  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
$\Delta\colon H^{k+1}(M)\cap H^1_0(M) \to H^{k-1}(M)$, $k\in \mathbb{N}_0$,
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. Nach. 2001). We begin with some
needed results on the geometry of manifolds with boundary and bounded geometry.
Let $\partial_D M \subset \partial M$ be an open and closed subset of the
boundary of $M$. We say that $(M, \partial_D M)$ has \emph{finite width} if, by
definition, $M$ is a manifold with boundary and bounded geometry such that the
distance $d(x, \partial_D M)$ from a point $x \in M$ to $\partial_D M \subset
\partial M$ is bounded uniformly in $x$ (and hence, in particular, $\partial_D
M$ intersects all connected components of $M$). For manifolds $(M, \partial_D
M)$ with finite width, we prove a Poincar\'e inequality for functions vanishing
on $\partial_D M$, thus generalizing an important result of Sakurai (Osaka J.
Math, 2017). The Poincar\'e 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 $(M, \partial_D
M)$ with finite width. The bounded geometry assumption then allows us to prove
the well-posedness of the Poisson problem with mixed boundary conditions in
higher Sobolev spaces $H^s(M)$, $s \ge 0$.
