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

2024/06/17 by Davide Bianchi, Bianchi, Davide, Batu Güneysu +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2406.11120

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 ∂ M and let C^∞_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: - C^∞_c(M) is dense in W^1,p(\mathringM) for all p∈ (1,∞); this generalizes a classical result by Aubin [2] for ∂ M=∅. - M admits a sequence of first order cut-off functions in C^∞_c(M); for ∂ M=∅ this result can be traced back to Gaffney [7]. - the Laplace-Beltrami operator with domain of definition C^∞_c(M) is essentially self-adjoint; this is a generalization of a classical result by Strichartz [20] for ∂ M=∅.

Related

• Sequences of Laplacian cut-off functions
• Density problems for second order Sobolev spaces and cut-off functions on manifolds with unbounded geometry
• Estimates for functions of the Laplacian on manifolds with bounded geometry
• Well-posedness of the Laplacian on manifolds with boundary and bounded geometry
• The Boundary between Compact and Noncompact Complete Riemann Manifolds