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

TL;DR

This paper investigates the shape and stability of charged liquid drops, proving the non-existence of global minimizers in a broad class and establishing the ball as the unique stable shape at low charges.

Contribution

It introduces a non-local isoperimetric model for charged drops, proving existence of minimizers in regular classes and uniqueness of the spherical shape at small charges.

Findings

Global minimizers do not exist in the broad class considered.

Existence of minimizers in more regular classes.

The sphere is the unique stable shape for small charges.

Abstract

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.