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Journal Articles Communications in Mathematical Physics Year : 2014

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 nonlocal 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 [25, 26] 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.
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Dates and versions

hal-00955169 , version 1 (04-03-2014)

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Alessio Figalli, Nicola Fusco, Francesco Maggi, Vincent Millot, Massimiliano Morini. Isoperimetry and stability properties of balls with respect to nonlocal energies. Communications in Mathematical Physics, 2014, 336 (1), ⟨10.1007/s00220-014-2244-1⟩. ⟨hal-00955169⟩
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