Existence and stability for a non-local isoperimetric model of charged liquid drops

Michael Goldman (1) , Matteo Novaga (2, 3) , Berardo Ruffini (4)

1 MPI-MiS - Max Planck Institute for Mathematics in the Sciences
2 Dipartimento di Matematica Pura e Applicata [Padova]
3 Dipartimento di Matematica [Pisa]
4 SNS - Scuola Normale Superiore di Pisa

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 $L^1$ 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.

Domaines: Equations aux dérivées partielles [math.AP]

Michael Goldman, Matteo Novaga, Berardo Ruffini. Existence and stability for a non-local isoperimetric model of charged liquid drops. Archive for Rational Mechanics and Analysis, 2015. ⟨hal-00878678v4⟩