×

The isoperimetric problem for nonlocal perimeters. (English) Zbl 1379.53013

Summary: 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.

MSC:

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
49Q20 Variational problems in a geometric measure-theoretic setting
35R11 Fractional partial differential equations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] L. Ambrosio, <em>Functions of Bounded Variation and Free Discontinuity Problems</em>,, Oxford Mathematical Monographs (2000) · Zbl 0957.49001
[2] L. A. Caffarelli, Nonlocal minimal surfaces,, Comm. Pure Appl. Math., 63, 1111 (2010) · Zbl 1248.53009
[3] A. Cesaroni, Minimizers for nonlocal perimeters of Minkowski type,, Arxiv preprint (2017)
[4] A. Cesaroni, Volume constrained minimizers of the fractional perimeter with a potential energy,, Discrete Contin. Dyn. Syst. S, 10, 715 (2017) · Zbl 1366.49054
[5] A. Chambolle, Nonlocal curvature flows,, Arch. Ration. Mech. Anal., 218, 1263 (2015) · Zbl 1328.35166
[6] M. Cicalese, Ground states of a two phase model with cross and self attractive interactions,, SIAM J. Math. Anal., 48, 3412 (2016) · Zbl 1352.49009
[7] E. Cinti, Quantitative flatness results and BV-estimates for stable nonlocal minimal surfaces,, Arxiv preprint (2016)
[8] A. Di Castro, Nonlocal quantitative isoperimetric inequalities,, Calc. Var. Partial Differential Equations, 54, 2421 (2015) · Zbl 1333.49061
[9] A. Figalli, Isoperimetry and stability properties of balls with respect to nonlocal energies,, Comm. Math. Phys., 336, 441 (2015) · Zbl 1312.49051
[10] M. Goldman, Volume-constrained minimizers for the prescribed curvature problem in periodic media,, Calc. Var. Partial Differential Equations, 44, 297 (2012) · Zbl 1241.49027
[11] M. Ludwig, Anisotropic fractional perimeters,, J. Differential Geom., 96, 77 (2014) · Zbl 1291.52013
[12] F. Maggi, <em>Sets of Finite Perimeter and Geometric Variational Problems, In: An introduction to Geometric Measure Theory,</em>, Cambridge Studies in Adavanced Mathematics (2012) · Zbl 1255.49074
[13] V. Maz’ya, <em>Lectures on Isoperimetric and Isocapacitary Inequalities in the Theory of Sobolev Spaces,</em>, Contemp. Math. (2003) · Zbl 1062.31009
[14] F. Riesz, Sur une inégalité intégrale. Journ,, London Math. Soc., 5, 162 (1930) · JFM 56.0232.02
[15] A. Visintin, Generalized coarea formula and fractal sets,, Japan J. Indust. Appl. Math., 8, 175 (1991) · Zbl 0736.49030
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.