Weighted Estimates in L for Laplace’s Equation on Lipschitz Domains

Let Ω ⊂ Rd, d ≥ 3, be a bounded Lipschitz domain. For Laplace’s equation ∆u = 0 in Ω, we study the Dirichlet and Neumann problems with boundary data in the weighted space L2(∂Ω, ωαdσ), where ωα(Q) = |Q−Q0|α, Q0 is a fixed point on ∂Ω, and dσ denotes the surface measure on ∂Ω. We prove that there exists ε = ε(Ω) ∈ (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 C1 domain, one may take ε = 2. The regularity for the Dirichlet problem with data in the weighted Sobolev space L1(∂Ω, ωαdσ) is also considered. Finally we establish the weighted L 2 estimates with general Ap weights for the Dirichlet and regularity problems.