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

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..