Neumann and mixed problems on manifolds with boundary and bounded geometry

Nadine Große, Victor Nistor

Published 2017-03-21Version 1

We prove regularity and well-posedness results for the mixed Dirichlet-Neumann problem for a second order, uniformly strongly elliptic differential operator on a manifold $M$ with boundary $\partial M$ and bounded geometry. Our well-posedness result for the Laplacian $\Delta_g := d^*d \ge 0$ associated to the given metric require the additional assumption that the pair $(M, \partial_D M)$ be of finite width (in the sense that the distance to $\partial_D M$ is bounded uniformly on $M$, where $\partial_D M$ is the Dirichlet part of the boundary). The proof is a continuation of the ideas in our previous paper on the Dirichlet problem on manifolds with boundary and bounded geometry (joint with Bernd Ammann). We also obtain regularity results for more general boundary conditions. Our results are formulated in the usual Sobolev spaces defined by the Riemannian metric on $M$.