Characterizations of the Sobolev space $\mathrm{H}^{1}$ on the boundary of a strong Lipschitz domain in 3-D

Abstract

In this work we investigate the Sobolev space $\mathrm{H}^{1}(\partial\Omega)$ on a strong Lipschitz boundary $\partial\Omega$, i.e., $\Omega$ is a strong Lipschitz domain. In most of the literature this space is defined via charts and Sobolev spaces on flat domains. We show that there is a different approach via differential operators on $\Omega$ and a weak formulation directly on the boundary that leads to the same space. This second characterization of $\mathrm{H}^{1}(\partial\Omega)$ is in particular of advantage, when it comes to traces of $\mathrm{H}(\operatorname{curl},\Omega)$ vector fields.