American Institute of Mathematical Sciences

Advanced
July  2013, 33(7): 2777-2790. doi: 10.3934/dcds.2013.33.2777

Asymptotics of the $s$-perimeter as $s\searrow 0$

Serena Dipierro 1, , Alessio Figalli 2, , Giampiero Palatucci 3, and  Enrico Valdinoci 4,

1. 

SISSA - International School for Advanced Studies, Sector of Mathematical Analysis Via Bonomea, 265, 34136 Trieste, Italy
2. 

University of Texas at Austin, Department of Mathematics, 2515 Speedway Stop C1200, Austin, TX 78712-1202
3. 

Università degli Studi di Parma, Dipartimento di Matematica Campus - Parco Area delle Scienze, 53/A, 43124 Parma, Italy
4. 

Università degli Studi di Milano, Dipartimento di Matematica Via Saldini, 50, 20133 Milano

Received  April 2012 Revised  August 2012 Published  January 2013

We deal with the asymptotic behavior of the $s$-perimeter of a set $E$ inside a domain $\Omega$ as $s\searrow0$. We prove necessary and sufficient conditions for the existence of such limit, by also providing an explicit formulation in terms of the Lebesgue measure of $E$ and $\Omega$. Moreover, we construct examples of sets for which the limit does not exist.
Keywords: Nonlinear problems, minimal surfaces., fractional Laplacian, fractional Sobolev spaces, nonlocal perimeter.
Mathematics Subject Classification: 49Q20, 49Q05, 28A75, 35S05, 60G2.
Citation: Serena Dipierro, Alessio Figalli, Giampiero Palatucci, Enrico Valdinoci. Asymptotics of the $s$-perimeter as $s\searrow 0$. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2777-2790. doi: 10.3934/dcds.2013.33.2777
References:
[1]

L. Ambrosio, G. De Philippis and L. Martinazzi, $\Gamma$-convergence of nonlocal perimeter functionals,, Manuscripta Math., 134 (2011), 377.  doi: 10.1007/s00229-010-0399-4.  Google Scholar

[2]

B. Barrios Barrera, A. Figalli and E. Valdinoci, Bootstrap regularity for integro-differential operators and its application to nonlocal minimal surfaces,, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), ().   Google Scholar

[3]

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

[4]

L. Caffarelli and P. E. Souganidis, Convergence of nonlocal threshold dynamics approximations to front propagation,, Arch. Ration. Mech. Anal., 195 (2010), 1.  doi: 10.1007/s00205-008-0181-x.  Google Scholar

[5]

L. Caffarelli and E. Valdinoci, Uniform estimates and limiting arguments for nonlocal minimal surfaces,, Calc. Var. Partial Differential Equations, 41 (2011), 203.  doi: 10.1007/s00526-010-0359-6.  Google Scholar

[6]

L. Caffarelli and E. Valdinoci, Regularity properties of nonlocal minimal surfaces via limiting arguments,, Preprint, (): 11.   Google Scholar

[7]

M. C. Caputo and N. Guillen, Regularity for non-local almost minimal boundaries and applications,, Preprint, ().   Google Scholar

[8]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces,, Bull. Sci. math., 136 (2012), 521.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[9]

G. Franzina and E. Valdinoci, Geometric analysis of fractional phase transition interfaces,, in, ().   Google Scholar

[10]

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.  doi: 10.1006/jfan.2002.3955.  Google Scholar

[11]

O. Savin and E. Valdinoci, Density estimates for a variational model driven by the Gagliardo norm,, Preprint, ().   Google Scholar

[12]

O. Savin and E. Valdinoci, $\Gamma$-convergence for nonlocal phase transitions,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 479.  doi: 10.1016/j.anihpc.2012.01.006.  Google Scholar

[13]

O. Savin and E. Valdinoci, Regularity of nonlocal minimal cones in dimension $2$,, Calc. Var. Partial Differential Equations, ().  doi: 10.1007/s00526-012-0539-7.  Google Scholar

[14]

A. Visintin, Nonconvex functionals related to multiphase systems,, SIAM J. Math. Anal., 21 (1990), 1281.  doi: 10.1137/0521071.  Google Scholar

[15]

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

show all references

References:
[1]

L. Ambrosio, G. De Philippis and L. Martinazzi, $\Gamma$-convergence of nonlocal perimeter functionals,, Manuscripta Math., 134 (2011), 377.  doi: 10.1007/s00229-010-0399-4.  Google Scholar

[2]

B. Barrios Barrera, A. Figalli and E. Valdinoci, Bootstrap regularity for integro-differential operators and its application to nonlocal minimal surfaces,, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), ().   Google Scholar

[3]

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

[4]

L. Caffarelli and P. E. Souganidis, Convergence of nonlocal threshold dynamics approximations to front propagation,, Arch. Ration. Mech. Anal., 195 (2010), 1.  doi: 10.1007/s00205-008-0181-x.  Google Scholar

[5]

L. Caffarelli and E. Valdinoci, Uniform estimates and limiting arguments for nonlocal minimal surfaces,, Calc. Var. Partial Differential Equations, 41 (2011), 203.  doi: 10.1007/s00526-010-0359-6.  Google Scholar

[6]

L. Caffarelli and E. Valdinoci, Regularity properties of nonlocal minimal surfaces via limiting arguments,, Preprint, (): 11.   Google Scholar

[7]

M. C. Caputo and N. Guillen, Regularity for non-local almost minimal boundaries and applications,, Preprint, ().   Google Scholar

[8]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces,, Bull. Sci. math., 136 (2012), 521.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[9]

G. Franzina and E. Valdinoci, Geometric analysis of fractional phase transition interfaces,, in, ().   Google Scholar

[10]

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.  doi: 10.1006/jfan.2002.3955.  Google Scholar

[11]

O. Savin and E. Valdinoci, Density estimates for a variational model driven by the Gagliardo norm,, Preprint, ().   Google Scholar

[12]

O. Savin and E. Valdinoci, $\Gamma$-convergence for nonlocal phase transitions,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 479.  doi: 10.1016/j.anihpc.2012.01.006.  Google Scholar

[13]

O. Savin and E. Valdinoci, Regularity of nonlocal minimal cones in dimension $2$,, Calc. Var. Partial Differential Equations, ().  doi: 10.1007/s00526-012-0539-7.  Google Scholar

[14]

A. Visintin, Nonconvex functionals related to multiphase systems,, SIAM J. Math. Anal., 21 (1990), 1281.  doi: 10.1137/0521071.  Google Scholar

[15]

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

[1]

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

Anouar Bahrouni, VicenŢiu D. RĂdulescu. On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 379-389. doi: 10.3934/dcdss.2018021
[3]

Yutong Chen, Jiabao Su. Resonant problems for fractional Laplacian. Communications on Pure & Applied Analysis, 2017, 16 (1) : 163-188. doi: 10.3934/cpaa.2017008
[4]

Younghun Hong, Yannick Sire. On Fractional Schrödinger Equations in sobolev spaces. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2265-2282. doi: 10.3934/cpaa.2015.14.2265
[5]

Stathis Filippas, Luisa Moschini, Achilles Tertikas. Trace Hardy--Sobolev--Maz'ya inequalities for the half fractional Laplacian. Communications on Pure & Applied Analysis, 2015, 14 (2) : 373-382. doi: 10.3934/cpaa.2015.14.373
[6]

Jinrong Wang, Michal Fečkan, Yong Zhou. Approximate controllability of Sobolev type fractional evolution systems with nonlocal conditions. Evolution Equations & Control Theory, 2017, 6 (3) : 471-486. doi: 10.3934/eect.2017024
[7]

Annalisa Cesaroni, Matteo Novaga. Volume constrained minimizers of the fractional perimeter with a potential energy. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 715-727. doi: 10.3934/dcdss.2017036
[8]

John R. Graef, Lingju Kong, Qingkai Kong, Min Wang. Positive solutions of nonlocal fractional boundary value problems. Conference Publications, 2013, 2013 (special) : 283-290. doi: 10.3934/proc.2013.2013.283
[9]

Irene Benedetti, Valeri Obukhovskii, Valentina Taddei. Evolution fractional differential problems with impulses and nonlocal conditions. Discrete & Continuous Dynamical Systems - S, 2020, 13 (7) : 1899-1919. doi: 10.3934/dcdss.2020149
[10]

Patrizia Pucci, Mingqi Xiang, Binlin Zhang. A diffusion problem of Kirchhoff type involving the nonlocal fractional p-Laplacian. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 4035-4051. doi: 10.3934/dcds.2017171
[11]

Siwei Duo, Hong Wang, Yanzhi Zhang. A comparative study on nonlocal diffusion operators related to the fractional Laplacian. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 231-256. doi: 10.3934/dcdsb.2018110
[12]

Ran Zhuo, Wenxiong Chen, Xuewei Cui, Zixia Yuan. Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1125-1141. doi: 10.3934/dcds.2016.36.1125
[13]

Dengfeng Lü, Shuangjie Peng. On the positive vector solutions for nonlinear fractional Laplacian systems with linear coupling. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3327-3352. doi: 10.3934/dcds.2017141
[14]

Chenchen Mou. Nonlinear elliptic systems involving the fractional Laplacian in the unit ball and on a half space. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2335-2362. doi: 10.3934/cpaa.2015.14.2335
[15]

Tadeusz Kulczycki, Robert Stańczy. Multiple solutions for Dirichlet nonlinear BVPs involving fractional Laplacian. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2581-2591. doi: 10.3934/dcdsb.2014.19.2581
[16]

Miaomiao Cai, Li Ma. Moving planes for nonlinear fractional Laplacian equation with negative powers. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4603-4615. doi: 10.3934/dcds.2018201
[17]

Juan-Luis Vázquez. Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 857-885. doi: 10.3934/dcdss.2014.7.857
[18]

Carsten Burstedde. On the numerical evaluation of fractional Sobolev norms. Communications on Pure & Applied Analysis, 2007, 6 (3) : 587-605. doi: 10.3934/cpaa.2007.6.587
[19]

Irene Benedetti, Luisa Malaguti, Valentina Taddei. Nonlocal problems in Hilbert spaces. Conference Publications, 2015, 2015 (special) : 103-111. doi: 10.3934/proc.2015.0103
[20]

Maoding Zhen, Jinchun He, Haoyun Xu. Critical system involving fractional Laplacian. Communications on Pure & Applied Analysis, 2019, 18 (1) : 237-253. doi: 10.3934/cpaa.2019013

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (43)
  • HTML views (0)
  • Cited by (24)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences