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

Authors: Augusto Ponce

Issue: Volume 62 (2013), Issue 4, 1055-1074


We prove that every closed set which is not $\sigma$-finite with respect to the Hausdorff measure $\mathcal{H}^{N-1}$ carries singularities of continuous vector fields in $\mathbb{R}^N$ for the divergence operator. We also show that finite measures which do not charge sets of $sigma$-finite Hausdorff measure $\mathcal{H}^{N-1}$ 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.