Weighted estimates in L 2 for Laplace's equation on lipschitz domains

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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 αdσ), 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 languageEnglish
Pages (from-to)2843-2870
Number of pages28
JournalTransactions of the American Mathematical Society
Volume357
Issue number7
DOIs
StatePublished - Jul 2005

Keywords

  • Laplace equation
  • Lipschitz domains
  • Weighted estimates

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics

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