Sobolev functions without compactly supported approximations
Abstract
A basilar property and a useful tool in the theory of Sobolev spaces is the density of smooth compactly supported functions in the space $W^{k,p}(\R^n)$ (i.e. the functions with weak derivatives of orders $0$ to $k$ in $L^p$). On Riemannian manifolds, it is well known that the same property remains valid under suitable geometric assumptions. However, on a complete noncompact manifold it can fail to be true in general, as we prove in this paper. This settles an open problem raised for instance by E. Hebey [\textit{Nonlinear analysis on manifolds: Sobolev spaces and inequalities}, Courant Lecture Notes in Mathematics, vol. 5, 1999, pp. 4849].
 Publication:

arXiv eprints
 Pub Date:
 April 2020
 arXiv:
 arXiv:2004.10682
 Bibcode:
 2020arXiv200410682V
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematics  Differential Geometry;
 Mathematics  Functional Analysis;
 46E35;
 53C20
 EPrint:
 9 pages. Minor typos corrected