On Estimates of Biharmonic Functions on Lipschitz and Convex Domains

Transcription

1 The Journal of Geometric Analysis Volume 16, Number 4, 2006 On Estimates of Biharmonic Functions on Lipschitz and Convex Domains By Zhongwei Shen ABSTRACT. Using Maz ya type integral identities with power weights, we obtain new boundary estimates for biharmonic functions on Lipschitz and convex domains in R n.forn 8, combined with a result in [18], these estimates lead to the solvability of the L p Dirichlet problem for the biharmonic equation on Lipschitz domains for a new range of p. In the case of convex domains, the estimates allow us to show that the L p Dirichlet problem is uniquely solvable for any 2 ε<p< and n Introduction Let be a bounded domain in R n with Lipschitz boundary. Let N denote the outward unit normal to. We consider the L p Dirichlet problem for the biharmonic equation, 2 u = 0 in, u = f W 1,p, N u on, u L p where W 1,p denotes the space of functions in L p whose first-order tangential derivatives are also in L p. We point out that the boundary values in 1.1 are taken in the sense of nontangential convergence a.e. with respect to the surface measure on. As such, one requires that the nontangential maximal function u is in L p. For n 2, the Dirichlet problem 1.1 with p = 2 was solved by Dahlberg, Kenig, and Verchota [2], using bilinear estimates for harmonic functions. The result was then extended to the case 2 ε<p<2 + ε by a real variable argument, where ε>0depends on n and. They also showed that the restriction p > 2 ε is necessary for general Lipschitz domains. In [13, 14], Pipher and Verchota proved that if n = 3 or 2, the L p Dirichlet problem 1.1 is uniquely solvable for the sharp range 2 ε<p. Moreover, they pointed out that 1.1 is not solvable in general for p>6ifn = 4, and for p>4ifn 5. Recently in [17, 18], for n 4 and p in a certain range, we established the solvability of the L p Dirichlet problem for higher order elliptic equations and systems, using a new approach via L 2 estimates and weak 1.1 Math Subject Classifications. 35J40. Key Words and Phrases. Biharmonic functions; Lipschitz domains; convex domains. Acknowledgements and Notes. Research supported by the NSF The Journal of Geometric Analysis ISSN

2 722 Zhongwei Shen reverse Hölder inequalities. In particular, we were able to solve the L p Dirichlet problem 1.1 in the following cases: 2 ε<p<6 + ε for n = 4, 2 ε<p<4 + ε for n = 5, 6, 7, ε<p<2 + n ε for n 8. This gives the sharp ranges of p for 4 n 7. It should be pointed out that the sharp range 2 ε<p<4 + ε for the case n = 6, 7 in 1.2 relies on a classical result of Maz ya [8, 9] on the boundary regularity of biharmonic functions in arbitrary domains. The approach we will use in this article is inspired by the work of Maz ya [8, 9, 11] we shall come back to this point later. We mention that if the domain is C 1, then 1.1 is uniquely solvable for all n 2 and 1 <p [1, 19, 14]. For related work on the L p Dirichlet problem for the polyharmonic equation and general higher-order equations and systems on Lipschitz domains, we refer the reader to [20, 15, 16, 6, 21, 17, 18]. The purpose of this article is twofold. First we study the case n 8 for which the question of the sharp ranges of p remains open for Lipschitz domains. Secondly we initiate the study of the L p Dirichlet problem 1.1 on convex domains. Note that any convex domain is Lipschitz, but may not be C 1. Let I Q, r = BQ, r and T Q, r = BQ, r where Q and r>0. Our starting point is the following theorem. Theorem 1.1. Let be a bounded Lipschitz domain in R n, n 4. Suppose that there exist constants C 0 > 0, R 0 > 0 and λ 0,n] such that for any 0 <r<r<r 0 and Q, r λ v 2 C 0 v 2, 1.3 R whenever v satisfies T Q,r T Q,R 2 v = 0 in, v = N v = 0 on I Q, R, v L 2. Then the L p Dirichlet problem 1.1 is uniquely solvable for 2 <p<2 + 4 n λ. 1.5 Moreover, the solution u satisfies u L p C{ t f L p + g L } p, 1.6 where t f denotes the tangential derivatives of f on. Theorem 1.1 is a special case of Theorem 1.10 in [18] for general higher-order homogeneous elliptic equations and systems with constant coefficients. It reduces the study of the L p Dirichlet problem to that of local L 2 estimates near the boundary. The main body of this article will be devoted to such estimates. In particular, we will prove that if n 8, then estimate 1.3 holds for some λ>λ n, where n n 2 n + 2 λ n =

3 On Estimates of Biharmonic Functions on Lipschitz and Convex Domains 723 We will also show that if is convex and n 4, then 1.3 holds for any 0 <λ<n. Consequently, by Theorem 1.1, we obtain the following. Main Theorem. Let be a bounded Lipschitz domain in R n. a If n 8, the L p Dirichlet problem 1.1 is uniquely solvable for 2 ε<p< ε. 1.8 n λ n b If n 4 and is convex, the L p Dirichlet problem 1.1 is uniquely solvable for 2 ε< p<. We remark that in the case of Laplace s equation u = 0, the Dirichlet problem in L p is uniquely solvable on convex domains for all 1 <p. This follows easily from the L boundary estimates on the first derivatives of the Green s functions. Whether a similar result the L boundary estimate on the second derivatives holds for biharmonic functions remains open for n 3 see [7] for the case n = 2. Note that part b of the Main Theorem as well as its proof gives the C α boundary estimate of u for any 0 <α<1. This seems to be the first regularity result for biharmonic functions on general convex domains in R n, n 4. As we mentioned earlier, our approach to estimate 1.3 is motivated by the work of Maz ya [8, 9, 11]. It is based on certain integral identities for 2 u u and 2 u u, 1.9 ρα 1 where ρ = x Q with Q fixed. See 2.10 and 3.1. These identities with power weights allow us to control the integrals u 2 +2 and u for certain values of α. We point out that integral identity 2.10 with α = n 4 appeared first in [8, 9], where it was used to establish a Wiener s type condition on the boundary continuity for the biharmonic equation 2 u = f on arbitrary domains in R n for n 7. Since the restriction n 7 in [8, 9] is related to the positivity of a quadratic form see 1.11 below, the idea to prove part a of the Main Theorem is to use the identity 2.10 for certain α<n 4in the case n 8. However, it should be pointed out that the main novelty of this article is the new identity 3.1, on which the proof of part b of the Main Theorem is based. This identity allows us to estimate the integrals in 1.10 on convex domains for any α<n 2. We remark that due to the lack of maximum principles for higher-order equations, identities such as 2.10 and 3.1 are valuable tools in the study of boundary regularities in nonsmooth domains. Finally, we mention that the results in [8, 9] were subsequently extended to the polyharmonic equation [12, 10] and general higher order elliptic equations [11]. Also, the related question of the positivity of the quadratic form λ u u R n x n 2λ for all real function u C0 R n, 1.11 has been studied systematically by Eilertsen [3, 4] for all λ 0,n/2. 2. Boundary estimates on Lipschitz domains The goal of this section is to prove part a of the Main Theorem. We begin with a Cacciopoli s inequality. Recall that for Q, T Q, R = BQ, R and I Q, R = BQ, R.

4 724 Zhongwei Shen We assume that 0 <R<R 0, where R 0 is a constant depending on so that for any Q, T Q, 4R 0 is given by the intersection of BQ, 4R 0 and the region above a Lipschitz graph, after a possible rotation. Lemma 2.1. Let u W 2,2 T Q, R for some Q and 0 <R<R 0. Suppose that 2 u = 0 in T Q, R and u = 0, u = 0 on I Q, R. Then 1 r 2 u u 2 C r 4 u 2, 2.1 T Q,r where 0 <r<r/4. T Q,r T Q,2r\T Q,r Proof. Let η be a smooth function on R n such that η = 1onBQ, r, supp η BQ, 2r and k η C/r k for 0 k 4. Since u W 2,2 T Q, R and u = 0, u = 0onI Q, R, we have uη 2 W 2,2 0. We will show that for any ε>0, 2 uη 2 2 ε 2 uη ε uη 2 2 r 2 + C 2.2 ε r 4 u 2. T Q,2r\T Q,r This, together with the Poincaré inequality uη 2 2 Cr 2 T Q,2r T Q,2r 2 uη 2 2, 2.3 yields the estimate 2.1. To prove 2.2, we use integration by parts and 2 u = 0inT Q, 2r to obtain 2 uη 2 2 = uη 2 2 { = uη 2 2 u uη 4}. 2.4 A direct computation shows that uη 2 uη 2 u uη 4 = u uη 2 η u η u u η 2 η 2 uu 2 η η 2 η 2. In view of 2.2, the integral of the first term in the right side of 2.5 can be handled easily by Hölder s inequality with an ε. The remaining terms may be handled by using integration by parts, together with the following observation. For terms with u x u i, like the third term in the right side of 2.5, we may write u u ψ = 1 2 u 2 ψ ψ u For terms with η 2 x u u i x j, like the second term, we use u u x j η 2 ψ = uη 2 x j uψ 2 uη 2 x j u u η 2 x j ψ u 2 η2 x j ψ. uψ u u x j ψ η 2 2.7

5 On Estimates of Biharmonic Functions on Lipschitz and Convex Domains 725 Finally, for the last term which contains η 2 uu, we note that η 2 uu ψ = The rest of the proof, which we omit, is fairly straightforward. η 2 u u ψ u u η 2 ψ η 2 u 2 ψ. 2.8 Remark 2.2. It follows from Lemma 2.1 that for any 0 <r<r/2and α R, ux 2 2 ux 2 + T Q,r x Q α+2 T Q,r x Q α C ux T Q,2r x Q α+4 This may be seen by writing T Q, r as j=0 T Q, 2 j r \ T Q, 2 j 1 r. The key step to establish estimate 1.3 relies on the following extension of an integral identity due to Maz ya [8, 9]. Lemma 2.3. Suppose that u C 2 and u = 0, u = 0 on. Then for any α R, u u = u 2 + 2α u 2 2αα + 2 ρα+2 u αα + 2n 2 αn 4 α u 2 +4, where ρ = x y and u =< ux, x y/ρ > with y c fixed. Proof. We will use the summation convention that the repeated indices are summed from 1 to n. First, note that u u = u u u 1 x j x j uu. Next it follows from integration by parts that 2 u u x j 1 1 x j = u 2 2 u u 2 1 x j x j Similarly, we have 1 1 uu = u u Substituting 2.12 and 2.13 into 2.11, we obtain u u = u 2 2 u u 2 1 x j x j + 1 u

6 726 Zhongwei Shen The desired formula 2.10 now follows from the fact that 2 1 x j = αρ α 2 δ ij + αα + 2x i y i x j y j ρ α 4, 2 1 = αα + 2n 2 αn 4 αρ α 4, 2.14 for any ρ = x y = 0. The proof is complete. Lemma 2.4. Under the same assumption as in Lemma 2.3, we have u u +1 = 1 n 4 α u 2 + α ρα+2 u ρα+2 Proof. It follows from integration by parts that u u +1 = u u x i y i +2 = 1 u 2 xi y i u 2 +2 u xi y i x j x j +2 = 1 2 n 4 α u 2 + α + 2 ρα+2 u Lemma 2.3, together with Lemma 2.4, allows us to estimate u 2 +4 and u 2 u +2 by u for certain values of α. Lemma 2.5. Let be a bounded Lipschitz domain in R n, n 5. Suppose that u C 2 and u = 0, u = 0 on. Then, if 0 <α n 4 and n 2 + 2nα 7α 2 8α >0, wehave u 2 u +2 C n,α u, 2.16 where C n,α > 0 depends only on n and α. Proof. We first use 2.15 for 0 <α n 4 to obtain n + α u u u +1 { u 2 } 1/2 { u 2 } 1/2 +2, where the Cauchy inequality is also used. It follows that 1 n + α2 4 u 2 +2 u

7 On Estimates of Biharmonic Functions on Lipschitz and Convex Domains 727 Since 0 <α n 4, in view of 2.10 and 2.17, we have u { } 1 u 4 n + α2 + 2α 2αα + 2 u 2 = 1 n 2 + 2nα 7α 2 8α u Thus, if n 2 + 2nα 7α 2 8α >0, by 2.10 again, 2α u 2 u +2 u + 2αα + 2 u 2 u C u. The proof is finished Remark 2.6. Let α = n 4. Then n 2 + 2nα 7α 2 8α = 4 n n 20 >0for n = 5, 6, 7. It follows that 2.16 holds for α = n 4 in the case n = 5, 6 or 7. This was the result obtained by Maz ya in [8, 9]. If n 8, then 2.16 holds for 0 <α<α n <n 4, where α n = 1 n n 7 2 n is the positive root of n 2 + 2nα 7α 2 8α = 0. Remark 2.7. If n 8 and α = α n given by 2.18, we observe that the sum of the first three terms on the right side of 2.10 is nonnegative, by an inspection of th proof of Lemma 2.5. It follows that u 2 u +4 C n u Since C0 2,2 is dense in W0, inequality 2.19 holds for any u W 2,2 0. We are now in a position to give the proof of part a of the Main Theorem. Theorem 2.8. Let be a bounded Lipschitz domain in R n, n 8. Then the L p Dirichlet problem 1.1 is uniquely solvable for 2 ε<p<2 + n λ 4 n + ε, where λ n = α n + 2 is given in 1.7. Proof. By Theorem 1.1, we only need to show that estimate 1.3 holds for some λ>λ n = α n +2. To this end, we fix Q and 0 <R<R 0, where R 0 is a constant depending on. Let v be a function on satisfying 1.4. Let η be a smooth function on R n such that η = 1onBQ, r, supp η BQ, 2rand k η C/r k for 0 k 4 where 0 <r<r/4. Since v = v N = 0on I Q, R and v L 2, by the regularity estimate v 2 C t v 2 established in [20], we know vη W 2,2 0. Thus, we may apply estimate 2.19 to u = vη with α = α n and ρ = x y, where y c. We obtain vη 2 +4 C vη vηρ α Using an identity similar to 2.5, vη vηρ α v vη 2 ρ α = v vηρ α η + 2 v η vηρ α 2 vηρ α η v vηρ α v η,

8 728 Zhongwei Shen and 2 v = 0in,weget { vη 2 +4 C v vηρ α η + 2 v η vηρ α 2 vηρ α η } v vηρ α v η Note that ρ α and its derivatives are uniformly bounded for y BQ, r/2 \ and x supp η {x R n : r x Q 2r}. It follows by a simple limiting argument that 2.21 holds for ρ = x Q. This gives vx 2 T Q,r x Q α+4 C { r α+4 v 2 + r 2 v 2 + r 4 2 v 2} T Q,2r C r α+4 v C 1 T Q,4r\T Q,2r T Q,4r\T Q,r vx 2 x Q α+4, where the second inequality follows from Cacciopoli s inequality 2.1. By filling the hole in 2.22, we obtain vx 2 T Q,r x Q α+4 C 1 vx 2 C T Q,4r x Q α+4. This implies that there exists δ>0such that vx 2 r δ T Q,r x Q α+4 C vx 2 R T Q,R/4 x Q α+4 r δ 1 C R R α+4 vx 2, T Q,R for any 0 <r<r/4, where the second inequality follows from Consequently, r α+4+δ vx 2 C vx 2. T Q,r R T Q,R This, together with Cacciopoli s inequality and Poincaré inequality, gives v 2 C T Q,r/2 r 2 v 2 C r α+4+δ T Q,r r 2 R r α+2+δ C v 2. R T Q,R T Q,R v 2 Thus, we have established estimate 1.3 for λ = α n δ = λ n + δ. The proof is finished. 3. Boundary estimates on convex domains In this section we give the proof of part b of the Main Theorem. By Theorem 1.1, it suffices to show that estimate 1.3 holds for any λ<n. To do this, the crucial step is to establish the

9 On Estimates of Biharmonic Functions on Lipschitz and Convex Domains 729 following new integral identity, u α + 4 n u 2 2 u u ρ α 1 = 2 u 2 < x y dσ,n > + 4α ρ ρα 1 ρ n α u + 2αα + 2n α 2 ρ n α u ρ n 2, ρ n 2 where u C 4 and u = 0, u = 0on. Recall that N denotes the outward unit normal to. Also in 3.1, as before, ρ = ρx = x y, u =< u, x y/ρ > with y c fixed. By a limiting argument, it is not hard to see that if α<n, 3.1 holds also for y. We will use 3.1 with α = n 2 for convex domain. The key observation is that if is convex, the boundary integral in 3.1 is nonnegative. This is because <P Q, NP > 0 for any P,Q. The proof of 3.1, which involves the repeated use of integration by parts, will be given through a series of lemmas. 3.1 Lemma 3.1. Suppose u C 2 and u = 0, u = 0 on. Then, for any α R, 2 u 2 u x j x j = 2 u 2 + αn α 1 u 2 ρα +2 αα + 2 u αα + 2n α 2n α 4 u , where the repeated indices are summed from 1 to n. 3.2 Proof. First we note that 2 u 2 u x j x j 2 u = x j { 2 u 1 x j + 2 u α + u 2 ρ α x j x j }. Next it follows from integration by parts and u = 0, u = 0on that 2 u 2 u α = u 2 ρ α, 3.4 x j x j and 2 u u 2 ρ α x j x j u = u 2 ρ α u 2 2 ρ α. x j x j 2 Substituting 3.4 and 3.5 into 3.3, we obtain 2 u 2 u x j x j = 2 u 2 u 2 ρ α u u 2 ρ α + 1 u 2 2 ρ α. x j x j 2 3.3

10 730 Zhongwei Shen The desired formula now follows from this and Lemma 3.2. Suppose u C 4 and u = 0, u = 0 on. Then, for any α R, 2 u u 1 = 1 2 u 2 < x y dσ,n > 2 ρ α + 4 n 2 u 2 2α u αn α u αα + 2n α u , 3.6 where ρ = x y with y c fixed. Proof. By translation we may assume that y = 0. Using integration by parts, we obtain 2 u u 3 u = xj 2 1 = 4 u x 2 i x2 j u x k xk 2 u x k xk 3 x 2 j u x k xk. 3.7 For the first term on the right side of 3.7, again from integration by parts, we have 3 u 2 u xj 2 xk x k = u 2 xk 2 x k + 2 u 2 < x ρ,n > 2 u 2 u x j x k x j dσ 1 xk, 3.8 where we also used the observation that u = 0on implies u u = 2 u 2 < x,n >. 3.9 N ρ For the second term on the right side of 3.7, we have 3 u xj 2 = u x k 2 u x j 2 u x k x j xk xk 2 u + u 2 x j x k x j xk. ρ 3.10 Substituting 3.8 and 3.10 into 3.7 and using 2 x j xj xk = δ ij α x ix j +2, = αδ ik x j + δ ij x k + δ jk x i ρ α 2 + αα + 2 x ix j x k +4, 3.11

11 On Estimates of Biharmonic Functions on Lipschitz and Convex Domains 731 we obtain 2 u u 1 = 1 2 u 2 < x 2 ρ,n > dσ α u 2 2α 2 u u x j x j α u u + αα + 2 ρα+1 Finally, we note that 2 2 u x j u x j α + 4 n 2 u 2 x i ρα+2 2 u u x j x i = u 2 ρα+2 xix j x k. x k ρα+4 = α + 2 n u 2 xi +2 +2, and 2 u u xix j x k x j x k +4 = 1 u u 2 x j x k = 1 2 α + 2 n u 2 xi x j x k The desired formula 3.6 follows by substituting 3.13, 3.14 as well as 2.15 into The proof is complete. Lemma 3.3. Suppose that u C 1 and u = 0 on. Then, for any α R, uρ n α 2 2 ρ n 2 = u n α2 u 2 4, 3.15 where ρ = x y with y c fixed. Proof. To see 3.15, we note that uρ n α 2 2 u = 2 ρ n α + n α u u ρn α n α2 u 2 ρ n α Also, using integration by parts and u = 0on,wehave n α u u ρn α 1 ρ n 2 = 1 n α u 2 2 = 1 2 n α2 u 2. xi 3.17 In view of 3.16, this gives We are now ready to prove the integral identity 3.1.

12 732 Zhongwei Shen Lemma 3.4. Let be a bounded Lipschitz domain in R n, n 2. Suppose that u C 4 and u = 0, u = 0 on. Then 3.1 holds for any α<nand any y. Proof. By the Lebesgue Dominated Convergence Theorem, it suffices to establish 3.1 for y c. To this end, we note that u 2 u 2 u u = x j x j from integration by parts. Thus, by 3.2 and 3.6, we have α + 4 n = + 4α u u 2 2 u u u, u 2 < x y dσ,n > ρ 1 u 2 αn α 22 u 2 ρα + 2αα + 2n α 2 u αα + 2n α 2n α 42 u 2 In view of 3.15, this gives the integral identity Next we will use 3.1 to derive estimate 1.3 on convex domains with smooth boundaries for any λ<n. Lemma 3.5. Let be a convex domain in R n, n 4 with smooth boundary. Let 0 <λ<n. Then there exist constants C 0 > 0 and R 0 > 0 depending only on n, λ, and the Lipschitz character of such that estimate 1.3 holds for any v satisfying 1.4. Proof. Let R 0 > 0 be a constant so that for any Q, T Q, 4R 0 is given by the intersection of BQ, 4R 0 and the region above a Lipschitz graph, after a possible rotation. Fix Q and 0 <R<R 0. Let v be a biharmonic function in such that v = N v = 0onI Q, R and v L 2. Since has smooth boundary, by the classical regularity theory for elliptic equations, v C 4 T Q, R/2. Let η be a smooth function on R n such that η = 1onBQ, R/8, supp η BQ, R/4 and k η C/R k for 0 k 4. Note that u = vη C 4 and u = 0, u = 0on. Thus, we may apply integral identity 3.1 to u with α = n 2 and y = Q. This gives { u 2 ρ n 2 C 4 u u ρ n u u } ρ n Since 2 v = 0in,wehave 2 u = 2 < v, η >+v η + {2 < v, η >+vη} Substituting 3.20 into 3.19 and using integration by parts as well as Cauchy inequali-

13 ty, we obtain On Estimates of Biharmonic Functions on Lipschitz and Convex Domains 733 u 2 ρ n 2 C { R n 2 2 v } 2 + v 2 T Q,R/4 R 2 + v 2 R 4 C R n T Q,R/2 v 2, 3.21 where we also used the Cacciopoli s inequality 2.1 and Poincaré inequality in the second inequality. Since suppu BQ, R, for any δ 0, n 2, wehave u 2 ρ n 2 1 R δ u 2 ρ n 2 δ δ2 4R δ u 2 ρ n δ, where the second inequality follows from 3.15 with α = n+2 δ, which also holds for y if α<n+ 2. In view of 3.21, this gives v 2 r n δ u 2 r n δ T Q,r T Q,r ρ n δ C δ v 2, R T Q,R for any 0 <r<r/8. Estimate 1.3 is thus proved for λ = n δ. Lemma 3.5, together with a well-known approximation argument, gives part b of the Main Theorem. Theorem 3.6. Let be a bounded convex domain in R n, n 4. Then the L p Dirichlet problem 1.1 is uniquely solvable for any 2 ε<p<. Proof. Let p>2and f W 1,p, g L p. We need to show that the unique solution u to the L 2 Dirichlet problem 1.1 satisfies estimate 1.6. To this end, we first note that by an approximation argument e.g., see [5] for Laplace s equation, we may assume that f, g C0 Rn. Next we approximate from outside by a sequence of convex domains { j } with smooth boundaries, 1 2. Let u j be the solution to the L 2 Dirichlet problem 1.1 on j with boundary data u j, u j N = f j,g j on j. By Lemma 3.5 and Theorem 1.1, we have uj L p C uj j L p j C { t f L p j + g L p j }, 3.22 where u j j denotes the nontangential maximal function of u j with respect to j, and C is a constant independent of j. Estimate 3.22 implies that the sequence { u j } is uniformly bounded on any compact subset of. It follows that there exist a subsequence, which we still denoted by { u j }, and a function u on such that u j converges to u uniformly on any compact subset of. It is easy to show that u is biharmonic in. Also by 3.22 and Fatou s Lemma, u K L p C { t f L p + g L } p, 3.23 where K is a compact subset of, and u K Q = sup{ ux : x K and x Q < 2 dist x, }. By the monotone convergence theorem, this gives the estimate 1.6 on. Finally, one may use L 2 estimates on u i u j L 2 j for i j as well as L2 regularity estimate, 2 u j j L 2 j C { 2 f L 2 j + g L 2 i } see [20] to show that u = f and N u = g on in the sense of nontangential convergence. We leave the details to the reader.

14 734 Zhongwei Shen Acknowledgments The author is indebted to Jill Pipher for bring the article [9] to his attention, and for several helpful discussions. The author also would like to thank Vladimir Maz ya for pointing out the relevance of the articles [8, 3, 4]. References [1] Cohen, J. and Gosselin, J. The Dirichlet problem for the biharmonic equation in a C 1 domain in the plane, Indiana Univ. Math. J. 325, , [2] Dahlberg, B., Kenig, C., and Verchota, G. The Dirichlet problem for the biharmonic equation in a Lipschitz domain, Ann. Inst. Fourier Grenoble 36, , [3] Eilertsen, S. On weighted positivity and the Wiener regularity of a boundary point for the fractional Laplacian, Ark. Mat. 38, 53 57, [4] Eilertsen, S. On weighted fractional integral inequalities, J. Funct. Anal. 185, , [5] Jerison, D. and Kenig, C. Boundary value problems on Lipschitz domains, MAA Studies in Math. 23, 1 68, [6] Kenig, C. Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems, CBMS Regional Conference Series in Math. 83, AMS, Providence, RI, [7] Kozlov, V. and Maz ya, V. G. Asymptotics formula for solutions to elliptic equations near the Lipschitz boundary, Ann. Mat. Pura Appl , , [8] Maz ya, V. G. On the behavior near the boundary of solutions to the Dirichelt problem for the biharmonic operator, Dokl. Akad. Nauk SSSR 235, 1977, , Russian. English transl. Soviet Math. Dokl. 18, , [9] Maz ya, V. G. Behavior of solutions to the Dirichlet problem for the biharmonic operator at a boundary point, Equadiff IV, Lecture Notes in Math. 703, , [10] Maz ya, V. G. On the Wiener type regularity of a boundary point for the polyharmonic operator, Appl. Anal., , [11] Maz ya, V. G. The Wiener test for higher order elliptic equations, Duke Math. J. 115, , [12] Maz ya, V. G. and Donchev, T. On the Wiener regularity of a boundary point for the polyharmonic operator, Dokl. Bolg. Akad. Nauk 36, 1983, , Russian. English transl. Amer. Math. Soc. Transl. 137, 53 55, [13] Pipher, J. and Verchota, G. The Dirichlet problem in L p for the biharmonic equation on Lipschitz domains, Amer. J. Math. 114, , [14] Pipher, J. and Verchota, G. A maximum principle for biharmonic functions in Lipschitz and C 1 domains, Comment. Math. Helv. 68, , [15] Pipher, J. and Verchota, G. Dilation invariant estimates and the boundary Garding inequality for higher order elliptic operators, Ann. of Math. 2142, 1 38, [16] Pipher, J. and Verchota, G. Maximum principle for the polyharmonic equation on Lipschitz domains, Potential Anal. 4, , [17] Shen, Z. The L p Dirichlet problem for elliptic systems on Lipschitz domains, Math. Res. Lett. 13, , [18] Shen, Z. Necessary and sufficient conditions for the solvability of the L p Dirichlet problem on Lipschitz domains, to appear in Math. Ann., [19] Verchota, G. The Dirichlet problem for the biharmonic equation in C 1 domains, Indiana Univ. Math. J. 36, , [20] Verchota, G. The Dirichlet problem for the polyharmonic equation in Lipschitz domains, Indiana Univ. Math. J. 39, , [21] Verchota, G. Potentials for the Dirichlet problem in Lipschitz domains, Potential Theory-ICPT94, Received April 2, 2005 Department of Mathematics, University of Kentucky, Lexington, KY shenz@ms.uky.edu Communicated by David Jerison