Minimisers of a general Riesz-type Problem

TL;DR

This paper investigates the minimization of a combined perimeter and non-local term for sets in Euclidean space, establishing existence, regularity, and uniqueness of minimizers, especially in two dimensions where balls are shown to be optimal under certain conditions.

Contribution

It provides new existence and regularity results for minimizers of a Riesz-type problem and proves the uniqueness of balls as minimizers in two dimensions under perimeter-dominated regimes.

Findings

Existence and regularity of minimizers established.

Balls are unique minimizers in 2D under certain conditions.

Results apply to a wide class of kernel functions g.

Abstract

We consider sets in $\mathbb R^N$ which minimise, for fixed volume, the sum of the perimeter and a non-local term given by the double integral of a kernel $g:\mathbb R^N\setminus\{0\}\to \mathbb R^+$. We establish some general existence and regularity results for minimisers. In the two-dimensional case we show that balls are the unique minimisers if the perimeter-dominated regime, for a wide class of functions $g$.