Extendability of functions with partially vanishing trace

TL;DR

This paper constructs a global Sobolev extension operator for functions vanishing on a part of the boundary, working under mild assumptions and providing sharp boundary regularity and local estimates.

Contribution

It introduces a novel global extension operator for Sobolev spaces with boundary vanishing conditions, avoiding localization and handling sharp boundary regularity.

Findings

Constructed a bounded Sobolev extension operator for partial boundary vanishing functions.

Provided homogeneous and local estimates for the extension operator.

Extended the analysis to Lipschitz function spaces with vanishing trace conditions.

Abstract

Let $\Omega \subseteq \mathbb{R}^d$ be open and $D\subseteq \partial\Omega$ be a closed part of its boundary. Under very mild assumptions on $\Omega$, we construct a bounded Sobolev extension operator for the Sobolev space $\mathrm{W}^{k , p}_D (\Omega)$, $1 \leq p < \infty$, which consists of all functions in $\mathrm{W}^{k , p} (\Omega)$ that vanish in a suitable sense on $D$. In contrast to earlier work, this construction is global and \emph{not} using a localization argument, which allows to work with a boundary regularity that is sharp at the interface dividing $D$ and $\partial \Omega \setminus D$. Moreover, we provide homogeneous and local estimates for the extension operator. Also, we treat the case of Lipschitz function spaces with a vanishing trace condition on $D$.