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

In this work, we investigate the Sobolev space H1(∂Ω)$\mathrm{H}^{1}(\partial \Omega)$ on a strongly Lipschitz boundary ∂Ω$\partial \Omega$, that is, Ω$\Omega$ 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 Ω$\Omega$ and a weak formulation directly on the boundary that leads to the same space. This second characterization of H1(∂Ω)$\mathrm{H}^{1}(\partial \Omega)$ is in particular of advantage, when it comes to traces of H(curl,Ω)$\mathrm{H}(\operatorname{curl},\Omega)$ vector fields.