Local and global minimality issues for a nonlocal isoperimetric problem on \(\mathbb{R}^N\) (Q269584)

Summary: 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 \(\alpha\) in the nonlocal term the ball is the unique minimizer as long as the problem has a solution.