# Well-posedness of the Laplacian on manifolds with boundary and bounded geometry

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Bernd Ammann, Nadine Große, Victor Nistor

**Well-posedness of the Laplacian 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 Poincaré 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 Poincaré 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 1.11.2016

Last update 29.6.2018