Rigidity of the ball for an isoperimetric problem with strong capacitary repulsion

Abstract

We consider a variational problem involving competition between surface tension and charge repulsion. We show that, as opposed to the case of weak (short-range) interactions where we proved ill-posedness of the problem in a previous paper, when the repulsion is stronger the perimeter dominates the capacitary term at small scales. In particular, we prove existence of minimizers for small charges as well as their regularity. Combining this with the stability of the ball under small $C^{1,\gamma }$ perturbations, this ultimately leads to the minimality of the ball for small charges. We cover in particular the borderline case of the $1$-capacity where both terms in the energy are of the same order.