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

TL;DR

This paper establishes that weak L2 regularity for partial tangential traces on Lipschitz domains is equivalent to strong regularity, providing a key density result relevant for Maxwell's equations with mixed boundary conditions.

Contribution

It generalizes previous density results to partial tangential traces on Lipschitz domains, addressing an open problem for Maxwell's equations with mixed boundary conditions.

Findings

Density of smooth fields in the trace space is proven.

Weak L2 regularity is equivalent to strong regularity for partial tangential traces.

Results apply to strongly Lipschitz domains, including unbounded cases.

Abstract

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 (possibly unbounded). 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 Belgacem, Bernardi, Costabel and Dauge 1997. This is especially important for Maxwell's equation with mixed boundary condition as we answer the open problem by Weiss and Staffans 2013 (Section 5) for strongly Lipschitz pairs.