On the normal trace space of extended divergence-measure fields
Christopher Irving
TL;DR
This work characterizes the distributional normal trace of extended divergence-measure fields on boundaries of sets in $\mathbb{R}^n$ by identifying the trace space with the Arens–Eells space $\text{AE}(\partial E)$ and by constructing a trace operator into the predual framework $\mathrm{Lip}_{\mathrm{b}}(\partial E)^{\ast}$. Central to the analysis is a refined pointwise description of the Anzellotti pairing $\overline{\nabla \phi \cdot {\bm F}}$ and Smirnov's curve-decomposition which together yield a robust, weak-${\ast}$ continuous trace and its surjectivity under mild domain regularity (locally rectifiably convex, with $\partial U = \partial \overline{U}^{c}$). The results enable extension theorems for divergence-measure fields and divergence-free measures, with both general $\mathcal{DM}^{\mathrm{ext}}$ and $\mathcal{DM}^1$ formulations; in particular, the $\mathcal{DM}^1$ setting admits $L^1$-type extensions. These contributions advance understanding of flux traces for highly singular fields and provide concrete tools for boundary-value problems in continuum mechanics and related PDEs.
Abstract
We characterise the normal trace space associated to extended (measure-valued) divergence-measure fields on the boundary of a set $E \subset \mathbb R^n$, as the Arens-Eells space $\mathrm{AE}(\partial E)$. Such a trace operator is constructed for any Borel set $E$, and under a mild regularity condition, which includes Lipschitz domains, this trace operator is shown to moreover be surjective. This relies in part on a new pointwise description of the Anzellotti pairing $\overline{\nabla φ\cdot {\boldsymbol F}}$ between a $\mathrm{W}^{1,\infty}$ function $φ$ and extended divergence-measure field ${\boldsymbol F}$. As an application, we prove extension theorems for divergence-measure fields and divergence-free measures. Results for $\mathrm{L}^1$-fields are also obtained.