Weighted estimates for elliptic systems on Lipschitz domains

Abstract

Let Ω be a bounded Lipschitz domain in R n , n ≥ 3. Let ω λ (Q) = IQ - Q 0 | λ , where Q 0 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 L 2 (∂Ω, ω λ dσ), where dσ denotes the surface measure on ∂Ω. We show that there exists e > 0 such that the Dirichlet problem is uniquely solvable for -min(2+e, n-1) < A < e, and the Neumann type problem is uniquely solvable if -e < A < min(2+e, n-1). The regularity for the Dirichlet problem with data in the weighted Sobolev space L 2 1 (∂Ω, ωλ dσ) is also considered.