Poincare inequality and well-posedness of the Poisson problem on manifolds with boundary and bounded geometry

Bernd Ammann, Nadine Große, Victor Nistor

Published 2016-11-01Version 1

Let M be a manifold with boundary and bounded geometry. We assume that M has "finite width," that is, that the distance $dist(x, \partial M)$ from any point $x \in M$ to the boundary $\partial M$ is bounded uniformly. Under this assumption, we prove that the Poincar\'e inequality for vector valued functions holds on $M$. We also prove a general regularity result for uniformly strongly elliptic equations and systems on general manifolds with boundary and bounded geometry. By combining the Poincar\'e inequality with the regularity result, we obtain---as in the classical case---that uniformly strongly elliptic equations and systems are well-posed on $M$ in Hadamard's sense between the usual Sobolev spaces associated to the metric. We also provide variants of these results that apply to suitable mixed Dirichlet-Neumann boundary conditions. We also indicate applications to boundary value problems on singular domains.