Geometry of minimizers for the interaction energy with mildly repulsive potentials

TL;DR

This paper investigates the structure of minimizers for interaction energies with mildly repulsive potentials, establishing conditions under which their supports are finite or bounded, with specific results in one-dimensional cases.

Contribution

It provides new insights into the support properties of local and global minimizers for mildly repulsive interaction potentials, including bounds and structural characterizations.

Findings

Support of local minimizers consists of isolated points.

Support of global minimizers is finite.

Quantitative bounds in one-dimensional cases.

Abstract

We show that the support of any local minimizer of the interaction energy consists of isolated points whenever the interaction potential is of class $C^2$ and mildly repulsive at the origin; moreover, if the minimizer is global, then its support is finite. In addition, for some class of potentials we prove the validity of a uniform upper bound on the cardinal of the support of a global minimizer. Finally, in the one-dimensional case, we give quantitative bounds.