  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.
