A note on some non-local variational problems

TL;DR

This paper investigates two non-local variational problems involving competing Riesz-like repulsive and attractive terms, proving that balls are the unique minimizers under certain volume constraints, extending existing results.

Contribution

It generalizes known results by establishing the uniqueness of ball minimizers for broader classes of non-local energies with competing interactions.

Findings

Balls are the unique minimizers for the studied functionals.

Results extend previous literature to more general energies.

The analysis applies to both fractional perimeter and positive-power-type attractive terms.

Abstract

We study two non-local variational problems that are characterized by the presence of a Riesz-like repulsive term that competes with an attractive term. The first functional is defined on the subsets of $\mathbb{R}^N$ and has the fractional perimeter $\mathcal{P}_s$ as attractive term. The second functional instead is defined on $L^1(\mathbb{R}^N;[0,1])$ and contains an attractive term of positive-power-type. For both of the functionals we prove that balls are the unique minimizers in the appropriate volume constraint range, generalizing the results already present in the literature for more specific energies.