Local and global minimality issues for a nonlocal isoperimetric problem on $\mathbb R^N$

Abstract

We consider a nonlocal isoperimetric problem defined in the whole space $\mathbb R^N$, 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. Finally we deduce that for small masses the ball is also the unique global minimizer, and that for small exponents a in the nonlocal term the ball is the unique minimizer as long as the problem has a solution.