Weighted estimates in 𝐿² for Laplace’s equation on Lipschitz domains

Abstract

Let Ω ⊂ R d \Omega \subset \mathbb {R}^{d} , d ≥ 3 d\ge 3 , be a bounded Lipschitz domain. For Laplace’s equation Δ u = 0 \Delta u=0 in Ω \Omega , we study the Dirichlet and Neumann problems with boundary data in the weighted space L 2 ( ∂ Ω , ω α d σ ) L^{2}(\partial \Omega ,\omega _{\alpha }d\sigma ) , where ω α ( Q ) = | Q − Q 0 | α \omega _{\alpha }(Q) =|Q-Q_{0}|^{\alpha } , Q 0 Q_{0} is a fixed point on ∂ Ω \partial \Omega , and d σ d\sigma denotes the surface measure on ∂ Ω \partial \Omega . We prove that there exists ε = ε ( Ω ) ∈ ( 0 , 2 ] \varepsilon =\varepsilon (\Omega )\in (0,2] such that the Dirichlet problem is uniquely solvable if 1 − d > α > d − 3 + ε 1-d>\alpha >d-3+\varepsilon , and the Neumann problem is uniquely solvable if 3 − d − ε > α > d − 1 3-d-\varepsilon >\alpha >d-1 . If Ω \Omega is a C 1 C^{1} domain, one may take ε = 2 \varepsilon =2 . The regularity for the Dirichlet problem with data in the weighted Sobolev space L 1 2 ( ∂ Ω , ω α d σ ) L^{2}_{1}(\partial \Omega ,\omega _{\alpha }d\sigma ) is also considered. Finally we establish the weighted L 2 L^{2} estimates with general A p A_{p} weights for the Dirichlet and regularity problems.