Large volume minimizers of a non local isoperimetric problem: theoretical and numerical approaches

TL;DR

This paper investigates the minimization of a non-local isoperimetric problem involving perimeter, Riesz potential, and an external potential, providing theoretical insights and a numerical method for large volume minimizers.

Contribution

It introduces an external potential to ensure existence of minimizers and develops a numerical approach to analyze large volume solutions.

Findings

Existence of minimizers for all volumes due to external potential.

Characterization of the geometry of large minimizers.

A numerical method for solving the variational problem.

Abstract

We consider the volume-constrained minimization of the sum of the perimeter and the Riesz potential. We add an external potential of the form $\|x\|^{\beta}$ that provides the existence of a minimizer for any volume constraint, and we study the geometry of big minimizers. Then we provide a numerical method to adress this variational problem.