u (x, ψ (x ) + s) p dx ds t p+α. This contradicts with estimates (3.7) and (3.9), since u is a solution of b ij i j u =, with b ij given by (3.24) [BLP], in {(x, x n ) : x < 3/2 and ψ (x ) < x n < ψ (x ) + 3/2} and u = on {(x, ψ (x )) : x < 3/2}. We are now in a position to give the proof of Theorem 3.. Proof of Theorem 3.. Let be given by Lemma 3.4. It suffices to establish estimate (3.2) for r =. Let L u = in D 2 and u = on 2 for some <. Let v(x) = u (θx) where < /θ. Then L v = in θ (3.26) { (x, x n ) : x < 2θ and θ ψ(θx ) < x n < θ ψ(θx ) + 2θ }. Let ψ θ (x ) = θ ψ(θx ). Note that ψ θ = ψ and ψ θ () =. It follows from Lemma 3.4 that t v(x, θ ψ(θx ) + s) p dx ds (3.27) x < t p+α By a change of variables, this gives (3.28) θt x <θ 2 x <2 t p+α 2θ x <2θ v(x, θ ψ(θx ) + s) p dx ds.. By covering {x : x < + θ} with balls of radius θ, we may deduce from (3.28) that (3.29) θt x <+θ C n t p+α 2θ x <+2θ,

12 2 ZHONGWEI SHEN where C n depends only on n. Now suppose that t k+ < t k for some k. Since /t j tk j, it follows from (3.29) with θ = t j that (3.3) t j+ x <+t j C n t p+α t j x <+t j 2, for j =,..., k. By choosing t small we may assume that C n t α α for some < α < α. This implies that (3.3) t j x < C(C n t p+α ) j 2 C(t j ) p+α 2 x <2 x <2, for j =,..., k, where t k /. It is not hard to see that this yields the desired estimate (3.2) for r =. 4. The case of C boundary Let ψ : R n R be a C function with compact support such that ψ() = ψ() =. Suppose that L u = in D 6r and u = in 6r. Then for any 2 < p <, { } /p { } /2 (4.) r n u p dx C D r r n u 2 dx, D 6r where C depends on µ, n, p, ω(t) as well as on the modulus of continuity of ψ, (4.2) η(t) = sup { ψ(x ) ψ(y ) : x y < t }. The proof of (4.) follows the same line of argument as in the case of Lipschitz boundary. We remark that the W,p boundary estimates for operators with VMO coefficients hold on C domains for any 2 < p < (see e.g. [AQ,S]). One also has a stronger Hölder estimate (4.3) u (x, ψ(x ) + s) C s β D 3/2 u dy for any < β <, where u is a solution of an elliptic equation with constant coefficients L u = in D 3/2 and u = on 3/2. This, together with (3.6), shows that for any 2 < p <, < t < and < α <, (4.4) t x < u (x, ψ(x ) + s) p dx ds Ct p+α 3 2 x < 3 2 u (x, ψ(x ) + s) p dx ds

13 HOMOGENIZATION PROBLEMS 3 where C > depends only on µ, n, p, α and function η(t). The rest of the proof is exactly the same as in the Lipschitz case. We omit the details. It worths pointing out that by Sobolev imbedding, estimate (4.) implies the uniform boundary C α estimate for any < α <, ( ) α { } /2 d(x) (4.5) u (x) C r r n u (y) 2 dy D 2r for x D r, where C > depends on µ, n, α as well as on functions ω(t) and η(t). 5. Proof of Theorem. and Corollary.2 By uniform interior W,p estimates in section 2 and uniform boundary estimates in section 3, the weak reverse Hölder inequality (.9) holds for p = 3 in the case n 3, and for p = 4 in the case n = 2. If Ω is a C domain, the interior estimates in section 2 and boundary estimates in section 4 give (.9) for any 2 < p <. It then follows from Theorem.3 that the Riesz transforms (L ) /2 are uniformly bounded on L p (Ω) for < p < 3 + δ in the case n 3, and for < p < 4 + δ in the case n = 2. If Ω is a C domain, the Riesz transforms are uniformly bounded on L p (Ω) for any < p <. This completes the proof of Theorem.. Let q > 2. Suppose that (L ) /2 is bounded on L p (Ω) for < p < q. By duality it follows that (L ) /2 is bounded on L p (Ω) for q < p <. Consequently, (L ) div is bounded on L p (Ω) for any q < p < q. Corollary.2 follows from this and Theorem.. [A] [Au] [AC] References G. Alexopoulos, La conjecture de Kato pour des opérateurs différentiels elliptiques, á coefficients périodiques, C. R. Acad. Sci. Paris Sér. I 32 (99), P. Auscher, On necessary and sufficient conditions for L p estimates of Riesz transform associated to elliptic operators on R n and related estimates, Preprint (24). P.Auscher and T. Couhlon, Riesz transform on manifolds and Poincaré inequalities, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 4 (25), [ACDH] P. Aucher, T. Coulhon, X.T. Duong, and S. Hofmann,, Riesz transforms on manifolds and heat kernel regularity, Ann. Sci. Ecole Norm. Sup. (4) 37 (24), [AT] [AT2] [AQ] [AL] [AL2] P. Auscher and Ph. Tchamitchian, Square Root Problem for Divergence Operators and Related Topics, Astérisque 249, Soc. Math. France, 998. P. Auscher and Ph. Tchamitchian, Square roots of elliptic second order divergence operators on strongly Lipschitz domains: L p theory, Math. Ann 32 (2), P. Auscher and M. Qafsaoui, Observation on W,p estimates for divergence elliptic equations with VMO coefficients, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (7) 5 (22), M. Avellaneda and F. Lin, Compactness methods in the theory of homogenization, Comm. Pure Appl. Math. 4 (987), M. Avellaneda and F. Lin, Homogenization of Poisson s kernel and applications to boundary control, J. Math. Pure Appl. 68 (989), -29.

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