Regularity for Vector Fields, Hodge Decomposition and Green Formula in Lipschitz Domains

Abstract

For a bounded Lipschitz domain $\Omega \subset \mathbb{R}^3$, neither necessarily simply connected nor with connected boundary $\Gamma$, we prove a new Hodge-type decomposition: every $u \in H(\operatorname{curl}, \Omega)$ splits uniquely into a gradient and a field in $H^{\frac12}(\Omega) \cap H(\operatorname{curl}, \Omega)$ with tangential $L^2$-trace. As applications we remove the additional topological hypotheses from Costabel’s $H^{\frac12}$-regularity theorem, and extend the classical Green formula to two $H(\operatorname{curl})$-fields. We define tangential differential operators on the non-smooth boundary $\Gamma$ and revisit the space $H^{-\frac12}(\operatorname{div}_\Gamma, \Gamma)$.