Conducting flat drops in a confining potential

TL;DR

This paper analyzes a geometric variational problem modeling charged liquid drops under external potential, characterizing energy behavior, existence, and regularity of minimizers depending on Coulomb interaction strength.

Contribution

It introduces a characterization of the semicontinuous energy envelope and establishes existence and regularity results for minimizers when Coulomb repulsion is below a critical threshold.

Findings

Characterization of the semicontinuous energy envelope.

Existence of minimizers under certain conditions.

Derivation of the Euler--Lagrange equation for regular critical points.

Abstract

We study a geometric variational problem arising from modeling two-dimensional charged drops of a perfectly conducting liquid in the presence of an external potential. We characterize the semicontinuous envelope of the energy in terms of a parameter measuring the relative strength of the Coulomb interaction. As a consequence, when the potential is confining and the Coulomb repulsion strength is below a critical value, we show existence and partial regularity of volume-constrained minimizers. We also derive the Euler--Lagrange equation satisfied by regular critical points, expressing the first variation of the Coulombic energy in terms of the normal $\frac12$-derivative of the capacitary potential.