On the variation of the fractional mean curvature under the effect of $C^{1, \alpha}$ perturbations

Matteo  Cozzi

doi:10.3934/dcds.2015.35.5769

Article Contents

2015, Volume 35, Issue 12: 5769-5786. Doi: 10.3934/dcds.2015.35.5769

This issue Previous Article Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains Next Article Short-time existence of the second order renormalization group flow in dimension three

On the variation of the fractional mean curvature under the effect of $C^{1, \alpha}$ perturbations

Matteo Cozzi^1,

1.
Dipartimento di Matematica "Federigo Enriques", Università degli Studi di Milano, Via Saldini 50, I-20133 Milano

Received: February 28, 2014

Published: November 2015

Abstract Related Papers Cited by

Abstract

In this brief note we study how the fractional mean curvature of order $s \in (0, 1)$ varies with respect to $C^{1, \alpha}$ diffeomorphisms. We prove that, if $\alpha > s$, then the variation under a $C^{1, \alpha}$ diffeomorphism $\Psi$ of the $s$-mean curvature of a set $E$ is controlled by the $C^{0, \alpha}$ norm of the Jacobian of $\Psi$. When $\alpha = 1$ we discuss the stability of these estimates as $s \rightarrow 1^-$ and comment on the consistency of our result with the classical framework.

Keywords:

Mathematics Subject Classification: Primary: 35R11, 28A75, 49Q05; Secondary: 26B10.

Citation:

Related Papers

Cited by

References

[1]	L. Ambrosio, G. De Philippis and L. Martinazzi, Gamma-convergence of nonlocal perimeter functionals, Manuscripta Math., 134 (2011), 377-403.doi: 10.1007/s00229-010-0399-4.
[2]	N. Abatangelo and E. Valdinoci, A notion of nonlocal curvature, Numer. Funct. Anal. Optim., 35 (2014), 793-815.doi: 10.1080/01630563.2014.901837.
[3]	G. Albanese, A. Fiscella and E. Valdinoci, Gevrey regularity for integro-differential operators, J. Math. Anal. Appl., 428 (2015), 1225-1238.doi: 10.1016/j.jmaa.2015.04.002.
[4]	G. Alberti and G. Bellettini, A nonlocal anisotropic model for phase transitions. Part I: The optimal profile problem, Math. Ann., 310 (1998), 527-560.doi: 10.1007/s002080050159.
[5]	G. Alberti and G. Bellettini, A non-local anisotropic model for phase transitions: Asymptotic behaviour of rescaled energies, European J. Appl. Math., 9 (1998), 261-284.doi: 10.1017/S0956792598003453.
[6]	B. Barrios, 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), 13 (2014), 609-639.
[7]	X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53.doi: 10.1016/j.anihpc.2013.02.001.
[8]	X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians II: Existence, uniqueness, and qualitative properties of solutions, Trans. Amer. Math. Soc., 367 (2015), 911-941.doi: 10.1090/S0002-9947-2014-05906-0.
[9]	L. Caffarelli, J.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math., 63 (2010), 1111-1144.doi: 10.1002/cpa.20331.
[10]	L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.doi: 10.1080/03605300600987306.
[11]	L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62 (2009), 597-638.doi: 10.1002/cpa.20274.
[12]	L. Caffarelli and L. Silvestre, Regularity results for nonlocal equations by approximation, Arch. Ration. Mech. Anal., 200 (2011), 59-88.doi: 10.1007/s00205-010-0336-4.
[13]	L. Caffarelli and E. Valdinoci, Uniform estimates and limiting arguments for nonlocal minimal surfaces, Calc. Var. Partial Differential Equations, 41 (2011), 203-240.doi: 10.1007/s00526-010-0359-6.
[14]	L. Caffarelli and E. Valdinoci, Regularity properties of nonlocal minimal surfaces via limiting arguments, Adv. Math., 248 (2013), 843-871.doi: 10.1016/j.aim.2013.08.007.
[15]	E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.doi: 10.1016/j.bulsci.2011.12.004.
[16]	S. Dipierro, A. Figalli, G. Palatucci and E. Valdinoci, Asymptotics of the s-perimeter as s ↘ 0, Discrete Contin. Dyn. Syst., 33 (2013), 2777-2790.doi: 10.3934/dcds.2013.33.2777.
[17]	A. Figalli and E. Valdinoci, Regularity and Bernstein-type results for nonlocal minimal surfaces, J. Reine Angew. Math., published online, (2015).doi: 10.1515/crelle-2015-0006.
[18]	N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, Berlin, 1972.
[19]	M. Ludwig, Anisotropic fractional perimeters, J. Differential Geom., 96 (2014), 77-93.
[20]	L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Ration. Mech. Anal., 98 (1987), 123-142.doi: 10.1007/BF00251230.
[21]	L. Modica and S. Mortola, Un esempio di $\Gamma$-convergenza, Boll. Un. Mat. Ital. B (5), 14 (1977), 285-299.
[22]	G. Palatucci, O. Savin and E. Valdinoci, Local and global minimizers for a variational energy involving a fractional norm, Ann. Mat. Pura Appl., 192 (2013), 673-718.doi: 10.1007/s10231-011-0243-9.
[23]	O. Savin and E. Valdinoci, $\Gamma$-convergence for nonlocal phase transitions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 479-500.doi: 10.1016/j.anihpc.2012.01.006.
[24]	O. Savin and E. Valdinoci, Regularity of nonlocal minimal cones in dimension 2, Calc. Var. Partial Differential Equations, 48 (2013), 33-39.doi: 10.1007/s00526-012-0539-7.
[25]	E. Valdinoci, A fractional framework for perimeters and phase transitions, Milan J. Math., 81 (2013), 1-23.doi: 10.1007/s00032-013-0199-x.