Sobolev and Besov Space Estimates for Solutions to Second Order PDE on Lipschitz Domains in Manifolds with Dini or Hölder Continuous Metric Tensors

Abstract

We examine solutions u = PIf to Δu − Vu = 0 on a Lipschitz domain Ω in a compact Riemannian manifold M, satisfying u = f on ∂Ω, with particular attention to ranges of (s, p) for which one has Besov-to-L p -Sobolev space results of the form and variants, when the metric tensor on M has limited regularity, described by a Hölder or a Dini-type modulus of continuity. We also discuss related estimates for solutions to the Neumann problem.