Characterizations of Sobolev Functions that vanish on a part of the boundary

TL;DR

This paper characterizes Sobolev functions that vanish on a boundary part, showing they can be approximated by smooth functions away from that part, and reviews equivalent conditions for such functions.

Contribution

It provides a comprehensive characterization of Sobolev functions vanishing on boundary parts, including approximation and trace conditions, in domains with extension properties.

Findings

Functions vanishing on boundary parts can be approximated by smooth functions away from those parts.

Multiple equivalent characterizations of such Sobolev functions are established.

The results apply to domains with Sobolev extension properties.

Abstract

Let $\Omega$ be a bounded domain in R n with a Sobolev extension property around the complement of a closed part D of its boundary. We prove that a function u $\in$ W 1,p ($\Omega$) vanishes on D in the sense of an interior trace if and only if it can be approximated within W 1,p ($\Omega$) by smooth functions with support away from D. We also review several other equivalent characterizations, so to draw a rather complete picture of these Sobolev functions vanishing on a part of the boundary.