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

The isoperimetric problem for nonlocal perimeters

Annalisa Cesaroni 1, and  Matteo Novaga 2,,

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.

Keywords: Fractional perimeter, nonlocal isoperimetric inequality, Poincaré inequality.
Mathematics Subject Classification: 53A10, 49Q20, 35R11.
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
References:
[1]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems Oxford Mathematical Monographs, 2000.  Google Scholar

[2]

L. A. Caffarelli, J.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math., 63 (2010), 1111-1144.  doi: 10.1002/cpa.20331.  Google Scholar

[3]

A. Cesaroni, S. Dipierro, M. Novaga and E. Valdinoci, Minimizers for nonlocal perimeters of Minkowski type, Arxiv preprint, 2017, arXiv: 1704.03195. Google Scholar

[4]

A. Cesaroni and M. Novaga, Volume constrained minimizers of the fractional perimeter with a potential energy, Discrete Contin. Dyn. Syst. S, 10 (2017), 715-727.  doi: 10.3934/dcdss.2017036.  Google Scholar

[5]

A. Chambolle, M. Morini and M. Ponsiglione, Nonlocal curvature flows, Arch. Ration. Mech. Anal., 218 (2015), 1263-1329.  doi: 10.1007/s00205-015-0880-z.  Google Scholar

[6]

M. Cicalese, L. De Luca, M. Novaga and M. Ponsiglione, Ground states of a two phase model with cross and self attractive interactions, SIAM J. Math. Anal., 48 (2016), 3412-3443.  doi: 10.1137/15M1033976.  Google Scholar

[7]

E. Cinti, J. Serra and E. Valdinoci, Quantitative flatness results and BV-estimates for stable nonlocal minimal surfaces, Arxiv preprint, 2016, arXiv: 1602.00540. Google Scholar

[8]

A. Di Castro, M. Novaga, B. Ruffini and E. Valdinoci, Nonlocal quantitative isoperimetric inequalities, Calc. Var. Partial Differential Equations, 54 (2015), 2421-2464.  doi: 10.1007/s00526-015-0870-x.  Google Scholar

[9]

A. Figalli, N. Fusco, F. Maggi, V. Millot and M. Morini, Isoperimetry and stability properties of balls with respect to nonlocal energies, Comm. Math. Phys., 336 (2015), 441-507.  doi: 10.1007/s00220-014-2244-1.  Google Scholar

[10]

M. Goldman and M. Novaga, Volume-constrained minimizers for the prescribed curvature problem in periodic media, Calc. Var. Partial Differential Equations, 44 (2012), 297-318.  doi: 10.1007/s00526-011-0435-6.  Google Scholar

[11]

M. Ludwig, Anisotropic fractional perimeters, J. Differential Geom., 96 (2014), 77-93.  doi: 10.4310/jdg/1391192693.  Google Scholar

[12]

F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems, In: An introduction to Geometric Measure Theory, Cambridge Studies in Adavanced Mathematics, vol. 135, Cambridge University Press, Cambridge, 2012. doi: 10.1017/CBO9781139108133.  Google Scholar

[13]

V. Maz'ya, Lectures on Isoperimetric and Isocapacitary Inequalities in the Theory of Sobolev Spaces, Contemp. Math., 338, Amer. Math. Soc., Providence, RI, 2003. Google Scholar

[14]

F. Riesz, Sur une inégalité intégrale. Journ, London Math. Soc., 5 (1930), 162-168.  doi: 10.1112/jlms/s1-5.3.162.  Google Scholar

[15]

A. Visintin, Generalized coarea formula and fractal sets, Japan J. Indust. Appl. Math., 8 (1991), 175-201.  doi: 10.1007/BF03167679.  Google Scholar

show all references

References:
[1]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems Oxford Mathematical Monographs, 2000.  Google Scholar

[2]

L. A. Caffarelli, J.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math., 63 (2010), 1111-1144.  doi: 10.1002/cpa.20331.  Google Scholar

[3]

A. Cesaroni, S. Dipierro, M. Novaga and E. Valdinoci, Minimizers for nonlocal perimeters of Minkowski type, Arxiv preprint, 2017, arXiv: 1704.03195. Google Scholar

[4]

A. Cesaroni and M. Novaga, Volume constrained minimizers of the fractional perimeter with a potential energy, Discrete Contin. Dyn. Syst. S, 10 (2017), 715-727.  doi: 10.3934/dcdss.2017036.  Google Scholar

[5]

A. Chambolle, M. Morini and M. Ponsiglione, Nonlocal curvature flows, Arch. Ration. Mech. Anal., 218 (2015), 1263-1329.  doi: 10.1007/s00205-015-0880-z.  Google Scholar

[6]

M. Cicalese, L. De Luca, M. Novaga and M. Ponsiglione, Ground states of a two phase model with cross and self attractive interactions, SIAM J. Math. Anal., 48 (2016), 3412-3443.  doi: 10.1137/15M1033976.  Google Scholar

[7]

E. Cinti, J. Serra and E. Valdinoci, Quantitative flatness results and BV-estimates for stable nonlocal minimal surfaces, Arxiv preprint, 2016, arXiv: 1602.00540. Google Scholar

[8]

A. Di Castro, M. Novaga, B. Ruffini and E. Valdinoci, Nonlocal quantitative isoperimetric inequalities, Calc. Var. Partial Differential Equations, 54 (2015), 2421-2464.  doi: 10.1007/s00526-015-0870-x.  Google Scholar

[9]

A. Figalli, N. Fusco, F. Maggi, V. Millot and M. Morini, Isoperimetry and stability properties of balls with respect to nonlocal energies, Comm. Math. Phys., 336 (2015), 441-507.  doi: 10.1007/s00220-014-2244-1.  Google Scholar

[10]

M. Goldman and M. Novaga, Volume-constrained minimizers for the prescribed curvature problem in periodic media, Calc. Var. Partial Differential Equations, 44 (2012), 297-318.  doi: 10.1007/s00526-011-0435-6.  Google Scholar

[11]

M. Ludwig, Anisotropic fractional perimeters, J. Differential Geom., 96 (2014), 77-93.  doi: 10.4310/jdg/1391192693.  Google Scholar

[12]

F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems, In: An introduction to Geometric Measure Theory, Cambridge Studies in Adavanced Mathematics, vol. 135, Cambridge University Press, Cambridge, 2012. doi: 10.1017/CBO9781139108133.  Google Scholar

[13]

V. Maz'ya, Lectures on Isoperimetric and Isocapacitary Inequalities in the Theory of Sobolev Spaces, Contemp. Math., 338, Amer. Math. Soc., Providence, RI, 2003. Google Scholar

[14]

F. Riesz, Sur une inégalité intégrale. Journ, London Math. Soc., 5 (1930), 162-168.  doi: 10.1112/jlms/s1-5.3.162.  Google Scholar

[15]

A. Visintin, Generalized coarea formula and fractal sets, Japan J. Indust. Appl. Math., 8 (1991), 175-201.  doi: 10.1007/BF03167679.  Google Scholar

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