## Abstract

Let M be a Riemannian manifold with a smooth boundary. The main question we address in this article is: "When is the Laplace-Beltrami operator Delta :Hk+1(M) boolean AND H-0(1)((M) -> Hk-1 (M), k is an element of N-0, invertible?" We consider also the case of mixed boundary conditions. The study of this main question leads us to the class of manifolds with boundary and bounded geometry introduced ...

## Abstract

Let M be a Riemannian manifold with a smooth boundary. The main question we address in this article is: "When is the Laplace-Beltrami operator Delta :Hk+1(M) boolean AND H-0(1)((M) -> Hk-1 (M), k is an element of N-0, invertible?" We consider also the case of mixed boundary conditions. The study of this main question leads us to the class of manifolds with boundary and bounded geometry introduced by Schick (Math. Nachr. 223 (2001), 103-120). We thus begin with some needed results on the geometry of manifolds with boundary and bounded geometry. Let partial derivative M-D subset of partial derivative M be an open and closed subset of the boundary of M. We say that (M, partial derivative M-D) has finite width if, by definition, M is a manifold with boundary and bounded geometry such that the distance dist (x, partial derivative M-D) from a point x subset of M to partial derivative M-D subset of partial derivative M is bounded uniformly in x (and hence, in particular, partial derivative M-D intersects all connected components of M). For manifolds (M, partial derivative M-D) with finite width, we prove a Poincare inequality for functions vanishing on partial derivative M-D, thus generalizing an important result of Sakurai (Osaka J. Math, 2017). The Poincare inequality then leads, as in the classical case to results on the spectrum of Delta with domain given by mixed boundary conditions, in particular, Delta is invertible for manifolds (M, partial derivative M-D) with finite width. The bounded geometry assumption then allows us to prove the well-posedness of the Poisson problem with mixed boundary conditions in the higher Sobolev spaces H-s(M), s >= 0.