Isoperimetry and stability properties of balls with respect to nonlocal energies

Abstract

We obtain a sharp quantitative isoperimetric inequality for nonlocal $s$-perimeters, uniform with respect to $s$ bounded away from $0$. This allows us to address local and global minimality properties of balls with respect to the volume-constrained minimization of a free energy consisting of a nonlocal $s$-perimeter plus a non-local repulsive interaction term. In the particular case $s=1$ the $s$-perimeter coincides with the classical perimeter, and our results improve the ones of Knüpfer and Muratov concerning minimality of balls of small volume in isoperimetric problems with a competition between perimeter and a nonlocal potential term. More precisely, their result is extended to its maximal range of validity concerning the type of nonlocal potentials considered, and is also generalized to the case where local perimeters are replaced by their nonlocal counterparts.

56 pages, 1 figure. This version features a new, short appendix which explains how to correct an error contained in the original manuscript. The authors wish to thank Marco Bonacini, Riccardo Cristoferi, and Ihsan Topaloglu for pointing out this error to them

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