## Abstract

Let Ω ⊂ ℝ ^{d}, d ≥ 3, be a bounded Lipschitz domain. For Laplace's equation Δu = 0 in Ω, we study the Dirichlet and Neumann problems with boundary data in the weighted space L ^{2}(δΩ,ω _{α}dσ), where ω _{α}(Q) = |Q -Q _{0}| ^{α}, Q _{0} is a fixed point on δΩ, and dσ denotes the surface measure on δΩ. We prove that there exists ε= ε(Ω) ε(0,2] such that the Dirichlet problem is uniquely solvable if 1 -d < α < d -3 + ε, and the Neumann problem is uniquely solvable if 3 -d - ε < α < d -1. If Ω is a C ^{1} domain, one may take ε= 2. The regularity for the Dirichlet problem with data in the weighted Sobolev space L _{1} ^{2} (δΩ, ωαdσ) is also considered. Finally we establish the weighted L ^{2} estimates with general A _{p} weights for the Dirichlet and regularity problems..

Original language | English |
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Pages (from-to) | 2843-2870 |

Number of pages | 28 |

Journal | Transactions of the American Mathematical Society |

Volume | 357 |

Issue number | 7 |

DOIs | |

State | Published - Jul 2005 |

## Keywords

- Laplace equation
- Lipschitz domains
- Weighted estimates

## ASJC Scopus subject areas

- General Mathematics
- Applied Mathematics

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