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

Abstract

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 ${\hat{C}}_c^{\infty }(M)$ 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: - ${\hat{C}}_c^{\infty }(M)$ is dense in $W^{1,p}(\mathring{M})$ for all $p\in (1,\infty )$; this generalizes a classical result by Aubin [2] for $\partial M=\varnothing$. - $M$ admits a sequence of first order cut-off functions in ${\hat{C}}_c^{\infty }(M)$; for $\partial M=\varnothing$ this result can be traced back to Gaffney [7]. - the Laplace-Beltrami operator with domain of definition ${\hat{C}}_c^{\infty }(M)$ is essentially self-adjoint; this is a generalization of a classical result by Strichartz [20] for $\partial M=\varnothing$.