Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Weighted estimates in $L^{2}$ for Laplace's equation on Lipschitz domains

Author: Zhongwei Shen
Journal: Trans. Amer. Math. Soc. 357 (2005), 2843-2870
MSC (2000): Primary 35J25
Published electronically: October 28, 2004
MathSciNet review: 2139930
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $\Omega \subset \mathbb{R}^{d}$, $d\ge 3$, be a bounded Lipschitz domain. For Laplace's equation $\Delta u=0$ in $\Omega $, we study the Dirichlet and Neumann problems with boundary data in the weighted space $L^{2}(\partial \Omega ,\omega _{\alpha }d\sigma )$, where $\omega _{\alpha }(Q) =\vert Q-Q_{0}\vert^{\alpha }$, $Q_{0}$ is a fixed point on $\partial \Omega $, and $d\sigma $ denotes the surface measure on $\partial \Omega $. We prove that there exists $\varepsilon =\varepsilon (\Omega )\in (0,2]$ such that the Dirichlet problem is uniquely solvable if $1-d<\alpha <d-3+\varepsilon $, and the Neumann problem is uniquely solvable if $3-d-\varepsilon <\alpha <d-1$. If $\Omega $ is a $C^{1}$ domain, one may take $\varepsilon =2$. The regularity for the Dirichlet problem with data in the weighted Sobolev space $L^{2}_{1}(\partial \Omega ,\omega _{\alpha }d\sigma )$ is also considered. Finally we establish the weighted $L^{2}$ estimates with general $A_{p}$weights for the Dirichlet and regularity problems.

References [Enhancements On Off] (What's this?)

  • [B] R. Brown, The Neumann problem on Lipschitz domains on Hardy spaces of order less than one, Pacific J. Math. 171(2) (1995), 389-407. MR 97m:35048
  • [CF] R.R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241-250.MR 50:10670
  • [CMM] R. Coifman, A. McIntosh and Y. Meyer, L'integrale de Cauchy definit un operateur borne sur $L^{2}$ pour les courbes Lipschitziennes, Ann. of Math. 116 (1982), 361-388. MR 84m:42027
  • [D1] B. Dahlberg, On estimates for harmonic measure, Arch. Rat. Mech. Anal. 65 (1977), 273-288. MR 57:6470
  • [D2] B. Dahlberg, On the Poisson integral for Lipschitz and $C^{1}$ domains, Studia Math. 66 (1979), 13-24. MR 81g:31007
  • [D3] B. Dahlberg, Weighted norm inequalities for the Lusin area integral and the nontangential maximal functions for functions harmonic in a Lipschitz domain, Studia Math. 67 (1980), 297-314. MR 82f:31003
  • [DK] B. Dahlberg and C. Kenig, Hardy spaces and the Neumann problem in $L^{p}$ for Laplace's equation in Lipschitz domains, Ann. of Math. 125 (1987), 437-466. MR 88d:35044
  • [FJR] E. Fabes, M. Jodeit, and N. Riviére, Potential techniques for boundary value problems on $C^{1}$-domains, Acta Math. 141 (1978), 165-186. MR 80b:31006
  • [FKS] E. Fabes, C. Kenig, R. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Diff. Eqs. 7 (1982), 77-116. MR 84i:35070
  • [FS] E. Fabes and D. Stroock, The $L^{p}$ integrability of Green's functions and fundamental solutions for elliptic and parabolic equations, Duke Math. J. 51 (1984), 977-1016. MR 86g:35057
  • [JK] D. Jerison and C. Kenig, The Neumann problem in Lipschitz domains, Bull. Amer. Math. Soc. 4 (1981), 203-207. MR 84a:35064
  • [K1] C. Kenig, Elliptic boundary value problems on Lipschitz domains, Beijing Lectures in Harmonic Analysis, Ann. of Math. Studies 112 (1986), 131-183. MR 88a:35066
  • [K2] C. Kenig, Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems, CBMS Regional Conference Series in Math., vol. 83, AMS, Providence, RI, 1994. MR 96a:35040
  • [KP1] C. Kenig and J. Pipher, The oblique derivative problem on Lipschitz domains with $L^{p}$ data, Amer. J. Math. 110 (1988), 715-738.MR 89i:35047
  • [KP2] C. Kenig and J. Pipher, The Neumann problem for elliptic equations with non-smooth coefficients, Invent. Math. 113 (1993), 447-509.MR 95b:35046
  • [KMR] V.A. Kozlov, V.G. Maz'ya and J. Rossmann, Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations, Mathematical Surveys and Monographs, vol. 85, AMS, Providence, RI, 2001.MR 2001i:35069
  • [M] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Soc. 165 (1972), 207-226. MR 45:2461
  • [PV] J. Pipher and G. Verchota, A maximum principle for biharmonic functions in Lipschitz and $C^{1}$ domains, Commen. Math. Helv. 68 (1993), 385-414.MR 94j:35030
  • [R] J. L. Rubio de Francia, Factorization theory and $A_{p}$weights, Amer. J. Math. 106 (1984), 533-547. MR 86a:47028a
  • [SW] E. Sawyer and R. L. Wheeden, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J. Math. 114(4) (1992), 813-874. MR 94i:42024
  • [Sh1] Z. Shen, Boundary value problems in Morrey spaces for elliptic systems on Lipschitz domains, Amer. J. Math 125 (2003), 1079-1115.
  • [Sh2] Z. Shen, Weighted estimates for elliptic systems on Lipschitz domains, to appear in Indiana Univ. Math. J.
  • [St1] E.M. Stein, Note on singular integrals, Proc. Amer. Math. Soc. 8 (1957), 250-254. MR 19:547b
  • [St2] E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Preceton, NJ, 1970. MR 44:7280
  • [St3] E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, Princeton, NJ, 1993.MR 95c:42002
  • [V1] G. Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace's equation, J. Funct. Anal. 59 (1984), 572-611.MR 86e:35038
  • [V2] G. Verchota, The Dirichlet problem for the biharmonic equation in $C^{1}$ domains, Indiana Univ. Math. J. 36 (1987), 867-895. MR 88m:35051

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35J25

Retrieve articles in all journals with MSC (2000): 35J25

Additional Information

Zhongwei Shen
Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506

Keywords: Laplace equation, Lipschitz domains, weighted estimates
Received by editor(s): October 20, 2002
Received by editor(s) in revised form: December 11, 2003
Published electronically: October 28, 2004
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society