November  2021, 41(11): 5439-5454. 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  November 2021 Early access  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 and Continuous Dynamical Systems, 2021, 41 (11) : 5439-5454. 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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[8]

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

[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.

[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. 

[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.

[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.

[13]

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

[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.

[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.

[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.

[17]

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

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[28]

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

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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[8]

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

[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.

[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. 

[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.

[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.

[13]

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

[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.

[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.

[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.

[17]

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

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[28]

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

[1]

Sabri Bahrouni, Hichem Ounaies. Embedding theorems in the fractional Orlicz-Sobolev space and applications to non-local problems. Discrete and Continuous Dynamical Systems, 2020, 40 (5) : 2917-2944. doi: 10.3934/dcds.2020155

[2]

Carsten Burstedde. On the numerical evaluation of fractional Sobolev norms. Communications on Pure and Applied Analysis, 2007, 6 (3) : 587-605. doi: 10.3934/cpaa.2007.6.587

[3]

Thaís Jordão, Xingping Sun. General types of spherical mean operators and $K$-functionals of fractional orders. Communications on Pure and Applied Analysis, 2015, 14 (3) : 743-757. doi: 10.3934/cpaa.2015.14.743

[4]

Massimiliano Ferrara, Giovanni Molica Bisci, Binlin Zhang. Existence of weak solutions for non-local fractional problems via Morse theory. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2483-2499. doi: 10.3934/dcdsb.2014.19.2483

[5]

Ritu Agarwal, Kritika, Sunil Dutt Purohit, Devendra Kumar. Mathematical modelling of cytosolic calcium concentration distribution using non-local fractional operator. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3387-3399. doi: 10.3934/dcdss.2021017

[6]

Umberto Biccari. Internal control for a non-local Schrödinger equation involving the fractional Laplace operator. Evolution Equations and Control Theory, 2022, 11 (1) : 301-324. doi: 10.3934/eect.2021014

[7]

Fausto Ferrari, Michele Miranda Jr, Diego Pallara, Andrea Pinamonti, Yannick Sire. Fractional Laplacians, perimeters and heat semigroups in Carnot groups. Discrete and Continuous Dynamical Systems - S, 2018, 11 (3) : 477-491. doi: 10.3934/dcdss.2018026

[8]

Giovanni Bellettini, Matteo Novaga, Shokhrukh Yusufovich Kholmatov. Minimizers of anisotropic perimeters with cylindrical norms. Communications on Pure and Applied Analysis, 2017, 16 (4) : 1427-1454. doi: 10.3934/cpaa.2017068

[9]

Entisar A.-L. Ali, G. Charlot. Local contact sub-Finslerian geometry for maximum norms in dimension 3. Mathematical Control and Related Fields, 2021, 11 (2) : 373-401. doi: 10.3934/mcrf.2020041

[10]

Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete and Continuous Dynamical Systems - S, 2021, 14 (6) : 1945-1966. doi: 10.3934/dcdss.2020469

[11]

Olusola Kolebaje, Ebenezer Bonyah, Lateef Mustapha. The first integral method for two fractional non-linear biological models. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 487-502. doi: 10.3934/dcdss.2019032

[12]

Joel Fotso Tachago, Giuliano Gargiulo, Hubert Nnang, Elvira Zappale. Multiscale homogenization of integral convex functionals in Orlicz Sobolev setting. Evolution Equations and Control Theory, 2021, 10 (2) : 297-320. doi: 10.3934/eect.2020067

[13]

Haim Brezis, Petru Mironescu. Composition in fractional Sobolev spaces. Discrete and Continuous Dynamical Systems, 2001, 7 (2) : 241-246. doi: 10.3934/dcds.2001.7.241

[14]

Qiyu Jin, Ion Grama, Quansheng Liu. Convergence theorems for the Non-Local Means filter. Inverse Problems and Imaging, 2018, 12 (4) : 853-881. doi: 10.3934/ipi.2018036

[15]

Gabriel Peyré, Sébastien Bougleux, Laurent Cohen. Non-local regularization of inverse problems. Inverse Problems and Imaging, 2011, 5 (2) : 511-530. doi: 10.3934/ipi.2011.5.511

[16]

Olivier Bonnefon, Jérôme Coville, Guillaume Legendre. Concentration phenomenon in some non-local equation. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 763-781. doi: 10.3934/dcdsb.2017037

[17]

Seckin Demirbas. Local well-posedness for 2-D Schrödinger equation on irrational tori and bounds on Sobolev norms. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1517-1530. doi: 10.3934/cpaa.2017072

[18]

Henri Berestycki, Nancy Rodríguez. A non-local bistable reaction-diffusion equation with a gap. Discrete and Continuous Dynamical Systems, 2017, 37 (2) : 685-723. doi: 10.3934/dcds.2017029

[19]

Matteo Focardi. Vector-valued obstacle problems for non-local energies. Discrete and Continuous Dynamical Systems - B, 2012, 17 (2) : 487-507. doi: 10.3934/dcdsb.2012.17.487

[20]

Chiu-Yen Kao, Yuan Lou, Wenxian Shen. Random dispersal vs. non-local dispersal. Discrete and Continuous Dynamical Systems, 2010, 26 (2) : 551-596. doi: 10.3934/dcds.2010.26.551

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (127)
  • HTML views (185)
  • Cited by (0)

Other articles
by authors

[Back to Top]