Trace operator on $H^1(Ω)$ for general open bounded domains
Robert Eymard, David Maltese, Thierry Gallouët, Yannick Vincent
TL;DR
This work develops a directional trace framework for $H^1(\\Omega)$ on general bounded open domains by introducing direction-dependent boundary measures $\\mu_\\theta$ and associated traces $\\gamma_\\theta u$ for $u\\in W_\\theta(\\Omega)$. A global trace space $L^2(\\partial\\Omega,(\\mu_\\theta)_{\\theta\\in\\mathcal{S}})$ is constructed, and the trace operator $\\mathrm{tr}$ is defined on the closed subspace $H_{tr}^1(\\Omega)$, yielding a robust integration-by-parts formula that remains valid beyond Lipschitz domains. The paper shows that $H_{tr}^1(\\Omega)$ is closed and contains the natural fill $\\widetilde{H}^1(\\Omega)$, with precise criteria and examples illustrating when $H_{tr}^1(\\Omega)$ coincides with $H^1(\\Omega)$ or not. These results provide a rigorous framework for variational problems on irregular domains and clarify trace behavior in the absence of standard boundary regularity, potentially connecting to harmonic measures through future work.
Abstract
In the case of any bounded open set $Ω\subset{\mathbb R}^d$ with boundary $\partialΩ$, we first construct a directional trace in any direction $θ$ of the unit sphere, for any $u\in L^2(Ω)$ whose the directional derivative $\partial_θu$ in the direction $θ$ belongs to $L^2(Ω)$. This directional trace is shown to belong to $L^2(\partialΩ,μ_θ)$, where $μ_θ$ is a measure supported by the closure of all points of $\partialΩ$ which are the extremity of an open segment directed by $θ$, included in $Ω$. This trace enables an integration by parts formula. We then show that the set $H_{\rm tr}^1(Ω)$ containing the elements of $H^1(Ω)$ whose the directional trace does not depend on $θ$ is closed. It therefore contains the closure of $H^1(Ω)\cap C^0(\overlineΩ)$ in $H^1(Ω)$. Examples where $H_{\rm tr}^1(Ω) = H^1(Ω)$ and $H_{\rm tr}^1(Ω) \neq H^1(Ω)$ are provided.