Calculus of Variations and Geometric Measure Theory

M. Bonacini - R. Cristoferi

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

created by bonacini on 19 Jul 2013
modified on 18 Jul 2014


Published Paper

Inserted: 19 jul 2013
Last Updated: 18 jul 2014

Journal: SIAM J. Math. Anal.
Volume: 46
Number: 4
Pages: 2310–2349
Year: 2014
Doi: 10.1137/130929898


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, and allows to address several global minimality issues.

Keywords: Local minimizers, second variation, global minimizers, nonlocal isoperimetri problem, minimality conditions