Continuity of critical points for 1-dimensional non-local energies

TL;DR

This paper proves that bounded critical points of a 1D Riesz energy with attractive-repulsive interactions are continuous within their support, under certain kernel growth conditions.

Contribution

It establishes the continuity of critical points for a class of non-local energies in one dimension, extending understanding of their regularity properties.

Findings

Critical points are continuous inside their support.

Continuity holds under specific kernel growth assumptions.

Results contribute to the theory of non-local energy minimizers.

Abstract

In this paper we deal with the bounded critical points of a Riesz energy of attractive-repulsive type in dimension 1. Under suitable assumptions on the growth of the kernel in the origin, we are able to prove that they are continuous inside their support.