Extendability of functions with partially vanishing trace

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 ${\mathrm{W}}_D^{k,p}(\Omega )$, $1\le p<\infty$, 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 $D$.

32 pages, 5 Figures. Completely revised manuscript, including extension of higher-order Sobolev spaces, an a-priori density result, homogeneous estimates, and additional results for Lipschitz spaces. To be submitted