American Institute of Mathematical Sciences

June  2018, 11(3): 425-440. doi: 10.3934/dcdss.2018023

The isoperimetric problem for nonlocal perimeters

 1 Department of Statistical Sciences, University of Padova, Via Cesare Battisti 141,35121 Padova, Italy 2 Department of Mathematics, University of Pisa, Largo Bruno Pontecorvo 5,56127 Pisa, Italy

* Corresponding author

Received  May 2017 Revised  August 2017 Published  October 2017

Fund Project: The authors were supported the Fondazione CaRiPaRo Project "Nonlinear Partial Differential Equations: Asymptotic Problems and Mean-Field Games", Project PRA 2017 of the University of Pisa "Problemi di ottimizzazione e di evoluzione in ambito variazionale", the INdAM-GNAMPA project "Tecniche EDP, dinamiche e probabilistiche per lo studio di problemi asintotici".

We consider a class of nonlocal generalized perimeters which includes fractional perimeters and Riesz type potentials. We prove a general isoperimetric inequality for such functionals, and we discuss some applications. In particular we prove existence of an isoperimetric profile, under suitable assumptions on the interaction kernel.

Citation: Annalisa Cesaroni, Matteo Novaga. The isoperimetric problem for nonlocal perimeters. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 425-440. doi: 10.3934/dcdss.2018023
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