Neumann cut-offs and essential self-adjointness on complete Riemannian manifolds with boundary

We generalize some fundamental results for noncompact Riemannian manfolds without boundary, that only require completeness and no curvature assumptions, to manifolds with boundary: let $M$ be a smooth Riemannian manifold with boundary $\partial M$ and let $\hat{C}^\infty_c(M)$ denote the space of smooth compactly supported cut-off functions with vanishing normal derivative, Neumann cut-offs. We show, among other things, that under completeness: - $\hat{C}^\infty_c(M)$ is dense in $W^{1,p}(\mathring{M})$ for all $p\in (1,\infty)$; this generalizes a classical result by Aubin [2] for $\partial M=\emptyset$. - $M$ admits a sequence of first order cut-off functions in $\hat{C}^\infty_c(M)$; for $\partial M=\emptyset$ this result can be traced back to Gaffney [7]. - the Laplace-Beltrami operator with domain of definition $\hat{C}^\infty_c(M)$ is essentially self-adjoint; this is a generalization of a classical result by Strichartz [20] for $\partial M=\emptyset$.