Isoperimetry and stability properties of balls with respect to nonlocal energies. (English) Zbl 1312.49051
Summary: 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 H. Knüpfer and C. B. Muratov [Commun. Pure Appl. Math. 66, No. 7, 1129–1162 (2013; Zbl 1269.49087); Commun. Pure Appl. Math. 67, No. 12, 1974–1994 (2014; Zbl 1302.49064)] 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.
MSC:
| 49Q10 | Optimization of shapes other than minimal surfaces |
| 49Q20 | Variational problems in a geometric measure-theoretic setting |
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