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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 theliterature 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 samespace. This second characterization of $\mathrm{H}^{1}(\partial\Omega)$ is inparticular of advantage, when it comes to traces of$\mathrm{H}(\operatorname{curl},\Omega)$ vector fields.

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