Weak equals strong L2 regularity for partial tangential traces on Lipschitz domains

Nathanael Skrepek, Dirk Pauly

Published 2023-09-26Version 1

We investigate the boundary trace operators that naturally correspond to $\mathrm{H}(\operatorname{curl},\Omega)$, namely the tangential and twisted tangential trace, where $\Omega \subseteq \mathbb{R}^{3}$. In particular we regard partial tangential traces, i.e., we look only on a subset $\Gamma$ of the boundary $\partial\Omega$. We assume both $\Omega$ and $\Gamma$ to be strongly Lipschitz. We define the space of all $\mathrm{H}(\operatorname{curl},\Omega)$ fields that possess a $\mathrm{L}^{2}$ tangential trace in a weak sense and show that the set of all smooth fields is dense in that space, which is a generalization of \cite{BeBeCoDa97}. This is especially important for Maxwell's equation with mixed boundary condition as we answer the open problem by Weiss and Staffans in \cite[Sec.~5]{WeSt13} for strongly Lipschitz pairs.