Minimality of the ball for a model of charged liquid droplets

TL;DR

This paper proves that, under a specific variational model, charged liquid droplets minimize their free energy by forming spherical shapes when the charge is small, establishing the ball as the unique optimal shape.

Contribution

It demonstrates the minimality and uniqueness of spherical droplets in a regularized model for small charges, extending previous theoretical results.

Findings

Charged droplets are spherical in the small charge regime.

The ball is the unique minimizer of the free energy in this regime.

The proof combines regularity results with the Selection Principle.

Abstract

We deduce that charged liquid droplets minimizing Debye-H\"uckel-type free energy are spherical in the small charge regime. The variational model was proposed by Muratov and Novaga in 2016 to avoid the ill-posedness of the classical one. By combining a recent (partial) regularity result with Selection Principle of Cicalese and Leonardi, we prove that the ball is the unique minimizer in the small charge regime.