Nonlocal quantitative isoperimetric inequalities

Agnese Di Castro, Matteo Novaga, Berardo Ruffini, Enrico Valdinoci

Research output: Contribution to journalArticle

18 Citations (Scopus)

Abstract

We show a quantitative-type isoperimetric inequality for fractional perimeters where the deficit of the t-perimeter, up to multiplicative constants, controls from above that of the s-perimeter, with s smaller than t. To do this we consider a problem of independent interest: we characterize the volume-constrained minimizers of a nonlocal free energy given by the difference of the t-perimeter and the s-perimeter. In particular, we show that balls are the unique minimizers if the volume is sufficiently small, depending on t-s, while the existence vs. nonexistence of minimizers for large volumes remains open. We also consider the corresponding isoperimetric problem and prove existence and regularity of minimizers for all s,t. When s=0 this problem reduces to the fractional isoperimetric problem, for which it is well known that balls are the only minimizers.

Original languageEnglish
Pages (from-to)2421-2464
Number of pages44
JournalCalculus of Variations and Partial Differential Equations
Volume54
Issue number3
DOIs
Publication statusPublished - 1 Nov 2015
Externally publishedYes

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