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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

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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

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