Minimizers for a one-dimensional interaction energy

TL;DR

This paper explicitly solves a one-dimensional probability measure minimization problem involving an interaction energy, revealing that minimizers can have unbounded densities, thus answering an open question in the field.

Contribution

It provides an explicit solution to a specific one-dimensional interaction energy minimization problem and establishes the nature of minimizers, addressing an unresolved issue in prior research.

Findings

Minimizers can be absolutely continuous with unbounded density.

The explicit form of minimizers is derived.

The work settles an open question about the nature of minimizers.

Abstract

We solve explicitly a certain minimization problem for probability measures in one dimension involving an interaction energy that arises in the modelling of aggregation phenomena. We show that in a certain regime minimizers are absolutely continuous with an unbounded density, thereby settling a question that was left open in previous works.