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

We consider a nonlocal isoperimetric problem defined in the whole space RN, whose nonlocal part is given by a Riesz potential with exponent a a ε0;N1. We show that critical configurations with positive second variation are local minimizers and satisfy a quantitative inequality with respect to the L1-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.