Characterizations of the Sobolev space H1 on the boundary of a strongly Lipschitz domain in 3-D

Abstract

In this work, we investigate the Sobolev space $H^1\left(\partial \Omega \right)$ on a strongly Lipschitz boundary $\partial \Omega$, that is, $\Omega$ is a strongly Lipschitz domain (not necessarily bounded). 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 $H^1\left(\partial \Omega \right)$ is in particular of advantage, when it comes to traces of $H(curl,\Omega )$ vector fields.