Weighted estimates for elliptic systems on Lipschitz domains

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Abstract

Let Ω be a bounded Lipschitz domain in ℝn, n ≥ 3. Let ωλ (Q) = |Q - Q0|λ, where Q0 is a fixed point on ∂Ω. For a second order elliptic system with constant coefficients on Ω, we study boundary value problems with boundary data in the weighted space L2(∂Ω, ωλ d σ), where dσ denotes the surface measure on ∂Ω. We show that there exists ε > 0 such that the Dirichlet problem is uniquely solvable for - min(2+ε, n-1) < λ < ε, and the Neumann type problem is uniquely solvable if -ε < λ < min(2+ε, n-1). The regularity for the Dirichlet problem with data in the weighted Sobolev space L12(∂Ω, ωλ dσ) is also considered. Indiana University Mathematics Journal

Original languageEnglish
Pages (from-to)1135-1154
Number of pages20
JournalIndiana University Mathematics Journal
Volume55
Issue number3
DOIs
StatePublished - 2006

Keywords

  • Elliptic systems
  • Lipschitz domains
  • Weighted spaces

ASJC Scopus subject areas

  • General Mathematics

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