On an isoperimetric problem with a competing non-local term. I. The planar case

TL;DR

This paper investigates a modified isoperimetric problem with a non-local repulsive term, analyzing existence, symmetry, and structure of minimizers depending on mass and kernel decay rate.

Contribution

It characterizes the existence, symmetry, and structure of minimizers for the non-local isoperimetric problem across different mass regimes and kernel decay rates.

Findings

Minimizers exist for small masses and are disks.

Minimizers do not exist for large masses due to fragmentation.

Energy scales linearly with mass at large masses.

Abstract

This paper is concerned with a study of the classical isoperimetric problem modified by an addition of a non-local repulsive term. We characterize existence, non-existence and radial symmetry of the minimizers as a function of mass in the situation where the non-local term is generated by a kernel given by an inverse power of the distance. We prove that minimizers of this problem exist for sufficiently small masses and are given by disks with prescribed mass below a certain threshold, when the interfacial term in the energy is dominant. At the same time, we prove that minimizers fail to exist for sufficiently large masses due to the tendency of the low energy configuration to split into smaller pieces when the non-local term in the energy is dominant. In the latter regime, we also establish linear scaling of energy with mass suggesting that for large masses low energy configurations consist of many roughly equal size pieces far apart. In the case of slowly decaying kernels we give a complete characterization of the minimizers.