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

TL;DR

The paper characterizes the Sobolev space $H^{1}(\partial\Omega)$ on strong Lipschitz boundaries in 3D via two approaches: the standard chart-based definition and a boundary-only weak formulation tied to tangential traces of $H(\operatorname{curl},\Omega)$ fields. It develops a chart-based lifting mechanism through strong Lipschitz charts that links the boundary tangential gradient to traces of volume gradients, proving the equivalence $\tilde{H}^{1}(\partial\Omega)=H^{1}(\partial\Omega)$ and $\widetilde{\nabla}_{\tau} f = \nabla_{\tau} f$. The results establish that a weak tangential trace of $\nabla F$ implies $F|_{\partial\Omega}\in H^{1}(\partial\Omega)$, and that the weak and strong boundary formulations coincide, resolving gaps in prior literature. This has practical impact for trace analysis of vector fields in electromagnetism and other areas where $H(\operatorname{curl},\Omega)$ traces are used, enabling boundary-focused methods without circular reasoning.

Abstract

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

Figures (1)

• Figure 1: Lipschitz boundary

Theorems & Definitions (41)

• Claim A
• Definition 2.1
• Lemma 3.1
• proof
• Definition 3.2
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Definition 3.5