The divergence equation with measure data

Abstract: We will investigate solvability results for the divergence equation $\mathrm{div}\, v=\mu$, when $\mu$ is a (signed) Radon measure in an open set $\Omega$, and one looks for solutions in weighted Lebesgue spaces in $\Omega$. A question of particular interest will be to understand how the geometry of the open set $\Omega$ allows the existence of an integrable weight $w$ yielding (constructive) solvability results in $L^\infty_{1/w}(\Omega)$ together with linear estimates on the norm of the constructed solutions.

The results we shall present were obtained in collaboration with E. Russ.