Weighted estimates for elliptic systems on Lipschitz domains

Let Ω be a bounded Lipschitz domain in ℝⁿ, n ≥ 3. Let ω_λ (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, ω_λ 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, ω_λ dσ) is also considered. Indiana University Mathematics Journal