Potential Theory on Lipschitz Domains in Riemannian Manifolds: Sobolev–Besov Space Results and the Poisson Problem

Abstract

We continue a program to develop layer potential techniques for PDE on Lipschitz domains in Riemannian manifolds. Building on Lp and Hardy space estimates established in previous papers, here we establish Sobolev and Besov space estimates on solutions to the Dirichlet and Neumann problems for the Laplace operator plus a potential, on a Lipschitz domain in a Riemannian manifold with a metric tensor smooth of class C1+γ, for some γ>0. We treat the inhomogeneous problem and extend it to the setting of manifolds results obtained for the constant-coefficient Laplace operator on a Lipschitz domain in Euclidean space, with the Dirichlet boundary condition, by D. Jerison and C. Kenig.