Extendability of functions with partially vanishing trace

Russell M. Brown, Robert Haller-Dintelmann, Patrick Tolksdorf

Published 2019-10-14Version 1

Let $\Omega \subset \mathbb{R}^d$ be an open set and $D$ be a closed part of its boundary. Under very mild assumptions on $\Omega$, we construct a bounded Sobolev extension operator for the Sobolev spaces $\mathrm{W}^{1 , p}_D (\Omega)$, $1 \leq p \leq \infty$, containing all $\mathrm{W}^{1 , p} (\Omega)$-functions that vanish in some sense on $D$. In comparison to other constructions of Brewster et. al. and Haller-Dintelmann et. al. the main focus of this work is to generalize the geometric conditions at the boundary dividing $D$ and $\partial \Omega \setminus D$ that ensure the existence of an extension operator.