Homogenization of Elliptic Boundary Value Problems in Lipschitz Domains

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1 Homogenization of Elliptic Boundary Value Problems in Lipschitz Domains arxiv: v1 [math.ap] 14 Aug 29 Carlos E. Kenig Zhongwei Shen Dedicated to the Memory of Björn Dahlberg Abstract In this paper we study the L p boundaryvalue problems for L(u) = in R d1, where L = div(a ) is a second order elliptic operator with real and symmetric coefficients. Assume that A is periodic in x d1 and satisfies some minimal smoothness condition in the x d1 variable, we show that the L p Neumann and regularity problems are uniquely solvable for 1 < p < 2δ. We also present a new proof of Dahlberg s theorem on the L p Dirichlet problem for 2 δ < p < (Dahlberg s original unpublished proof is given in the Appendix). As the periodic and smoothness conditions are imposed only on to regions above Lipschitz graphs. Consequently, by localization techniques, we obtain uniform L p estimates for the Dirichlet, Neumann and regularity problems on bounded Lipschitz domains for a family of second order elliptic operators arising in the theory of homogenization. The results on the Neumann and regularity problems are new even for smooth domains. the x d1 variable, these results extend directly from R d1 MSC 2: 35J25. Keywords: Boundary value problem; Periodic coefficient;lipschitz domain. 1 Introduction Let L = div(a ) be a second order elliptic operator defined in R d1 = { X = (x,t) R d R }, d 2. We will always assume that the (d1) (d1) coefficient matrix and satisfies the ellipticity condition, A = A(X) = (a i,j (X)) is real and symmetric, (1.1) µ ξ 2 a ij (X)ξ i ξ j 1 µ ξ 2 for all X,ξ R d1, (1.2) Supported in part by NSF grant DMS Supported in part by NSF grant DMS

2 where µ >. In this paper we shall be interested in boundary value problems for L(u) = in the upper half-space R d1 = R d (, ) with L p boundary data, under the assumption that the coefficients are periodic in the t variable, A(x,t1) = A(x,t) for (x,t) R d1. (1.3) More precisely, we will study the solvabilities of the L p Dirichlet problem (D) p { L(u) = in Ω = R d1, u = f L p ( Ω) n.t. on Ω and (u) L p ( Ω), (1.4) the L p Neumann problem (N) p L(u) = in Ω = R d1, u ν = f Lp ( Ω) on Ω and N( u) L p ( Ω), (1.5) where u denote the conormal derivative associated with operator L, and the ν Lp regularity problem (R) p {L(u) = in Ω = R d1, (1.6) u = f Ẇ1,p ( Ω) n.t. on Ω and N( u) L p ( Ω). Here (u) denotes the usual nontangential maximal function of u and N( u) a generalized nontangantial maximal function of u. By u = f n.t. on Ω we mean that u(x) converges to f(p) as X P nontangentially for a.e. P Ω. Under the periodic condition (1.3) as well as some (necessary) local solvability conditions on L, we will show that the L p Dirichlet problem is uniquely solvable for 2 δ < p <, and the L p Neumann and regularity problems are uniquely solvable for 1 < p < 2δ. Furthermore, the solution to the Dirichlet problem satisfies the estimate (u) p C u p, while the solutions to the L p Neumann and regularity problems satisfy N( u) p C u ν p and N( u) p C x u p, respectively. These results extend the work on the L p boundary value problems in R d1 for second order elliptic operators with t-independent coefficients (or in bounded star-like Lipschitz domains for operators with radially independent coefficients). As we shall discuss later, since the local solvability conditions may be deduced from certain minimal smoothness conditions in the t- variable and our periodic condition is only imposed on the t-variable, the L p global estimates on (u) and N( u) extend directly from R d1 to the regions above Lipschitz graphs. As a consequence, by well known localization techniques, we obtain uniform estimates for the L p boundary value problems in bounded Lipschitz domains for a family of second order elliptic operators arising in the theory of homogenization. We should point out that the result mentioned above for the Dirichlet problem is in fact due to the late B. Dahlberg [9] (unpublished, personal communication). His proof, which is given in the Appendix for the sake of reference, depends on the ingenious use of Green s functions and harmonic measures. As in the case of his celebrated theorem on the Dirichlet problem for u = in Lipschitz domains [1, 11], Dahlberg s method does not extend to the Neumann and regularity problems. Motivated by the work of Jerison and Kenig [16, 17, 18], we then seek to establish the global L 2 Rellich type estimate, u ν 2 x u 2 (1.7) 2

3 for suitable solutions of elliptic operators with t-periodic coefficients. In Section 3 we develop some new integral identities which play a crucial role in the proof of (1.7) and which may be regarded as the Rellich identities for operators with t-periodic coefficients. Indeed, in the case of constant or t-independent coefficients, Rellich identities are usually derived by using integration by parts on a form involving u. The basic insight here is to replace the t t derivative of u by the difference Q(u)(x,t) = u(x,t1) u(x,t). The t-periodicity of A(x,t) is used in the fact that Q commutes with L. In particular, Q(u) is a solution whenever u is a solution. We further remark that these integral identities allow us to control the near-boundary integral 1 u(x,t) 2 dxdt, (1.8) R d by the integral of u ν 2 or x u 2 on R d. The desired estimate (1.7) then follows by local solvability conditions. With (1.7) at our disposal, we are able to give another proof of Dahlberg s theorem on the Dirichlet problem in Section 4 and more importantly, solve the L 2 regularity problem in Section 5 and the L 2 Neumann problem in Section 8. Furthermore, using the L 2 estimates and following the approach developed in [13] and [21], we establish the solvability of the L p regularity and Neumann problems for 1 < p < 2 in Sections 6 and 9. Although the range 2 < p < 2δ may be also treated by the real variable arguments used in [13, 21], we choose to use a relatively new real variable approach, based on the weak reverse Hölder inequalities [25, 26]. In particular it allows us to show that for elliptic operators with coefficients satisfying (1.1)-(1.2), if the L 2 regularity problem (R) 2 is solvable and 1 1 = 1, then the Dirichlet p q problem (D) q and the regularity problem (R) p are equivalent. There exists an extensive literature on boundary value problems with minimal smoothness assumptions on the coefficients or on the boundary of the domain in question. In particular the L p Dirichlet, regularity and Neumann problems for u = in Lipschitz domains, which are closely related to the problems we investigate here, were well understood thanks to [1,11,16,17,18,29,13]. DeepresultsontheL p Dirichlet problemforageneral secondorder elliptic equation L(u) = may be found in [8, 12, 14, 15] (see [19] for further references). The L p regularity and Neumann problems for L(u) = were formulated and studied in [21, 22]. If the coefficient matrix A is t-independent, the L p solvability of the Neumann and regularity problems in the upper half-space was essentially established in [21] for the sharp range 1 < p < 2δ, although the paper only treats the case of radially independent coefficients in the unit ball. We also mention that the L 2 boundary value problems for some operators with t-independent complex coefficients was recently studied by the method of layer potentials in [1]. Related work on L p boundary value problems for operators with t-independent, but non-symmetric coefficients may be found in [2, 23]. Regarding the conditions on the t variable, it worths mentioning that the global estimate (u) p C f p may fail for any p, even if the matrix A(x,t) satisfies (1.1)-(1.2) and is C in R d1. Since the local solvability condition on L may be deduced from certain minimal smoothness assumption in the t variable on the coefficient matrix A, as we alluded to earlier, Dahlberg s theorem on the Dirichlet problem and our results on the Neumann and regularity problems extend readily to the case where D = { (x,t) R d1 : t > ψ(x) } (1.9) 3

4 is the region above a Lipschitz graph, using the bi-lipschitzian map (x,t) (x,t ψ(x)) from D to R d1. Indeed, let and assume that η(ρ) = sup { A(x,s 1 ) A(x,s 2 ) : x R d and s 1 s 2 ρ } (1.1) 1 ( ) 2 η(ρ) dρ <. (1.11) ρ The following three theorems are proved in Section 1. The constant δ in Theorems 1.1, 1.2 and 1.3 depends only on d, µ, η(ρ) and x ψ. Theorem 1.1. Let L = div(a ) with coefficient matrix satisfying (1.1), (1.2), (1.3) and (1.11). Let D be given by (1.9). Then there exists δ (,1) such that for any f L p ( D) with 2 δ < p <, there exists a unique solution to L(u) = in D with the property that (u) L p ( D) and u = f n.t. on D. Moreover, the solution u satisfies the estimate (u) p C p f p, where C p depends only on d, p, µ, η(ρ) and x ψ. Theorem 1.2. Under the same assumptions on L and D as in Theorem 1.1, there exists δ > such that given any g L p ( D) with 1 < p < 2δ, there exists a solution, unique up to constants, to L(u) = in D such that N( u) L p ( D) and u = g on D. Moreover, ν the solution satisfies the estimate N( u) p C p g p, where C p depends only on d, p, µ, η(ρ) and x ψ. Let tan u denote the tangential derivatives of u on Ω. We say f Ẇ1,p ( Ω) if f L p loc ( Ω) and tanf L p ( Ω). Theorem 1.3. Under the same assumptions on L and D as in Theorem 1.1, there exists δ > such that given any f Ẇ1,p ( D) with 1 < p < 2δ, there exists a unique solution to L(u) = in D such that N( u) L p ( D) and u = f n.t. on D. Moreover, the solution satisfies the estimate N( u) p C p tan f p, where C p depends only on d, p, µ, η(ρ) and x ψ. As we indicated earlier, Theorem 1.1 aswell astheorem 1.4below aredue tob. Dahlberg [9]. However, under the t-periodic condition, Theorems 1.2 and 1.3 are new even for operators with smooth coefficients on smooth domains. We further point out that by well known localization techniques, Theorems 1.1, 1.2 and 1.3 allow one to establish uniform estimates for the L p boundary value problems in Lipschitz domains for a family of second order elliptic operators arising in the theory of homogenization (see e.g. [7]). More precisely, we consider L ε = div ( A ( ) ) X, ε >, (1.12) ε where the matrix A(X) is periodic with respect to the standard lattice, A(X Z) = A(X) for any X R d1, Z Z d1. (1.13) 4

5 In view of (1.11), we will assume that A(X) is continuous and its modulus of continuity satisfies 1 (ω(ρ)) 2 dρ <, (1.14) ρ where ω(ρ) = sup { A(X) A(Y) : X,Y R d1 and X Y ρ }. The proofs of the following three theorems are given in Section 11. Theorem 1.4. Assume that the coefficient matrix A(X) is real, symmetric, and satisfies the ellipticity condition (1.2), the periodicity condition (1.13) and the smoothness condition (1.14). Let Ω be a bounded Lipschitz domain in R d1. Then there exist constants δ (,1) and C > independent of ε, such that if f L p ( Ω) for some 2 δ < p <, then the unique solution to the L p Dirichlet problem: L ε (u ε ) = in Ω, u ε = f n.t. on Ω, (u ε ) L p ( Ω), satisfies the estimate (u ε ) p C f p. Let u ν ε denote the conomal derivative associated with the operator L ε. Theorem 1.5. Let Ω be a bounded Lipschitz domain in R d1. Assume that A(X) satisfies the same assumption as in Theorem 1.4. Then there exist constants δ (,1) and C > independent of ε, such that given any g L p ( Ω) with 1 < p < 2δ and mean value zero, the unique solution to the L p u Neumann problem: L ε (u ε ) = in Ω, ε ν ε = g on Ω, N( u ε ) L p ( Ω), satisfies the estimate N( u ε ) p C g p. Theorem 1.6. Let Ω be a bounded Lipschitz domain in R d1. Assume that A(X) satisfies the same assumption as in Theorem 1.4. Then there exist constants δ (,1) and C > independent of ε >, such that given any f W 1,p ( Ω) with 1 < p < 2 δ, the unique solution to the L p regularity problem: L ε (u ε ) = in Ω, u ε = f n.t. on Ω, N( u ε ) L p ( Ω) satisfies the estimate N( u ε ) p C { } tan f p Ω 1 1 d f p. We remark that Dahlberg s work on operators with periodic coefficients was inspired by a series of remarkable papers [2, 3, 4, 5, 6] of M. Avellaneda and F. Lin on elliptic homogenization problems. In particular, the uniform estimate (u ε ) p C f p in Theorem 1.4 for 1 < p < was established on C 1,γ domains by Avellaneda and Lin in [2, 3, 6] for second order elliptic equations and systems in divergence form with C α periodic coefficients. RelatedworkontheboundednessofRiesztransforms (L ε ) 1/2 onlipschitzandc 1 domains may be found in [28]. However, to the authors best knowledge, the uniform L p estimates for periodic operators have not been studied before for the regularity and Neumann problems; Theorems 1.5 and 1.6 are new even for smooth domains. Finally we mention that as in the case of operators with constant coefficients, the Rellich estimates we develop in this paper, can be used to solve the L p boundary value problems by the method of layer potentials. This would enable us to represent the solutions in Theorems 1.4, 1.5 and 1.6 in terms of layer potentials with density functions uniformly bounded in appropriate spaces. Even more importantly, the method of layer potentials can be applied to elliptic systems and would allow us to establish uniform L p estimates for elliptic systems with periodic coefficients in nonsmooth domains. This line of ideas will be fully developed in a forthcoming paper [24]. 5

6 Acknowledgment. The first named author is indebted to the late Björn Dahlberg for sharing withhim theproofoftheorem 4.1ontheDirichlet problem. Our workwas motivatedinpart by Dahlberg s theorem. Dahlberg died suddenly on January 3, 1998 and did not publish his proof. As we mentioned earlier, we include a version of his original proof in the Appendix for future reference. We dedicate this paper to the memory of Björn Dahlberg. 2 Notations and Preliminaries We will use X,Y,Z to denote points in R d1 and x,y,z points in R d. Balls in R d are denoted by B(x,r) = {y R d : y x < r}. By T(x,r) we mean the cylinder B(x,r) (,r). We will use for the gradient in R d1 and x for the gradient in R d. The classical Dirichlet problem in Ω = R d1. Let f C(R d ) such that f(x) as x. Then there exists a unique weak solution to L(u) = in R d1 such that u C(R d1 [, )), u(x,) = f(x) for x R d and u is bounded in R d1. Moreover, the solution satisfies sup{ u(x,t) : x R d,t > } f. The uniqueness follows from Louiville s Theorem by a reflection argument. To establish the existence, onemayassumethatf. LetΩ k = T(,k) = B(,k) (,k)andϕacontinuous decreasing function on [, ) such that ϕ 1, ϕ(s) = 1 for s (1/2) and ϕ(s) = for s > (3/4). Let u k be the solution to the classical Dirichlet problem for L(u) = in T(,k) with boundary data f(x)ϕ( x /k) on {(x,) : x < k} and zero otherwise. Since u k u k1 on Ω k and u k L (Ω k ) f, it follows that u(x) = lim k u k (x) exists and is finite for any x R d1 and u L (R d1 ) f. Also, we may deduce from the estimate sup Ω l u k u m C(u k u m )(,l/2), for k > m > l that as k, u k converges to u uniformly on Ω l for any l > 1. This implies that L(u) = in Ω = R d1, u is continuous in Ω and u(x,) = f(x). By the Riesz representation theorem, there exists a family of regular Borel measures {ω X : X R d1 } on R d, called L-harmonic measures, such that u(x) = f(y)dω X (y). (2.1) R d Let v k be the solution to the classical Dirichlet problem in R d1 with data ϕ( x /k). It follows from the uniqueness that v k (X) 1 as k for any X R d1. This shows that ω X is a probability measure, as in the case of bounded domains. Furthermore, let ωk X denote the L-harmonic measure for the cylinder B(,k) (,k). It is not hard to see that ωk X(B) ωx (B) as k, for any ball B R d and X R d1. Green s function on Ω = R d1. Let G k (X,Y) denote the Green s function for L on the domain Ω k = T(,k). Since G k (X,Y) G k1 (X,Y) for X,Y Ω k, the Green s function for L on Ω = R d1 may be defined by G(X,Y) = lim k G k (X,Y) for X,Y R d1 and X Y. 6

7 By well known properties of G k, we have G(X,Y) C X Y d 1 and G(X,Y) Ct α X Y d 1α (2.2) for X = (x,t), Y R d1, where C and α are positive constants depending only on d and µ. Since ω X k (B(y,r)) rd 1 G k (X,(y,r)) for any X Ω k \T(y,2r) [8], by letting k, we obtain ω X (B(y,r)) r d 1 G(X,(y,r)) for any X / T(y,2r). (2.3) Two properties of nonnegative weak solutions. Let u be a nonnegative weak solution of L(u) = in T(x,2r) for some x R d and r >. Suppose that u C(T(x,2r)) and u(x,) = on B(x,2r). Then we have the boundary Harnack inequality, u(x,t) Cu(x,r) for any (x,t) T(x,r). (2.4) We will also need the following comparison principle, u(x, t) v(x,t) Cu(x,r) v(x,r) for any (x,t) T(x,r), (2.5) where u, v are two nonnegative weak solutions in T(x,2r) such that u,v C(T(x,2r)) and u(x,) = v(x,) = on B(x,2r) (see e.g. [8]). The constants in (2.4) and (2.5) depend only on d and µ. Difference operator Q. Define Q(u)(x,t) = u(x,t1) u(x,t) for (x,t) R d1. (2.6) Clearly, Q( u x i ) = x i Q(u) for i = 1,...,d1. It is also easy to verify that Q(uv) Q(u)Q(v) = uq(v) vq(u) (2.7) and Q(u)(y,s)dyds= t1 u(y,s)dyds 1 T(x,t) t B(x,t) B(x,t) u(y, s) dyds. (2.8) Nontangential maximal functions. For a function u defined in R d1, we let ( N(u)(x) = sup 1 t B(x, t) 3t 2 t 2 B(x,t) 1/2 u(y,s) dyds) 2 : < t <. (2.9) This is a variant of the usual nontangential maximal function (u) which is defined by (u) (x) = sup { u(y,t) : < t < and y x < t }. (2.1) 7

8 Clearly, N(u)(x) (u) (x). If u is a weak solution to L(u) = in R d1 or any function that has the property ( 1/2 1 u(x,t) C u(y,s) dyds) 2, r d1 B((x,t),r) whenever B((x,t),r) R d1, then N(u) p (u) p for any < p <. We also need to introduce N r (u) and (u) r which are defined by restricting the variable t in (2.9) and (2.1) respectively to < t < r/2. The definitions of N(u) and (u) extend naturally to regions above Lipschitz graphs and to bounded Lipschitz domains. Weconcludethissectionwithalemmawhichallowsustoestimate N( u) p by N 4 ( u) p and (Q(u)) p. Lemma 2.1. Let u C 1 (T(,3R)) be a solution to L(u) = in T(,3R) for some R > 1. Let f(x) = u(x,). Then for any x B(,R), ( ) [N4 ] N R ( u)(x) CM ( u) x f (Q(u)) R χb(,3r) (x), (2.11) where M denotes the Hardy-Littlewood maximal operator on R d and C depends only on d and µ. Proof. We begin by fixing (x,t ) R d1 with x B(,R) and t (2,R/2). Note that by Cacciopoli s inequality and interior regularity of weak solutions, ( 1 t d1 s t < t 2 C t ( 1 C t d2 t d1 B(x,t ) ) 1/2 u(y,s) 2 dyds s t <α 1 t s t <α 2 t B(x,β 2 t ) B(x,β 1 t ) ) 1/2 u(y,s) E 2 dyds u(y,s) E dyds, (2.12) where (1/2) < α 1 < α 2 < (3/4), 1 < β 1 < β 2 < (3/2) and E R. To estimate the right-hand side of (2.12), we use u(y,s) E u(y,s) u(y,) u(y,) E and choose E to be the average of u(y,) over B(x,β 1 t ). By Poincaré s inequality, we have 1 u(y,) E dyds t d2 s t <α 2 t B(x,β 2 t ) C t d x f dx CM ( ) (2.13) x f χ B(,3R) (x ). B(x,β 2 t ) Next we observe that for any s (,2t ), 1 u(y,s) u(y,) Ct (Q(u)) R (y) u(y, t) dt. (2.14) 8

9 In view of (2.12), the first term in the right-hand side of (2.14) can be handled easily by M ( (Q(u)) R χ B(,3R)) (x ). Thus it remains to estimate 1 t d1 B(x,β 2 t ) This may be done by observing that C N 4 ( u)dx B(x,4t ) 1 1 u(y, t) dtdy. B(x,2t ) u(y, s) dyds. Remark 2.2. It follows from (2.11) that for 1 < p <, N R ( u) p dx C x u(x,) p dxc B(,R) B(,3R) C ( Q(u) ) R p dx B(,3R) B(,3R) N 4 ( u) p dx (2.15) for any R > 1. Thus, if u C 1 (R d [, )) and L(u) = in R d1, we may let R in (2.15) to obtain N( u) p dx C x u(x,) p dxc N 4 ( u) p dx R d R d R d C ( (2.16) Q(u) ) p dx, R d where C depends only on d, p and µ. 3 Integral identities In this section we develop some new integral identities that will play a key role in our approach to the L 2 boundary value problems in R d1 for elliptic equations with t-periodic coefficients. Lemma 3.1. Let L = div(a ) with coefficients satisfying conditions (1.1)-(1.2). Let Ω R d1 be a bounded Lipschitz domain and u W 1,2 (Ω) a variational solution to the Neumann problem: L(u) = in Ω, u = g ν L2 ( Ω). Assume that u W 1,2 (Ω 1 ) where Ω 1 contains Ω and {(x,t1) : (x,t) Ω}. Then Ω a ij Q ( u u ) dx x i x j = 2 gq(u)dσ, Ω Ω a ij Q ( u x i ) Q ( u x j ) dx where indices i,j are summed from 1 to d1 and we identify x d1 with t. 9 (3.1)

10 Proof. Since Q(u) W 1,2 (Ω), it follows from the definition of variational solutions that u gq(u)dσ = a ij Q ( u ) dx Ω Ω x j x i = 1 { u a ij Q ( u) u Q ( } u ) dx, 2 x j x i x i x j Ω where we have used the symmetry condition a ij = a ji in the second equation. In view of (2.7), this gives = ΩgQ(u)dσ 1 { a ij Q ( u u } ) ( u ( u ) Q ) Q dx, 2 Ω x i x j x i x j from which the desired identity follows. In the rest of this section we will assume that a ij satisfy the periodic condition (1.3). To justify the use of integration by parts, we will also assume that both a ij and u are sufficiently smooth. However all constants C in this section depend only on d and µ. The following lemma is crucial in our argument for the L 2 Neumann problem. Lemma 3.2. Let L = div(a ) with C coefficients satisfying (1.1), (1.2) and (1.3). Let R > 1 and u C 1 (Ω) be a solution to L(u) = in Ω = T(x,3R) and u = g on ν B(x,3R) {}. Then 1 B(x,R) u 2 dxdt C where C depends only on d and µ. B(,r) B(x,2R) g 2 dx C R T(x,3R)\T(x,R) u 2 dxdt, (3.2) Proof. We may assume that x =. Let Ω r = T(,r) = B(,r) (,r) for r (R,2R). By the periodicity assumption (1.3) and (2.8), we have a ij Q ( u u ) dxdt = Q ( u a ij u ) dxdt Ω r x i x j Ω r x i x j 1 u = a ij u r1 u dxdt a ij u dxdt. x i x j x i x j r B(,r) It then follows from (3.1) that 1 B(,r) = 2 u a ij u dxdt a ij Q ( u) ( u ) Q dxdt x i x j Ω r x i x j Ω r u ν Q(u)dσ r1 r B(,r) a ij u x i u x j dxdt, (3.3) 1

11 for r (R,2R). By the Cauchy inequality, this leads to 1 u 2 dxdt δ Q(u)(x,) 2 dxc δ B(,r) B(,r) C u(x,r) 2 dxc for any < δ < 1. Using B(,r) r B(,r) C C B(,r) r1 r r B(,r) B(,r) Q(u)(x,t) 2 dx Q(u) 2 dσdt we obtain from (3.4) that 1 u 2 dxdt C B(,R) B(,r) u 2 dxdtc Q(u) 2 dσdt, t1 t r1 B(,2R) r1 C C B(,r) B(,r) g(x) 2 dxc r 3R B(,2R) B(,r) B(,r) r g(x) 2 dx Q(u)(x,r) 2 dx B(,r) u 2 dxds, u 2 dσdt, B(,2R) u 2 dxdt u 2 dσdt, u 2 dσdt u(x,r) 2 dx (3.4) (3.5) for r (R,2R). The desired estimate now follows by integrating both sides of (3.5) with respect to r over the interval (R,2R). Remark 3.3. Under the same assumption as in Lemma 3.2, we also have 6 u 2 dxdt C g 2 dx C u 2 dxdt. (3.6) R B(x,R) B(x,2R) T(x,3R)\T(x,R) To see (3.6), one simply replaces Q(u) with Q(u) = u(x,t6) u(x,t) in the proof. The following lemma is the key in our approach to the L 2 regularity problem. Lemma 3.4. Under the same assumptions on L as in Lemma 3.2, there exists C = C(d,µ) > such that 1 u 2 dxdt C x f 2 dx C u 2 dxdt, (3.7) R B(x,R) B(x,2R) T(x,3R)\T(x,R) where u C 1 (T(x,3R)) C 2 (T(x,3R)) is a solution to L(u) = in T(x,3R) and u = f on B(x,3R) {}, R > 1. 11

12 Proof. Wemayassumethatx =. AsintheproofofLemma3.2, weletω r = B(,r) (,r) for r (R,2R). By the periodicity assumption, Q(u) is a solution in Ω r. It follows that for r (R,2R), u Ω r ν Q(u)dσ = u Ω r ν Q(u)dσ 1 { = un i a ij (x,ts) u } (x,ts) dsdσ(x,t) Ω r s x j { 1 = un i a ij (x,ts) u } (3.8) (x,ts)ds dσ(x, t) Ω r t x j = u ( { n i Ω r t n ) 1 d1 a ij (x,ts) u } (x,ts)ds dσ(x, t) x i x j { ( = ni Ω r t n ) 1 d1 u a ij (x,ts) u } (x,ts)ds dσ(x,t). x i x j Consequently, by the Cauchy inequality, we obtain u Ω r ν Q(u)dσ 1 C tan u(x,t) u(x,ts) dσ(x,t)ds Ω r 1 δ u(x,t) 2 dxdtc δ x u(x,) 2 dx C C B(,r) B(,r) r1 u(x,r) 2 dxc B(,r) B(,r) r1 u(x,t) 2 dσ(x)dt, r B(,r) u(x,t) 2 dxdt (3.9) for any < δ < 1, where tan denotes the tangential gradient on Ω r. This, together with (3.3), implies that 1 C B(,R) B(,2R) 3R C u 2 dxdt x f 2 dxc B(,r) B(,2R) u 2 dσdtc u(x,r) 2 dx r1 r B(,2R) u 2 dxdt, (3.1) for r (R, 2R). The estimate (3.7) follows by integrating both sides of (3.1) with respect to r over the interval (R,2R). 12

13 Remark 3.5. By replacing Q(u) with Q(u)(x,t) = u(x,t6) u(x,t) intheproof oflemma 3.4, we may obtain 6 u 2 dxdt C x f 2 dx C u 2 dxdt. (3.11) R B(x,R) B(x,2R) T(x,3R)\T(x,R) 4 The L p Dirichlet problem: a theorem of Dahlberg To solve the L p Dirichlet problem, we will impose a local solvability condition on L: for any x R d and < r 1, B(x,r) lim sup t ( u(x,t) t ) 2 dx C r 3 T(x,2r) u(x,t) 2 dxdt, (4.1) whenever u W 1,2 (T(x,4r)) is a nonnegative weak solution of L(u) = in T(x,4r) such that u C(T(x,4r)) and u(x,) = on B(x,4r). Recall that for Z R d1 \T(x,2t), ω Z (B(x,t)) B(x, t) G((x,t),Z), (4.2) t where ω Z denotes the L-harmonic measure on R d1 and G(X,Z) the Green s function on R d1. By applying the condition (4.1) to u(x) = G(X,Z) for Z R d1 \ T(x,4r), we may deduce that ω Z is absolutely continuous with respect to the Lebesgue measure on R d. Furthermore, the kernel function satisfies the reverse Hölder inequality K(x,Z) := lim t ω Z (B(x,t)) B(x, t) ( ) 1/2 1 K(y,Z) 2 dy C K(y, Z) dy, (4.3) B B B B for B = B(x,r) with < r 1, if Z / T(x,2r). In fact, by the comparison principle (2.5), conditions (4.1) and (4.3) are equivalent. The goal of this section is to show that with the additional periodic condition (1.3), the reverse Hölder inequality (4.3) holds without the restriction r 1. As a consequence, the L p Dirichlet problem for L(u) = in R d1 is solvable for 2 δ < p <, under the assumptions (1.1), (1.2), (1.3) and (4.1). The following theorem is due to B. Dahlberg [9]. Theorem 4.1. Let L = div(a ) be an elliptic operator satisfying conditions (1.1), (1.2), (1.3) and (4.1). Then there exists δ = δ(d,µ,c ) (,1) such that given any f L p (R d ) with 2 δ < p <, there exists a unique weak solution to L(u) = in R d with the property that u = f n.t. on R d and (u) L p (R d ). Moreover the solution satisfies the estimate (u) p C p f p, where C p depends only on d, p, µ and the constant C in (4.1). The key step in the proof of Theorem 4.1 is the following lemma. 13

14 Lemma 4.2. Under the same assumption on L as in Theorem 4.1, estimate (4.1) and hence (4.3) hold for r > 1. Proof. We will prove this lemma under the additional assumption that the coefficient matrix A(X) is sufficiently smooth. The assumption allows us to use estimate (3.7). For the sake of completeness and future reference we shall provide a version of Dahlberg s original proof of Lemma 4.2 in the Appendix. We point out that our proofs of the theorems stated in the Introduction, given in Sections 1 and 11, involve an approximation argument and do not rely on Dahlberg s proof. The case 1 < r 1 follows easily by covering B(x,r) with balls of radius 1. If r > 1, we also cover B(x,r) with a sequence of balls {B(x k,1)} of radius 1 such that B(x,r) B(x k,1) B(x,r 1) and χ B(xk,1) C. k k Let I denote the left-hand side of (4.1). It follows from the assumption (4.1) (with r = 1) and boundary Harnack inequality (2.4) that I C u(x) 2 dx C u(x) 2 dx k T(x k,2) k T(x k,1) 1 (4.4) u(x) 2 dx C u(x,t) 2 dxdt. k T(x k,1) B(x,r1) In view of Lemma 3.4, we obtain I C r C r 3 T(x,3.5r) T(x,2r) u 2 dxdt C r 3 u 2 dxdt, T(x,3.6r) u 2 dxdt where we also used Cacciopoli s inequality as well as the Harnack inequality (2.4). For 1 < p <, we say the L p Dirichlet problem (D) p for L(u) = in R d1 is solvable if for any f C c (R d ), the solution to the classical Dirichlet problem for L(u) = in R d1 with boundary data f satisfies the estimate (u) p C f p. Lemma 4.3. Under the same assumption on L as in Theorem 4.1, the L p Dirichlet problem for L(u) = in R d1 is solvable for 2 δ < p <. Moreover, we have (u) p C f p for 2 δ < p <, where δ = δ(d,µ,c ) (,1) and C = C(d,µ,C,p) >. Proof. By the self-improving property of the reverse Hölder inequality, there exists q > 2 such that for B = B(x,r) and Z = (z,z d1 ) R d1 \T(x,4r), { } 1/q 1 K(x,Z) q dx C K(x,Z)dx = C ωz (B) C(z d1) α, (4.5) B B B B B r dα where we used (2.3) and (2.2). Since u(z) = K(x,Z)f(x)dx, R d 14

15 one may deduce from (4.5) that (u) (x) C { M( f p )(x) } 1/p for any x R d, where p = q < 2 and M denotes the Hardy-Littlewood maximal operator on R d. This gives (u) p C f p for any p > p = 2 δ. We need the following Cacciopoli-type lemma in the proof of uniqueness. Lemma 4.4. Let u, v be two weak solutions in B(x,2R) (r/2,3r) where R r. Suppose that either u or v is nonnegative. Then 2r r B(x,R) where C depends only on d and µ. u v dxdt C r 3r r 2 B(x,2R) u v dxdt, Proof. Assume that u. Let X B(x,R) (r,2r). Using Cauchy inequality and Cacciopoli s inequality, we obtain B(X, r 4 ) u v dy C r C r ( B(X,.5r) sup u B(X,.5r) ) 1/2 ( u 2 dy B(X,.5r) B(X,.5r) v dy C r v 2 dy B(X,.5r) ) 1/2 u v dy, where we used the Harnack s inequality in the last step. The desired estimate now follows by covering B(x,R) (r,2r) with balls of radius r/4. The proof for the case that v is similar. The next theorem concerns the uniqueness in the L p Dirichlet problem in R d1. Theorem 4.5. Let L = div(a ) with coefficients satisfying (1.1)-(1.2). Suppose that dω Z /dx = K(,Z) L q (R d ) for all Z R d1 and some q > 1. Let u be a weak solution to L(u) = in R d1 such that (u) L p (R d ), where p = q. Assume that u(x,t) as t for a.e. x R d. Then u in R d1. Proof. Fix Z R d1. Let G(X) = G(X,Z) be the Green s function with pole at Z. Choose ϕ(x) C (B(,l/2)) and ψ(t) C (1/(2l),2l) such that ϕ = 1 in B(,l/4), ϕ C/l and ψ = 1 on (1/l,l), ψ Cl on (1/(2l),1/l), ψ C/l on (l,2l). Then G ( ) u(z) = a ij uϕψ dy x j x i = R d1 R d1 a ij G x j u x i ( ϕψ ) dy R d1 u ( ) G a ij ϕψ dy. x i x j 15

16 This gives 1 l u(z) Cl 1 2l y < l 2 2l C l C l 2 Cl 2 l 1 3l C l = I 1 I 2 I 3, l l ( G u G u ) dy 1 l y <l 2l y < l 2 l 4 < y <l 2 ( G u G u ) dy ( G u G u ) dy G u dy C 3l l 2 l 2 G u dydt l 8 < y <l t y <l G u dy (4.6) where we used Lemma 4.4 for the second inequality. To estimate I 2, we note that G(Y) C(Z)/l d 1α for Y B(,l) (l/2,3l), where α >. This leads to I 2 C l dα Next we observe that B(,3l) I 1 C (u) dy Cl α d p (u) p as l. (4.7) y <l M ( K(,Z) ) M 2/l (u)dy, (4.8) where M denotes the Hardy-Littlewood maximal operator on R d and M r (u)(x) = sup { u(x,t) : < t < r }. (4.9) Since K(,Z) L p (R d ), we have I 1 C M 2/l (u) p. By the assumption, M 2/l (u)(x) for a.e. x R d as l. Since M 2/l (u) (u) L p (R d ), we obtain M 2/l (u) p as l. Finally, note that as l, I 3 C M ( K(,Z) ) { (u) dy C l 4 < y <l y > l 4 (u) p dy} 1/p. Thus we have proved that I 1 I 2 I 3 as l Hence u(z) = for any Z R d1. Theorem 4.1 is a direct consequence of Lemma 4.3 and the following theorem. Theorem 4.6. Let L = div(a ) with coefficients satisfying (1.1)-(1.2). Suppose that the L p Dirichlet problem for L(u) = in R d1 is solvable for some 1 < p <. Then for any f L p (R d ), there exists a unique weak solution in R d1 such that (u) L p (R d ) and u = f n.t. on R d. Moreover the solution satisfies the estimate (u) p C f p. 16

17 Proof. Let f C c (R d ) and u(x) = fdω X be the solution to the classical Dirichlet R d problem with data f. Since u(x,t) C (u) dy Ct d t d p (u) p Ct d p f p, B(x,t) itfollowsthatω X isabsolutelycontinuous withrespect todxonr d andk(,x) = dω X /dx L p (R d ). In view of Theorem 4.5, this gives the uniqueness in Theorem 4.6. To establish the existence, we let f L p (R d ) and choose f k C c (R d ) so that f k f in L p (R d ). Let u k be the solution to the classical Dirichlet problem in R d1 with data f k. It follows from Lemma 4.3 that (u k ) p C f k p and (u l u k ) p C f l f k p. This implies that u k converges to a weak solution u, uniformly on any compact subset of R d1. Standard limiting arguments show that (u) p C f p and u = f n.t. on R d. 5 L 2 estimates for the regularity problem Definition 5.1. Let 1 < p <. We say that the L p regularity problem (R) p for L(u) = in R d1 is solvable, if for any f C(R 1 d ), the unique solution to the classical Dirichlet problem with boundary data f satisfies the estimate N( u) p C x f p. In this section we study the L 2 regularity problem in R d1 under the condition that the coefficient matrix is periodic in the t direction. As in the case of the Dirichlet problem, we need a local solvability condition: for any x R d and < r 1, { Nr ( u) 2 dx C 2 1 B(x,r) B(x,2r) x f 2 dx 1 r T(x,2r) } u 2 dxdt, (5.1) whenever u W 1,2 (T(x,4r)) is a weak solution to L(u) = in T(x,4r) such that u C(T(x,4r)) and u = f C 1 (B(x,4r)) on B(x,4r) {}. We will also assume that the coefficient matrix A(X) is sufficiently smooth. However, constants C in all estimates will depend only on d, µ and C 1 in (5.1). We begin by observing that since u(x,t) u(x,) CtN 2t ( u)(x) (5.2) (see [21], pp ), condition (5.1) implies the local condition (4.1) for the Dirichlet problem. It also follows from (5.1) that { } N 4 ( u) 2 dx C x f 2 dx u 2 dxdt. (5.3) B(x,2) T(x,6) B(x,1) Furthermore, if L(u) = in R d1, u C(R d [, )) and u = f C(R 1 d ) on R d, we may integrate both sides of (5.3) with respect to x over R d to obtain { 6 } N 4 ( u) 2 dx C x f 2 dx u 2 dxdt. (5.4) R d R d R d 17

18 Theorem 5.2. Let L = div(a ) with smooth coefficients satisfying conditions (1.1), (1.2), (1.3). Also assume that L satisfies the condition (5.1). Then the L 2 regularity problem for L(u) = in R d1 is solvable. Moreover the solution u satisfies the estimate N( u) 2 C x f 2, where C depends only on d, µ and the constant C 1 in (5.1). In view of (3.11), we start with a decay estimate. Lemma 5.3. Under the same assumption on L as in Theorem 5.2, we have R u 2 dxdt as R, (5.5) T(,1R)\T(,R) where u is a classical solution to L(u) = in R d1 such that u = f C c (R d ) on R d. Proof. By Cacciopoli s inequality, R u 2 dxdt C T(,1R)\T(,R) R C T(,11R)\T(,R/2) x R 1 (u) 2 dx u 2 dxdt (5.6) for R sufficiently large. Since (u) L 2 (R d ), the right-hand side of (5.6) goes to zero as R. Proof of Theorem 5.2. Let f C 1(Rd ) and u be the solution to the classical Dirichlet problem with data f. It follows from Remark 2.2 and (5.4) that { N( u) 2 dx C x f 2 dx N 4 ( u) 2 dx ( Q(u) ) } 2 dx R d R d R d R { d 6 C x f 2 dx u 2 dxdt ( } (5.7) Q(u) ) 2 dx. R d R d R d To estimate the integral of u 2 over R d (,6), we let R in (3.11). In view of (5.5), we obtain 6 u 2 dxdt C x f 2 dx. (5.8) R d R d Finally, to handle the term with Q(u) in (5.7), we observe that u is also the unique solution to the L 2 Dirichlet problem for L(u) = in R d1 with boundary data f. This implies that (Q(u)) L 2 (R d ) and by Theorem 4.1, ( Q(u) ) 1 2 dx C Q(u) 2 dx C u 2 dxdt R d R d R d (5.9) C x f 2 dx, R d where we used (5.8) in the last step. The desired estimate N( u) 2 C x f 2 now follows from (5.7), (5.8) and (5.9). We omit the proof of the next lemma and refer the reader to [21] for an analogous result (Theorem 3.1, pp ) in the case of the unit ball. 18

19 Lemma 5.4. Let L = div(a ) be an elliptic operator satisfying (1.1)-(1.2). Suppose that L(u) = in R d1 and N r ( u) L p (R d ) for some 1 < p < and r >. Then u converges n.t. to f on R d and x f L p (R d ). Moreover, let V i (x,t) = 1 t B(x, t) 3t 2 t 2 B(x,t) u y i dyds for i = 1,...,d, then V i (x,t) converges weakly to f x i in L p (R d ) as t. In particular, x f p N( u) p. The next theorem concerns the uniqueness of solutions to the L p regularity problem. Theorem 5.5. Let L = div(a ) with coefficients satisfying (1.1)-(1.2). Let u be a weak solution to L(u) = in R d1 such that N( u) L p (R d ) for some 1 p < d, where α > 1 α is given by (2.2). Then u in R d1, if u(x,t) as t for a.e. x R d. Proof. The proof begins in the same way as that of Theorem 4.5. As such, this leads to u(z) I 1 I 2 I 3, where I 1, I 2 and I 3 are defined in (4.6). To estimate I 1, we use the estimate (2.2) and the observation that u(x,t) CtN( u)(x) for a.e. x R d. This yields that I 1 = Cl 2 2 l 1 2l y <l G u dy C l α as l, where we used the assumption p < d 1 α. Similarly, I 2 = C 3l l 2 l 2 y <l R d N( u)(y)dy (y,) Z d 1α Cl α N( u) p G u dydt C N( u)dy Cl 1 α d l d 1α p N( u) p y <l which also tends to zero as l, since p < d. Finally, note that 1 α I 3 = C l 2l G u dydt l 4 < y <2l t C N( u) dy, l d 1α y <2l as l. Thus we have proved that I 1 I 2 I 3 as l. Hence u(z) = for any Z R d1. Recall that f Ẇ1,p (R d ) if f W 1,p loc (Rd ) and x f L p (R d ). The following theorem provides the existence of solutions with boundary data in Ẇ1,p (R d ). Theorem 5.6. Let L = div(a ) with coefficients satisfying (1.1)-(1.2). Suppose that the L p regularity problem for L(u) = in R d1 is solvable. Then for any f Ẇ1,p (R d ), there exists a solution u to L(u) = in R d1 such that N( u) p C f p and u = f n.t. on R d. Proof. Let f Ẇ1,p (R d ) and choose f k C 1 (Rd ) such that x f k x f in L p (R d ). Let v k be the solution to the classical Dirichlet problem in R d1 with boundary data f k. By 19

20 assumption, we have N( v k ) p C x f k p and N( v k v l ) p C x (f k f l ) p. It follows that for any m > 1, (v k v l ) in L 2 (Ω m ) as k,l, where { Ω m = (x,t) R d1 : x < 3m and 1 } 3m < t < 3m. Thus there exists u m W 1,2 (Ω m ) such that v k α(k,m) u m in W 1,2 (Ω m ) as k, where α(k,m) is the average of v k over Ω m. Clearly, L(u m ) = in Ω m. By a limiting argument, we also have Ñm( u m ) L p (B(,m)) C x f p, (5.1) where the nontangential maximal function Ñm( u) is defined in a manner similar to N m, but the variable t in (2.9) is restricted to (1/m) < t < m. Next, since u m1 u m is constant in Ω m, one may define a function u W 1,p loc (Rd1 ) such that u u m is constant in Ω m for any m 1. It follows that u is a weak solution in R d1. Moreover, in view of (5.1), we have Ñm( u) L p (B(,m)) C x f p. (5.11) Letting m in (5.11) gives N( u) p C x f p. Similar argument also gives N( u v k ) p C x (f f k ) p. (5.12) Finally we note that by Lemma 5.4, u converges n.t. to h Ẇ1,p (R d ) and the average of u x i over B(x,t) (t/2,3t/2) converges weakly to h x i in L p (R d ) as t. This, together with the weak convergence of the average of v k x i to f k x i and estimate (5.12), implies that h x i = f x i on R d for i = 1,...,d. It follows that f h is constant. Thus, by subtracting a constant, we obtain u = f n.t. on R d. We conclude this section with two more theorems on the consequences of the solvability of the regularity problem. Theorem 5.7. Let L = div(a ) with coefficients satisfying (1.1)-(1.2). Let 1 < p,q < and 1 1 = 1. Then for L(u) = in Rd1 p q, the solvability of the L p regularity problem implies the solvability of the L q Dirichlet problem. Theorem 5.8. Let L = div(a ) with coefficients satisfying (1.1)-(1.2). Suppose that the L 2 regularity problem for L(u) = in R d1 is solvable. Let u W 1,2 (T(x,8r)) be a weak solution in T(x,8r) such that N r ( u) L 2 (B(x,4r)) and u = f on B(x,4r). Then Nr ( u) 2 dx C 2 x f 2 dx C u 2 dx. (5.13) B(x,r) B(x,2r) r T(x,2r) We omit the proof of both theorems and refer the reader to Theorems 5.4 and 5.19 in [21], where the analogous results were proved in the case of the unit ball. We point out that Theorem 5.19 in [21] was stated for solutions in the whole domain. However an inspection of its proof, which extends readily to the case of R d1, shows that the conclusion holds for local solutions. Theorem 5.8 shows that the local solvability condition (5.1) is necessary. 2

21 6 L p estimates for the regularity problem In this section we study the L p regularity problem in R d1 for 1 < p < 2δ. Theorem 6.1. Let L = div(a u) with coefficients satisfying (1.1)-(1.2). Suppose that (R) 2 for L(u) = in R d1 is solvable. Then (R) p for L(u) = in R d1 is solvable for 1 < p < 2δ. In the case of the unit ball, Kenig and Pipher [21] proved that the solvability of (R) q implies the solvability of (R) p for 1 < p < q δ. Although it is possible to extend their results to the case of the upper half-space and thus obtain the L p solvability for 1 < p < 2δ from the L 2 solvability, we will provide a more direct proof. For the range 1 < p < 2, we follow the approach used in [13] (also see [21]) by proving N( u) 1 C for solutions of the L 2 regularity problem with Hat 1 data. Our approach to the range 2 < p < 2 δ, which is based on a real variable argument developed by Shen [25, 26], is different from that used in [13, 21]. The rather general approach reduces the problem to certain weak reverse Hölder estimates. In particular, our argument shows that if (R) 2 for L(u) = in R d1 is solvable, then the solvability of (D) q for some q < 2 implies the solvability of (R) p, where 1 1 = 1. p q We start with a comparison principle. Lemma 6.2. Let u,v C(T(x,3r)) be two weak solutions in T(x,3r) such that u(x,) = v(x,) = on B(x,3r). Suppose that v is nonnegative in T(x,3r). Then u(x,t) C ( ) 1/2 v(x, t) 1 u(y,s) 2 dyds v(x,r) r d1 T(x,3r) for any (x,t) T(x,r), where C depends only on d and µ. Proof. See Lemma 2.5 in [27]. The following is a localization result similar to Theorem 5.8. Note that here we assume the solvability of (D) 2 instead of the solvability of (R) 2. Theorem 6.3. Let L be an elliptic operator with coefficients satisfying (1.1)-(1.2). Assume that the L 2 Dirichlet problem for L(u) = in R d1 is solvable. Then N r ( u) 2 dx C u(x,t) 2 dxdt, (6.1) r 3 B(x,r) T(x,4r) where u C(T(x,6r)) is a weak solution of L(u) = in T(x,6r) and u(x,) = on B(x,6r). 21

22 Proof. Let v(x,t) be the Green s function for L on R d1 with pole at X = (x,1r). Fix x B(x,r ) and < t < r. By Cacciopoli s inequality, 2 3t 1 2 u 2 C 2t dyds u(y,s) 2 dyds t B(x, t) t B(x,t) t 3 B(x,t) t B(x,2t) 2 4 C 2t v(y, s) 2 1 t 3 B(x,t) t B(x,2t) v(x 4,r) dyds u(y,s) 2 dyds r d1 T(x,4r) ( ) 2 v(x,t) Cr 2d 2 1 u(y,s) 2 dyds, t r d1 T(x,4r) where we have used Lemma 6.2 and v(x,r) r 1 d. Let ω denote the L-harmonic measure for R d1, evaluated at (x,1r). Since v(x, t) t ω(b(x,t)) B(x, t) CM ( dω) χ B(x,2r) (x) dx where M denotes the Hardy-Littlewood maximal operator on R d, we obtain N r ( u)(x) Cr d 1 M ( dω) 1/2 1 χ B(x,2r) (x) ( u(y,s) dyds) 2. (6.2) dx r d1 T(x,4r) Since the L 2 Dirichlet problem for L(u) = in R d1 is solvable, the reverse Hölder inequality (4.3)forK = dω holdsonb(x dx,2r). ThedesiredestimatenowfollowsbytheL 2 boundedness of M and (4.3). Remark 6.4. Suppose that (D) p for L(u) = in R d1 is solvable for some p < 2. Then the reverse Hölder inequality (4.5) holds for q = p. It follows from (6.2) and (4.5) that ( ) 1/q 1 N r d r ( u) q dx C ( ) 1/2 1 u(y,s) 2 dyds B(x,r) r r d1 T(x,4r) ( ) 1/2 1 C u(y,s) 2 dyds (6.3) r d1 ( 1 C r d B(x,6r) T(x,4r) N( u) 2 dx) 1/2. A simple geometric observation shows that for x B(x,r), N( u)(x) N r ( u)(x) C N( u) dx. (6.4) r d In view of (6.3), this gives ( 1/q 1 N( u) dx) q C r d B(x,r) B(x,6r) ( 1/2 1 N( u) dx) 2. (6.5) r d B(x,6r) We will need this weak reverse Hölder estimate to treat the case 2 < p < 2 δ for the regularity problem. 22

23 Theorem 6.5. Under the same conditions on L as in Theorem 6.1, the L p regularity problem in R d1 is solvable for 1 < p < 2. Proof. We follow the approach in [13] and prove that N( u) 1 C for solutions of the L 2 regularity problem with Hat 1 data. To this end, let f be a Lipschitz function such that supp(f) B(x,r) and x f Cr d for some r > and x R d. Let u be the solution ofthel 2 regularityproblemwithdataf. BytheL 2 estimate N( u) 2 C x f 2, onehas N( u) L 1 (B(x,Cr)) C. Thus it suffices to show that if R r and B(y,1R) B(x,r) =, ( r ) α N( u) dx C (6.6) B(y,R) R for some α >. This follows from Theorem 6.3 and the pointwise estimate where X = (x,r). We omit the details. u(x) f ω X (B(x,r)) Cr 1 d ω X (B(x,r)) Cr α CG(X,X ) X X d 1α, If Ω is a bounded Lipschitz domain, it was proved in [27] that the solvability of (R) p for L(u) = in Ω is equivalent to the solvability of (D) p for L(u) = in Ω, provided that (R) p for L(u) = in Ω is solvable for some p > 1. We extend this result to the case Ω = R d1. Theorem 6.6. Let L = div(a ) with coefficients satisfying (1.1)-(1.2). Suppose that (R) 2 for L(u) = in R d1 is solvable. Let p > 2 and q = p. Then (R) p for L(u) = in R d1 is solvable if and only if (D) q for L(u) = in R d1 is solvable. The proof of Theorem 6.6 relies on a real variable argument which may be formulated as follows. Theorem 6.7. Let F L 2 (R d ) and g L p (R d ) for some 2 < p < q. Also assume that g has compact support. Suppose that for each ball B in R d, there exist two functions F B and R B such that F F B R B on 2B, and 1 F B 2 1 dx E 1 sup { 1 2B g 2 dx, 2B 2B B B B B } { 1/q ( 1 R B q dx E 2 βb 2B βb F 2 dx) 1/2 sup B B ( ) } 1/2 1 g 2 dx, B B where E 1,E 2 > and β > 2. Then F L p (R d ) and F p C g p, where C depends only on d, p, q, E 1, E 2 and β. Proof. This is a simple consequence of Theorem 3.2 in [26]. Indeed, it follows from the theorem that { } 1/p { 1/2 { 1/p F p dx C B(,r) 1 p 1 2 F dx} 2 C g dx} p. (6.7) B(,r) B(,Cr) The estimate F p C g p follows by letting r in (6.7). 23 B(,Cr)

24 Proof of Theorem 6.6. In view of Theorem 5.7, we only need to show that the solvability of (D) q implies the solvability of (R) p. More precisely, it suffices to show that N( u) p C x f p, if u is the solution of the classical Dirichlet problem with data f C 1(Rd ). This will be done by applying Theorem 6.7 to the functions F = N( u) and g = x f. To this end, we fix a ball B = B(x,r) in R d. Choose ϕ C 1(B(x,8r)) such that ϕ = 1 in B(x,7r) and ϕ Cr 1. Let λ be the average of f over B(x,8r). Write u λ = vw, where v is the solution of the L 2 regularity problem with boundary data (f λ)ϕ. We now let F B = N( v) and R B = N( w). Clearly, F = N( u) F B R B. By the L 2 regularity estimate, 1 2B 2B F B 2 dx 1 N( u) 2 dx C ( ) x (f λ)ϕ 2 dx B R B d R d C x f 2 1 dx C sup g 2 dx, B B(x,8r) B B B B where we have used the Poincaré inequality. This gives the estimate needed for F B. To verify the condition on R B = N( w), we note that w is the solution of the L 2 regularity problem with data (f λ)(1 ϕ). Consequently, w = on B(x,7r) and we may use the weak reverse Hölder estimate (6.5). This leads to { 1/q { 1 1 R B dx} q = 2B 2B 2B 2B { } 1/2 1 C N( w) 2 dx B B(x,12r) { 1/2 1 C N( u) dx} 2 C B B(x,12r) { 1 C B(x,12r) B(x,12r) } 1/q N( w) q dx { 1 B F 2 dx} 1/2 C sup B B B(x,12r) { 1 B } 1/2 N( v) 2 dx } 1/2 g 2 dx, B which gives the estimate needed for R B. Therefore, by Theorem 6.7, we obtain N( u) p C x f p for 2 < p < q. Finally, since the weak reverse Hölder estimate (6.5) is selfimproving, the argument above in fact gives N( u) p C x f p for 2 < p < qδ. This finishes the proof. 7 A Neumann function on R d1 Throughout this section we will assume that L = div(a ) with coefficients satisfying (1.1)-(1.2). Definition 7.1. Let g L 1 loc (Rd ). We call u W 1,2 loc (Rd1 ) a weak solution to the Neumann problem, L(u) = in Ω = R d1 and u ν = g on Ω, (7.1) 24

25 if u L 1 (T(,R)) for any R > 1 and for any ϕ C 1 (R d1 ). R d1 a ij u x j ϕ x i dx = R d gϕdx (7.2) To construct weak solutions to (7.1) we introduce a Neumann function. Let E = (e ij ) be the (d1) (d1) diagonal matrix with e 11 = = e dd = 1 and e (d)(d1) = 1, and { Ã = Ã(x,t) = A(x,t) if t >, (7.3) EA(x, t)e if t <, where A(x,t) is the coefficient matrix for L. Let Γ = Γ(X,Y) denote the fundamental solution for the operator L = div(ã ) on Rd1 with pole at X. Using the fact that L(v) = if v(x,t) = u(x, t) and L(u) =, one may show that Γ(X,Y ) = Γ(X,Y) for any X,Y R d1, (7.4) where X = (x, t) for X = (x,t). We now define N(X,Y) = Γ(X,Y) Γ(X,Y ) for X,Y R d1. (7.5) Clearly, N(X,Y) = N(Y,X). Since X Y X Y for X,Y R d1, we also have C N(X,Y) X Y d 1, (7.6) C X Z α N(X,Y) N(Z,Y) X Y d 1α, for any X,Y,Z R d1, where C >, α > depend only on d and µ. Let N(X,y) = N(X,(y,)) for y R d. The next lemma shows that N(X,Y) is a Neumann function for the operator L on R d1. Lemma 7.2. (a) For 1 p 2, let g L p (R d ) and N(X,y)g(y)dy for d 3 or d = 2, 1 p < 2, R u(x) = d ( N(X,y) N(X,y) ) g(y)dy for d = 2 and p = 2, R d (7.7) where X = (,1) R d1. Then u is a weak solution to the Neumann problem on R d1 with data g. (b) Let F C c (R d1 ) and w(x) = R d1 N(X,Y)F(Y)dY. (7.8) 25

26 Then R d1 w ϕ a ij dx = FϕdX (7.9) x j x i R d1 for any ϕ C 1 (R d1 ). (c) If g L 2 (R d ) and has compact support, then the solution u defined by (7.7) belongs to W 1,2 (T(,R)) for any R 1. Proof. Part (a) follows from ϕ(y) = 2 R d1 for Y = (y,) and ϕ C 1(Rd1 ). To see this, one uses } ϕ ϕ(y) = ã ij { Γ(X,Y) dx, R x d1 j x i a ij x j { Γ(X,Y) } ϕ x i dx (7.1) where ϕ is the even reflection of ϕ from R d1 to R d1, i.e., ϕ(x) = ϕ(x) if x d1 and ϕ(x) = ϕ(x ) if x d1 <. To see (b), we write w(x) = Γ(X,Y) F(Y)dY R d1 where F is the even reflection of F and deduce (7.9) from w ϕ ã ij dx = F ϕdx. R x d1 j x i R d1 where Using Part (c) follows by duality from part (b). Let h C 1 (R d1 ). Note that R d1 u h(x)dx = 2 x j g(y)dy R d R d1 = 2 g(y)v j (y)dy, R d v j (y) = R d1 Γ(X,(y,)) h x j dx. Γ x j (X,(y,))h(X)dX v j L 2 (B(,R)) C R v j W 1,2 (T(,R)) C R h L 2 (R d1 ), we may conclude by duality that u L 2 (T(,R)) for any R > 1. Definition 7.3. Let 1 < p <. We say the L p Neumann problem (N) p for R d1 is solvable if for any g L c (R d ), the weak solution u(x) = R d N(X,y)f(y)dy satisfies N( u) p C g p. 26

27 The solvability of (N) p yields the existence of weak solutions of the Neumann problem with data in L p (R d ). Theorem 7.4. Let 1 < p <. Suppose that (N) p for L(u) = in R d1 is solvable. Then for any g L p (R d ), there exists a weak solution u to the Neumann problem with data g such that N( u) p C g p. Proof. Let g L p (R d ). Choose g k C c (R d ) such that g k g in L p (R d ). Let u k (X) = N(X,y)g R d k (y)dy. Then N( v k ) p C g k p and N( v k v l ) p C g k g l p. By a limiting argument similar to that in the proof of Theorem 5.6, there exists u W 1,2 loc (Rd1 such that N( u) p C g p and N( u k u) p C g k g p. Note that the last inequality implies u k u in L p (T(,R)) for any R > 1. It follows that u is a weak solution to the Neumann problem with data g. Remark 7.5. Supposethat(N) p forl(u) = inr d1 issolvable. Itfollowsfromtheproofof Theorem 7.4 that if 1 < p 2, the weak solution given by (7.7) satisfies N( u) p C g p. Lemma 7.6. Let u W 1,2 loc (T(,2R)). Assume that R d1 ) a ij u x j ψ x i dx = (7.11) for any ψ C(B(,2R) ( R,R)). 1 Then ( ) β ( 1/2 X Y 1 u(x) u(y) CR u dx) 2 (7.12) R R d1 T(,2R) for any X,Y T(,R), where C > and β (,1] depend only on d and µ. Proof. Estimate (7.12) follows from the De Giorgi - Nash estimate by a reflection argument. The next theorem addresses the question of uniqueness. Theorem 7.7. Let u W 1,2 loc (Rd1 ) and N( u) L p (R d ) for some 2 p < d, where β 1 β is given by (7.12). Suppose that (7.11) holds for any ψ C(R 1 d1 ). Then u is constant in R d1. Proof. This follows readily from the estimate (7.12). We end this section with a result on the implication of N( u) L p (R d ) on the Neumann data. Theorem 7.8. Let u W 1,2 loc (Rd1 ) be a weak solution of L(u) = in R d1. Suppose that N( u) L p (R d ) for some 1 < p <. Then u is a weak solution to the Neumann problem on R d1 with data g for some g L p (R d ). Furthermore, g p C N( u) p and as ρ. 1 ρ 2ρ ρ a (d1)j (x,t) u x j dt g weakly in L p (R d ), 27