Sobolev functions without compactly supported approximations

Veronelli, Giona

doi:10.2140/apde.2022.15.1991

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Sobolev functions without compactly supported approximations

Giona Veronelli

Vol. 15 (2022), No. 8, 1991–2002

DOI: 10.2140/apde.2022.15.1991

Abstract

A basic property and 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 here. This settles an open problem raised for instance by E. Hebey (Nonlinear analysis on manifolds: Sobolev spaces and inequalities, Courant Lect. Notes Math. 5 (1999), 48–49).

Keywords

Sobolev spaces on manifolds, manifolds with unbounded geometry, density problems, Calderón–Zygmund inequalities

Mathematical Subject Classification

Primary: 46E35, 53C20

Milestones

Received: 25 June 2020

Revised: 25 February 2021

Accepted: 6 April 2021

Published: 10 February 2023

Authors

Giona Veronelli
Dipartimento di Matematica e Applicazioni Università di Milano Bicocca Milano Italy

Vol. 15, No. 8, 2022