On minimizers of an isoperimetric problem with long-range interactions under a convexity constraint

TL;DR

This paper investigates the equilibrium shapes of charged liquid drops under convexity constraints, proving regularity of minimizers, uniqueness of spherical solutions for small charge, and analyzing their asymptotic behavior as charge increases.

Contribution

It establishes well-posedness, regularity, and asymptotic properties of minimizers in a variational model with long-range Coulomb interactions under convexity constraints.

Findings

Minimizers are C1,1-regular for Coulombic interactions in 2D.

Balls are the unique minimizers for small charge.

As charge increases, the shape of minimizers exhibits specific asymptotic behavior.

Abstract

We study a variational problem modeling the behavior at equilibrium of charged liquid drops under convexity constraint. After proving well-posedness of the model, we show C 1,1-regularity of minimizers for the Coulombic interaction in dimension two. As a by-product we obtain that balls are the unique minimizers for small charge. Eventually, we study the asymptotic behavior of minimizers, as the charge goes to infinity.