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

Abstract

We investigate the boundary trace operators that naturally correspond to $\mathrm{H}(\mathrm{curl},\Omega )$, 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 $\mathrm{H}(\mathrm{curl},\Omega )$ 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.

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