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

Moritz Egert; Patrick Tolksdorf

doi:10.3934/dcdss.2017037

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2017, Volume 10, Issue 4: 729-743. Doi: 10.3934/dcdss.2017037

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

Moritz Egert^1, , and
Patrick Tolksdorf^2,

1.
Laboratoire de Mathématiques d'Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France
2.
Fachbereich Mathematik, Technische Universität, Darmstadt, Schlossgartenstr. 7,64289 Darmstadt, Germany

^* Corresponding author: Moritz Egert
^* Corresponding author: Moritz Egert

Received: July 2016

Revised: September 2016

Published: July 2017

The first author was supported by a public grant as part of the FMJH. The second author was supported by "Studienstiftung des deutschen Volkes"

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Abstract

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ with a Sobolev extension property around the complement of a closed part $D$ of its boundary. We prove that a function $u \in {\rm{W}}^{1,p}(\Omega)$ vanishes on $D$ in the sense of an interior trace if and only if it can be approximated within ${\rm{W}}^{1,p}(\Omega)$ 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.

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Mathematics Subject Classification: Primary: 46E35, 31B25; Secondary: 26B30.

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Figure 1. The dyadic 'skeleton' of $\Omega$ is obtained from the square $[0,1] \times [1,2]$ by iteratively attaching a total number of $2^j$ disjoint squares of side length $2^{-j}$ at the bottom of the existing construction. The domain $\Omega$ is then constructed by blowing up the line segments to appropriately sized open rectangles

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