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Global compactness results for nonlocal problems
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 |
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.
References:
[1] |
L. Ambrosio, N. Fusco and D. Pallara,
Functions of Bounded Variation and Free Discontinuity Problems Oxford Mathematical Monographs, 2000. |
[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. |
[3] |
A. Cesaroni, S. Dipierro, M. Novaga and E. Valdinoci, Minimizers for nonlocal perimeters of Minkowski type, Arxiv preprint, 2017, arXiv: 1704.03195. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[11] |
M. Ludwig,
Anisotropic fractional perimeters, J. Differential Geom., 96 (2014), 77-93.
doi: 10.4310/jdg/1391192693. |
[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. |
[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. |
[14] |
F. Riesz,
Sur une inégalité intégrale. Journ, London Math. Soc., 5 (1930), 162-168.
doi: 10.1112/jlms/s1-5.3.162. |
[15] |
A. Visintin,
Generalized coarea formula and fractal sets, Japan J. Indust. Appl. Math., 8 (1991), 175-201.
doi: 10.1007/BF03167679. |
show all references
References:
[1] |
L. Ambrosio, N. Fusco and D. Pallara,
Functions of Bounded Variation and Free Discontinuity Problems Oxford Mathematical Monographs, 2000. |
[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. |
[3] |
A. Cesaroni, S. Dipierro, M. Novaga and E. Valdinoci, Minimizers for nonlocal perimeters of Minkowski type, Arxiv preprint, 2017, arXiv: 1704.03195. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[11] |
M. Ludwig,
Anisotropic fractional perimeters, J. Differential Geom., 96 (2014), 77-93.
doi: 10.4310/jdg/1391192693. |
[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. |
[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. |
[14] |
F. Riesz,
Sur une inégalité intégrale. Journ, London Math. Soc., 5 (1930), 162-168.
doi: 10.1112/jlms/s1-5.3.162. |
[15] |
A. Visintin,
Generalized coarea formula and fractal sets, Japan J. Indust. Appl. Math., 8 (1991), 175-201.
doi: 10.1007/BF03167679. |
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