An isoperimetric problem with a Coulombic repulsion and attractive term

Abstract

We study an energy given by the sum of the perimeter of a set, a Coulomb repulsion term of the set with itself and an attraction term of the set to a point charge. We prove that there exists an optimal radius r0 such that if r < r0 the ball Br is a local minimizer with respect to any other set with same measure. The global minimality of balls is also addressed.