Removable singularities for div v = f in weighted Lebesgue spaces

TL;DR

This paper characterizes removable singularities for divergence equations in weighted Lebesgue spaces, linking them to weighted Hausdorff measure and capacity under certain conditions on the weight.

Contribution

It extends existing results by providing a characterization of removable sets for divergence in weighted spaces using weighted Hausdorff measure and capacity.

Findings

Removable sets are characterized by vanishing weighted Hausdorff measure.

Extension of results to $L^p_{1/w}$ spaces using capacity.

Generalization of prior work by Phuc, Torres, Silhavy, and the first author.

Abstract

Let $w\in L^1\_{loc}(\R^n)$ be apositive weight. Assuming that a doubling condition and an $L^1$ Poincar\'e inequality on balls for the measure $w(x)dx$, as well as a growth condition on $w$, we prove that the compact subsets of $\R^n$ which are removable for the distributional divergence in $L^{\infty}\_{1/w}$ are exactly those with vanishing weighted Hausdorff measure. We also give such a characterization for $L^p\_{1/w}$, $1\textless{}p\textless{}+\infty$, in terms of capacity. This generalizes results due to Phuc and Torres, Silhavy and the first author.