Characterizations of the existence and removable singularities of divergence-measure vector fields Nguyen Cong PhucMonica Torres 35F0528A1226B2026B1235L65weakly differentiable vector fieldsdivergence-measure vector fieldsGauss-Green theoremgeometric measurescapacitiesRiesz transformremovable singularities 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 $\mu$. Indiana University Mathematics Journal 2008 text pdf 10.1512/iumj.2008.57.3312 10.1512/iumj.2008.57.3312 en Indiana Univ. Math. J. 57 (2008) 1573 - 1598 state-of-the-art mathematics http://iumj.org/access/