  Let $M$ be a smooth manifold with boundary $\partial M$ and bounded geometry,
$\partial_D M \subset \partial M$ be an open and closed subset, $P$ be a second
order differential operator on $M$, and $b$ be a first order differential
operator on $\partial M \smallsetminus \partial_D M$. We prove the regularity
and well-posedness of the mixed Robin boundary value problem $$Pu = f \mbox{ in
} M,\ u = 0 \mbox{ on } \partial_D M,\ \partial^P_\nu u + bu = 0 \mbox{ on }
\partial M \setminus \partial_D M$$ under some natural assumptions. Our
operators act on sections of a vector bundle $E \to M$ with bounded geometry.
Our well-posedness result is in the Sobolev spaces $H^s(M; E)$, $s \geq 0$. The
main novelty of our results is that they are formulated on a non-compact
manifold.
  We include also some extensions of our main result in different directions.
First, the finite width assumption is required for the Poincar\'{e} inequality
on manifolds with bounded geometry, a result for which we give a new, more
general proof. Second, we consider also the case when we have a decomposition
of the vector bundle $E$ (instead of a decomposition of the boundary). Third,
we also consider operators with non-smooth coefficients, but, in this case, we
need to limit the range of $s$. Finally, we also consider the case of uniformly
strongly elliptic operators. In this case, we introduce a \emph{uniform Agmon
condition} and show that it is equivalent to the G\aa rding inequality. This
extends an important result of Agmon (1958).
