doi: 10.3934/dcds.2021083

Fractional perimeters on the sphere

1. 

Technische Universität Wien, Institut für Diskrete Mathematik und Geometrie, Wiedner Hauptstraße 8-10/1046, 1040 Vienna, Austria

2. 

Goethe-Universität Frankfurt am Main, Institut für Mathematik, Robert-Mayer-Straße 10, 60325 Frankfurt am Main, Germany

* Corresponding author: Andreas Kreuml

Received  December 2020 Revised  March 2021 Published  May 2021

Fund Project: The second author is partially funded by DFG project BE 2484/5-2

This note treats several problems for the fractional perimeter or $ s $-perimeter on the sphere. The spherical fractional isoperimetric inequality is established. It turns out that the equality cases are exactly the spherical caps. Furthermore, the convergence of fractional perimeters to the surface area as $ s \nearrow 1 $ is proven. It is shown that their limit as $ s \searrow -\infty $ can be expressed in terms of the volume.

Citation: Andreas Kreuml, Olaf Mordhorst. Fractional perimeters on the sphere. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021083
References:
[1]

E. Arbeiter and M. Zähle, Kinematic relations for Hausdorff moment measures in spherical spaces, Math. Nachr., 153 (1991), 333-348.  doi: 10.1002/mana.19911530129.  Google Scholar

[2]

W. Beckner, Sobolev inequalities, the Poisson semigroup, and analysis on the sphere $S^n$, Proc. Nat. Acad. Sci. U.S.A., 89 (1992), 4816-4819.  doi: 10.1073/pnas.89.11.4816.  Google Scholar

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J. Bourgain, H. Brézis and P. Mironescu, Another look at Sobolev spaces, in Optimal Control and Partial Differential Equations, IOS, Amsterdam, (2001), 439–455.  Google Scholar

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J. BourgainH. Brézis and P. Mironescu, Limiting embedding theorems for $W^{s, p}$ when $s\uparrow1$ and applications, J. Anal. Math., 87 (2002), 77-101.  doi: 10.1007/BF02868470.  Google Scholar

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C. BucurL. Lombardini and E. Valdinoci, Complete stickiness of nonlocal minimal surfaces for small values of the fractional parameter, Ann. Inst. H. Poincaré Anal. Non Linéaire, 36 (2019), 655-703.  doi: 10.1016/j.anihpc.2018.08.003.  Google Scholar

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E. De Giorgi, Definizione ed espressione analitica del perimetro di un insieme, Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (8), 14 (1953), 390-393.   Google Scholar

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S. DipierroA. FigalliG. Palatucci and E. Valdinoci, Asymptotics of the $s$-perimeter as $s\searrow0$, Discrete Contin. Dyn. Syst., 33 (2013), 2777-2790.  doi: 10.3934/dcds.2013.33.2777.  Google Scholar

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R. L. Frank and R. Seiringer, Non-linear ground state representations and sharp Hardy inequalities, J. Funct. Anal., 255 (2008), 3407-3430.  doi: 10.1016/j.jfa.2008.05.015.  Google Scholar

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S. Glasauer, Integralgeometrie Konvexer Körper im Sphärischen Raum, PhD Thesis, Universiät Freiburg, 1995. Google Scholar

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D. Hug and C. Thäle, Splitting tessellations in spherical spaces, Electron. J. Probab., 24 (2019), Paper No. 24, 60 pp. doi: 10.1214/19-EJP267.  Google Scholar

[15]

A. Kreuml, The anisotropic fractional isoperimetric problem with respect to unconditional unit balls, Commun. Pure Appl. Anal., 20 (2021), 783-799.  doi: 10.3934/cpaa.2020290.  Google Scholar

[16]

A. Kreuml and O. Mordhorst, Fractional Sobolev norms and BV functions on manifolds, Nonlinear Anal., 187 (2019), 450-466.  doi: 10.1016/j.na.2019.06.014.  Google Scholar

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M. Ludwig, Anisotropic fractional perimeters, J. Differential Geom., 96 (2014), 77–93, URL http://projecteuclid.org/euclid.jdg/1391192693.  Google Scholar

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M. Ludwig, Anisotropic fractional Sobolev norms, Adv. Math., 252 (2014), 150-157.  doi: 10.1016/j.aim.2013.10.024.  Google Scholar

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E. Lutwak, The Brunn-Minkowski-Firey theory. Ⅱ. Affine and geominimal surface areas, Adv. Math., 118 (1996), 244-294.  doi: 10.1006/aima.1996.0022.  Google Scholar

[20]

E. Lutwak, D. Yang and G. Zhang, $L_p$ affine isoperimetric inequalities, J. Differential Geom., 56 (2000), 111–132, URL http://projecteuclid.org/euclid.jdg/1090347527.  Google Scholar

[21]

E. Lutwak, D. Yang and G. Zhang, Sharp affine $L_p$ Sobolev inequalities, J. Differential Geom., 62 (2002), 17–38, URL http://projecteuclid.org/euclid.jdg/1090425527.  Google Scholar

[22]

F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2012, URL https://books.google.at/books?id=Qc2LO2PD10gC. doi: 10.1017/CBO9781139108133.  Google Scholar

[23]

V. Maz'ya and T. Shaposhnikova, On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal., 195 (2002), 230-238.  doi: 10.1006/jfan.2002.3955.  Google Scholar

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R. Schneider and W. Weil, Stochastic and Integral Geometry, Probability and its Applications (New York), Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-78859-1.  Google Scholar

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V. N. Sudakov and B. S. Cirel'son, Extremal properties of half-spaces for spherically invariant measures, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 41 (1974), 14–24,165, Problems in the theory of probability distributions, Ⅱ.  Google Scholar

[26]

E. Werner, The $p$-affine surface area and geometric interpretations, in IV International Conference in "Stochastic Geometry, Convex Bodies, Empirical Measures & Applications to Engineering Science", Vol. II (eds. R. Schneider and M. I. Stoka), Circ. Mat. Palermo, 70 (2002), 367–382.  Google Scholar

[27]

E. Werner and D. Ye, New $L_p$ affine isoperimetric inequalities, Adv. Math., 218 (2008), 762-780.  doi: 10.1016/j.aim.2008.02.002.  Google Scholar

[28]

J. A. Wieacker, Translative Poincaré formulae for Hausdorff rectifiable sets, Geom. Dedicata, 16 (1984), 231-248.  doi: 10.1007/BF00146833.  Google Scholar

show all references

References:
[1]

E. Arbeiter and M. Zähle, Kinematic relations for Hausdorff moment measures in spherical spaces, Math. Nachr., 153 (1991), 333-348.  doi: 10.1002/mana.19911530129.  Google Scholar

[2]

W. Beckner, Sobolev inequalities, the Poisson semigroup, and analysis on the sphere $S^n$, Proc. Nat. Acad. Sci. U.S.A., 89 (1992), 4816-4819.  doi: 10.1073/pnas.89.11.4816.  Google Scholar

[3]

F. Besau and E. M. Werner, The spherical convex floating body, Adv. Math., 301 (2016), 867-901.  doi: 10.1016/j.aim.2016.07.001.  Google Scholar

[4]

C. Borell, The Brunn-Minkowski inequality in Gauss space, Invent. Math., 30 (1975), 207-216.  doi: 10.1007/BF01425510.  Google Scholar

[5]

J. Bourgain, H. Brézis and P. Mironescu, Another look at Sobolev spaces, in Optimal Control and Partial Differential Equations, IOS, Amsterdam, (2001), 439–455.  Google Scholar

[6]

J. BourgainH. Brézis and P. Mironescu, Limiting embedding theorems for $W^{s, p}$ when $s\uparrow1$ and applications, J. Anal. Math., 87 (2002), 77-101.  doi: 10.1007/BF02868470.  Google Scholar

[7]

C. BucurL. Lombardini and E. Valdinoci, Complete stickiness of nonlocal minimal surfaces for small values of the fractional parameter, Ann. Inst. H. Poincaré Anal. Non Linéaire, 36 (2019), 655-703.  doi: 10.1016/j.anihpc.2018.08.003.  Google Scholar

[8]

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

[9]

J. Dávila, On an open question about functions of bounded variation, Calc. Var. Partial Differential Equations, 15 (2002), 519-527.  doi: 10.1007/s005260100135.  Google Scholar

[10]

E. De Giorgi, Definizione ed espressione analitica del perimetro di un insieme, Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (8), 14 (1953), 390-393.   Google Scholar

[11]

S. DipierroA. FigalliG. Palatucci and E. Valdinoci, Asymptotics of the $s$-perimeter as $s\searrow0$, Discrete Contin. Dyn. Syst., 33 (2013), 2777-2790.  doi: 10.3934/dcds.2013.33.2777.  Google Scholar

[12]

R. L. Frank and R. Seiringer, Non-linear ground state representations and sharp Hardy inequalities, J. Funct. Anal., 255 (2008), 3407-3430.  doi: 10.1016/j.jfa.2008.05.015.  Google Scholar

[13]

S. Glasauer, Integralgeometrie Konvexer Körper im Sphärischen Raum, PhD Thesis, Universiät Freiburg, 1995. Google Scholar

[14]

D. Hug and C. Thäle, Splitting tessellations in spherical spaces, Electron. J. Probab., 24 (2019), Paper No. 24, 60 pp. doi: 10.1214/19-EJP267.  Google Scholar

[15]

A. Kreuml, The anisotropic fractional isoperimetric problem with respect to unconditional unit balls, Commun. Pure Appl. Anal., 20 (2021), 783-799.  doi: 10.3934/cpaa.2020290.  Google Scholar

[16]

A. Kreuml and O. Mordhorst, Fractional Sobolev norms and BV functions on manifolds, Nonlinear Anal., 187 (2019), 450-466.  doi: 10.1016/j.na.2019.06.014.  Google Scholar

[17]

M. Ludwig, Anisotropic fractional perimeters, J. Differential Geom., 96 (2014), 77–93, URL http://projecteuclid.org/euclid.jdg/1391192693.  Google Scholar

[18]

M. Ludwig, Anisotropic fractional Sobolev norms, Adv. Math., 252 (2014), 150-157.  doi: 10.1016/j.aim.2013.10.024.  Google Scholar

[19]

E. Lutwak, The Brunn-Minkowski-Firey theory. Ⅱ. Affine and geominimal surface areas, Adv. Math., 118 (1996), 244-294.  doi: 10.1006/aima.1996.0022.  Google Scholar

[20]

E. Lutwak, D. Yang and G. Zhang, $L_p$ affine isoperimetric inequalities, J. Differential Geom., 56 (2000), 111–132, URL http://projecteuclid.org/euclid.jdg/1090347527.  Google Scholar

[21]

E. Lutwak, D. Yang and G. Zhang, Sharp affine $L_p$ Sobolev inequalities, J. Differential Geom., 62 (2002), 17–38, URL http://projecteuclid.org/euclid.jdg/1090425527.  Google Scholar

[22]

F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2012, URL https://books.google.at/books?id=Qc2LO2PD10gC. doi: 10.1017/CBO9781139108133.  Google Scholar

[23]

V. Maz'ya and T. Shaposhnikova, On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal., 195 (2002), 230-238.  doi: 10.1006/jfan.2002.3955.  Google Scholar

[24]

R. Schneider and W. Weil, Stochastic and Integral Geometry, Probability and its Applications (New York), Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-78859-1.  Google Scholar

[25]

V. N. Sudakov and B. S. Cirel'son, Extremal properties of half-spaces for spherically invariant measures, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 41 (1974), 14–24,165, Problems in the theory of probability distributions, Ⅱ.  Google Scholar

[26]

E. Werner, The $p$-affine surface area and geometric interpretations, in IV International Conference in "Stochastic Geometry, Convex Bodies, Empirical Measures & Applications to Engineering Science", Vol. II (eds. R. Schneider and M. I. Stoka), Circ. Mat. Palermo, 70 (2002), 367–382.  Google Scholar

[27]

E. Werner and D. Ye, New $L_p$ affine isoperimetric inequalities, Adv. Math., 218 (2008), 762-780.  doi: 10.1016/j.aim.2008.02.002.  Google Scholar

[28]

J. A. Wieacker, Translative Poincaré formulae for Hausdorff rectifiable sets, Geom. Dedicata, 16 (1984), 231-248.  doi: 10.1007/BF00146833.  Google Scholar

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