A charged liquid drop model with Willmore energy

TL;DR

This paper introduces a modified liquid drop model incorporating Willmore energy to ensure well-posedness, and demonstrates that small charges lead to spherical droplet shapes.

Contribution

It proposes a new variational model with Willmore energy for electrified drops, resolving ill-posedness and analyzing shape stability at small charges.

Findings

Adding Willmore energy stabilizes the model.

Small charge droplets are spherical.

The model is well-posed with the added energy.

Abstract

We consider a variational model of electrified liquid drops, involving competition between surface tension and charge repulsion. Since the natural model happens to be ill-posed, we show that by adding to the perimeter a Willmore-type energy, the problem turns back to be well-posed. We also prove that for small charge the droplets is spherical.