On divergence operators: Free space and vanishing charges

Abstract

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