Asymptotics of the s-perimeter as s\searrow 0

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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

Abstract
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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.
[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), ().
[3]	L. Caffarelli, J.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces,, Comm. Pure Appl. Math., 63 (2010), 1111. doi: 10.1002/cpa.20331.
[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.
[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.
[6]	L. Caffarelli and E. Valdinoci, Regularity properties of nonlocal minimal surfaces via limiting arguments,, Preprint, (): 11.
[7]	M. C. Caputo and N. Guillen, Regularity for non-local almost minimal boundaries and applications,, Preprint, ().
[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.
[9]	G. Franzina and E. Valdinoci, Geometric analysis of fractional phase transition interfaces,, in, ().
[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.
[11]	O. Savin and E. Valdinoci, Density estimates for a variational model driven by the Gagliardo norm,, Preprint, ().
[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.
[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.
[14]	A. Visintin, Nonconvex functionals related to multiphase systems,, SIAM J. Math. Anal., 21 (1990), 1281. doi: 10.1137/0521071.
[15]	A. Visintin, Generalized coarea formula and fractal sets,, Japan J. Industrial Appl. Math., 8 (1991), 175. doi: 10.1007/BF03167679.

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.
[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), ().
[3]	L. Caffarelli, J.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces,, Comm. Pure Appl. Math., 63 (2010), 1111. doi: 10.1002/cpa.20331.
[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.
[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.
[6]	L. Caffarelli and E. Valdinoci, Regularity properties of nonlocal minimal surfaces via limiting arguments,, Preprint, (): 11.
[7]	M. C. Caputo and N. Guillen, Regularity for non-local almost minimal boundaries and applications,, Preprint, ().
[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.
[9]	G. Franzina and E. Valdinoci, Geometric analysis of fractional phase transition interfaces,, in, ().
[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.
[11]	O. Savin and E. Valdinoci, Density estimates for a variational model driven by the Gagliardo norm,, Preprint, ().
[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.
[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.
[14]	A. Visintin, Nonconvex functionals related to multiphase systems,, SIAM J. Math. Anal., 21 (1990), 1281. doi: 10.1137/0521071.
[15]	A. Visintin, Generalized coarea formula and fractal sets,, Japan J. Industrial Appl. Math., 8 (1991), 175. doi: 10.1007/BF03167679.

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