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 $\mathrm{H}(\mathrm{curl},\Omega )$ vector fields.

arXiv comment12 pages, 1 figure

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