Regularity of the minimiser of one-dimensional interaction energies

TL;DR

This paper establishes the uniqueness and regularity of solutions for a class of nonlocal interaction energy minimisation problems and related singular integral equations, with applications across various physical and mathematical fields.

Contribution

It proves that the minimisation problems and singular integral equations share a unique solution, offering new insights into regularity and positivity of solutions.

Findings

Unique solution for both minimisation and integral equations

New regularity results for the minimiser

Positivity results for solutions to integral equations

Abstract

We consider both the minimisation of a class of nonlocal interaction energies over non-negative measures with unit mass and a class of singular integral equations of the first kind of Fredholm type. Our setting covers applications to dislocation pile-ups, contact problems, fracture mechanics and random matrix theory. Our main result shows that both the minimisation problems and the related singular integral equations have the same unique solution, which provides new regularity results on the minimiser of the energy and new positivity results on the solutions to singular integral equations.