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

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