Calculus of Variations and Geometric Measure Theory

A. Figalli - N. Fusco - F. Maggi - V. Millot - M. Morini

Isoperimetry and stability properties of balls with respect to nonlocal energies

created by maggi on 03 Mar 2014
modified by figalli on 28 Nov 2014


Accepted Paper

Inserted: 3 mar 2014
Last Updated: 28 nov 2014

Journal: Comm. Math. Phys.
Year: 2014


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 Knu ̈pfer 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.