Neumann Cut-Offs and Essential Self-adjointness on Complete Riemannian Manifolds with Boundary

• DOI: 10.1007/s12220-025-01972-y
• Publication Date: 2025-03-26T23:46:58Z
• Authors: Davide Bianchi; Batu Güneysu; Alberto G. Setti
• Keywords: Riemannian Geometry

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

$$\partial M$$

∂ M

and let

$$\hat{C}^\infty _c(M)$$

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:

$$\hat{C}^\infty _c(M)$$

C ^

c ∞

( M )

is dense in

$$W^{1,p}(\mathring{M})$$

W

1 , p

(

M ˚

)

for all

$$p\in (1,\infty )$$

p ∈ ( 1 , ∞ )

; this generalizes a classical result by Aubin (Bull. Sci. Math. 100:149–173, 1976) for

$$\partial M=\emptyset $$

∂ M = ∅

.

M admits a sequence of first order cut-off functions in

$$\hat{C}^\infty _c(M)$$

C ^

c ∞

( M )

; for

$$\partial M=\emptyset $$

∂ M = ∅

this result can be traced back to Gaffney (Ann. Math. (2) 60(1):140–145, 1954).

the Laplace–Beltrami operator with domain of definition

$$\hat{C}^\infty _c(M)$$

C ^

c ∞

( M )

is essentially self-adjoint; this is a generalization of a classical result by Strichartz (J. Funct. Anal. 52(1):48–79, 1983) for

$$\partial M=\emptyset $$

∂ M = ∅

.