We consider a nonlocal isoperimetric problem defined in the whole space , whose nonlocal part is given by a Riesz potential with exponent . We show that critical configurations with positive second variation are local minimizers and satisfy a quantitative inequality with respect to the -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.
Cite this article
Marco Bonacini, Riccardo Cristoferi, Local and global minimality issues for a nonlocal isoperimetric problem on . Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 27 (2016), no. 1, pp. 37–50DOI 10.4171/RLM/721