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 derivative M and let C-c(infinity)(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: - C-c(infinity)(M) is dense in W1,p(M & ring;) for all p is an element of(1,infinity); this generalizes a classical result by Aubin [2] for partial derivative M=& empty;. - M admits a sequence of first order cut-off functions in C-c(infinity)(M); for partial derivative M=& empty; this result can be traced back to Gaffney [7]. - the Laplace-Beltrami operator with domain of definition C-c(infinity)(M) is essentially self-adjoint; this is a generalization of a classical result by Strichartz [20] for partial derivative M=& empty;.