Potential theory on Lipschitz domains in Riemannian manifolds: The case of Dini metric tensors

Abstract:

We study the applicability of the method of layer potentials in the treatment of boundary value problems for the Laplace-Beltrami operator on Lipschitz sub-domains of Riemannian manifolds, in the case when the metric tensor $g_{jk} dx_j\otimes dx_k$ has low regularity. Under the assumption that \[ |g_{jk}(x)-g_{jk}(y)|\leq C \omega (|x-y|),\] where the modulus of continuity $\omega$ satisfies a Dini-type condition, we prove the well-posedness of the classical Dirichlet and Neumann problems with $L^p$ boundary data, for sharp ranges of $p$’s and with optimal nontangential maximal function estimates.