Equilibrium shapes of charged droplets and related problems: (mostly) a review

TL;DR

This paper reviews recent findings on the equilibrium shapes of charged liquid drops, discusses the ill-posedness of the variational model, and presents new results on charge distribution and the non-existence of optimal conducting drops.

Contribution

It introduces new proofs for the existence of optimal charge distributions and demonstrates the non-existence of optimal conducting drops under certain conditions.

Findings

Existence of an optimal charge distribution for conducting drops in external fields.

Ill-posedness of the natural variational model for charged drops.

Non-existence of an optimal conducting drop in the studied setting.

Abstract

We review some recent results on the equilibrium shapes of charged liquid drops. We show that the natural variational model is ill-posed and how this can be overcome by either restricting the class of competitors or by adding penalizations in the functional. The original contribution of this note is twofold. First, we prove existence of an optimal distribution of charge for a conducting drop subject to an external electric field. Second, we prove that there exists no optimal conducting drop in this setting.