On divergence operators: Free space and vanishing charges

TL;DR

This paper investigates the existence and regularity of solutions to the divergence equation in critical function spaces using localized topologies, with applications to specific classes of functions and domains.

Contribution

It introduces a framework using localized topologies to establish existence and regularity results for divergence equations in critical spaces, characterizing admissible right-hand sides.

Findings

Characterization of $F$ for which solutions exist with bounded norms

Application of theory to specific function spaces and domains

Examples of admissible $F$ in various cases

Abstract

We use localized topologies to prove existence and optimal regularity results for the divergence equation $\mathrm{div} (v) = F$ in critical cases $v \in L_1(\Omega;\mathbb{R}^m)$ or $v \in C_0(\Omega;\mathbb{R}^m)$, i.e. we characterize those $F$ for which a solution $v$ exists whose norm is bounded by an appropriate norm of $F$. We assume $\Omega$ satisfies a Poincar\'e inequality or an extension property. We apply the general theory to give examples of admissible $F$ in each case.