On the normal trace space of extended divergence-measure fields

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.