IUMJ

Title: Characterizations of the existence and removable singularities of divergence-measure vector fields

Authors: Nguyen Cong Phuc and Monica Torres

Issue: Volume 57 (2008), Issue 4, 1573-1598

Abstract:

We study the solvability and removable singularities of the equation $\div F = \mu$, with measure data $\mu$, in the class of continuous or $L^{p}$ vector fields $F$, where $1 \leq p \leq \infty$. In particular, we show that, for a signed measure $\mu$, the equation $\div F = \mu$ has a solution $F \in L^{\infty}(\mathbb{R}^{n})$ if and only if $| \mu(U) | \leq C\mathcal{H}^{n-1}(\partial U)$ for any open set $U$ with smooth boundary. For non-negative measures $\mu$, we obtain explicit characterizations of the solvability of $\div F = \mu$ in terms of potential energies of $\mu$ for $p \neq \infty$, and in terms of densities of $\mu$ for continuous vector fields. These existence results allow us to characterize the removable singularities of the corresponding equation $\div F = \mu$ with signed measures $\mu$.