Optimal distribution of oppositely charged phases: perfect screening and other properties

We study the minimum energy configuration of a uniform distribution of negative charge subject to Coulomb repulsive self-interaction and attractive interaction with a fixed positively charged domain. After having established existence and uniqueness of a minimizing configuration, we prove charge neutrality and the complete screening of the Coulomb potential exerted by the positive charge, and we discuss the regularity properties of the solution. We also determine, in the variational sense of $\Gamma$-convergence, the limit model when the charge density of the negative phase is much higher than the positive one.