Local and global minimality results for a nonlocal isoperimetric problem on R^N

TL;DR

This paper investigates a nonlocal isoperimetric problem in ^N involving Riesz potentials, establishing conditions under which the ball is a local minimizer and providing criteria for global minimality based on volume thresholds.

Contribution

It introduces a criterion linking second variation positivity to local minimality and determines explicit volume thresholds for the ball's minimality in a nonlocal setting.

Findings

Critical configurations with positive second variation are local minimizers.

A quantitative inequality with respect to the $L^1$-norm is established.

Explicit critical volume thresholds for the ball's local minimality are derived.

Abstract

We consider a nonlocal isoperimetric problem defined in the whole space $\R^N$, whose nonlocal part is given by a Riesz potential with exponent $\alpha\in(0, N-1)$. We show that critical configurations with positive second variation are local minimizers and satisfy a quantitative inequality with respect to the $L^1$-norm. This criterion provides the existence of a (explicitly determined) critical threshold determining the interval of volumes for which the ball is a local minimizer, and allows to address several global minimality issues.