Extension theorems for external cusps with minimal regularity

Gabriel Acosta and Ignacio Ojea

Vol. 259 (2012), No. 1, 1–39

DOI: 10.2140/pjm.2012.259.1

Abstract

Sobolev functions defined on certain simple domains with an isolated singular point (such as power type external cusps) can not be extended in standard, but in appropriate weighted spaces. In this article we show that this result holds for a large class of domains that generalizes external cusps, allowing minimal boundary regularity. The construction of our extension operator is based on a modification of reflection techniques originally developed for dealing with uniform domains. The weight involved in the extension appears as a consequence of the failure of the domain to comply with basic properties of uniform domains, and it turns out to be a quantification of that failure. We show that weighted, rather than standard spaces, can be treated with our approach for weights that are given by a monotonic function either of the distance to the boundary or of the distance to the tip of the cusp.

Keywords

extension theorems, external cusp, weighted Sobolev spaces

Mathematical Subject Classification 2010

Primary: 46E35

Milestones

Received: 13 February 2012

Accepted: 27 February 2012

Published: 31 August 2012