Singularities of the divergence of continuous vector fields and uniform Hausdorff estimates

TL;DR

The paper characterizes the singularities of divergence of continuous vector fields in Euclidean space, linking them to the measure-theoretic properties of sets and measures via Hausdorff measures and content.

Contribution

It establishes a connection between the non-sigma-finite sets and the presence of divergence singularities, and shows measures not charging sigma-finite sets can be approximated by divergence of continuous fields.

Findings

Sets not sigma-finite with respect to H^{N-1} carry divergence singularities.

Measures avoiding sigma-finite H^{N-1} sets can be approximated by divergence of continuous vector fields.

The main tool involves measure approximation via Hausdorff content.

Abstract

We prove that every closed set which is not sigma-finite with respect to the Hausdorff measure H^{N-1} carries singularities of continuous vector fields in the Euclidean space R^N for the divergence operator. We also show that finite measures which do not charge sets of sigma-finite Hausdorff measure H^{N-1} can be written as an L^1 perturbation of the divergence of a continuous vector field. The main tool is a property of approximation of measures in terms of the Hausdorff content.