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

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Bernd Ammann, Nadine Große, Victor Nistor

**Poincaré inequality and well-posedness of the Poisson problem on manifolds with boundary and bounded geometry**
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Let M be a manifold with boundary and bounded geometry. We assume that M has "finite width", that is, that the distance from any point to the boundary is bounded uniformly. Under this assumption, we prove that the Poincar&ecuate; 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&ecuate; 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.

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The Paper was written on 24.10.2016

Last update 1.11.2016