Inserted: 30 oct 2013
Last Updated: 3 nov 2015
Journal: Arch. Rat. Mech. Anal.
We consider a variational problem related to the shape of charged liquid drops at equilibrium. We show that this problem never admits global minimizers with respect to L1 perturbations preserving the volume. This leads us to study it in more regular classes of competitors, for which we show existence of minimizers. We then prove that the ball is the unique solution for sufficiently small charges.