Short-time existence of the second order renormalization group flow in dimension three
Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains
December 2015, 35(12): 5769-5786. doi: 10.3934/dcds.2015.35.5769
On the variation of the fractional mean curvature under the effect of $C^{1, \alpha}$ perturbations
Matteo Cozzi 1,
Dipartimento di Matematica "Federigo Enriques", Università degli Studi di Milano, Via Saldini 50, I-20133 Milano, Italy
Received March 2014 Published May 2015
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: $C^{1, Fractional mean curvature, \alpha}$ diffeomorphisms, fractional Laplacian, quantitative Implicit Function Theorem., non-local equations, $s$-perimeter.
Mathematics Subject Classification: Primary: 35R11, 28A75, 49Q05; Secondary: 26B1.
Citation: Matteo Cozzi. On the variation of the fractional mean curvature under the effect of $C^{1, \alpha}$ perturbations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5769-5786. doi: 10.3934/dcds.2015.35.5769
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
N. Abatangelo and E. Valdinoci, A notion of nonlocal curvature,, Numer. Funct. Anal. Optim., 35 (2014), 793. doi: 10.1080/01630563.2014.901837. Google Scholar
G. Albanese, A. Fiscella and E. Valdinoci, Gevrey regularity for integro-differential operators,, J. Math. Anal. Appl., 428 (2015), 1225. doi: 10.1016/j.jmaa.2015.04.002. Google Scholar
G. Alberti and G. Bellettini, A nonlocal anisotropic model for phase transitions. Part I: The optimal profile problem,, Math. Ann., 310 (1998), 527. doi: 10.1007/s002080050159. Google Scholar
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. doi: 10.1017/S0956792598003453. Google Scholar
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. Google Scholar
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. doi: 10.1016/j.anihpc.2013.02.001. Google Scholar
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. doi: 10.1090/S0002-9947-2014-05906-0. Google Scholar
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
L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, Comm. Partial Differential Equations, 32 (2007), 1245. doi: 10.1080/03605300600987306. Google Scholar
L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations,, Comm. Pure Appl. Math., 62 (2009), 597. doi: 10.1002/cpa.20274. Google Scholar
L. Caffarelli and L. Silvestre, Regularity results for nonlocal equations by approximation,, Arch. Ration. Mech. Anal., 200 (2011), 59. doi: 10.1007/s00205-010-0336-4. Google Scholar
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
L. Caffarelli and E. Valdinoci, Regularity properties of nonlocal minimal surfaces via limiting arguments,, Adv. Math., 248 (2013), 843. doi: 10.1016/j.aim.2013.08.007. Google Scholar
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
S. Dipierro, A. Figalli, G. Palatucci and E. Valdinoci, Asymptotics of the s-perimeter as s ↘ 0,, Discrete Contin. Dyn. Syst., 33 (2013), 2777. doi: 10.3934/dcds.2013.33.2777. Google Scholar
A. Figalli and E. Valdinoci, Regularity and Bernstein-type results for nonlocal minimal surfaces,, \emph{J. Reine Angew. Math.}, (2015). doi: 10.1515/crelle-2015-0006. Google Scholar
N. S. Landkof, Foundations of Modern Potential Theory,, Springer-Verlag, (1972). Google Scholar
M. Ludwig, Anisotropic fractional perimeters,, J. Differential Geom., 96 (2014), 77. Google Scholar
L. Modica, The gradient theory of phase transitions and the minimal interface criterion,, Arch. Ration. Mech. Anal., 98 (1987), 123. doi: 10.1007/BF00251230. Google Scholar
L. Modica and S. Mortola, Un esempio di $\Gamma$-convergenza,, Boll. Un. Mat. Ital. B (5), 14 (1977), 285. Google Scholar
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. doi: 10.1007/s10231-011-0243-9. Google Scholar
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
O. Savin and E. Valdinoci, Regularity of nonlocal minimal cones in dimension 2,, Calc. Var. Partial Differential Equations, 48 (2013), 33. doi: 10.1007/s00526-012-0539-7. Google Scholar
E. Valdinoci, A fractional framework for perimeters and phase transitions,, Milan J. Math., 81 (2013), 1. doi: 10.1007/s00032-013-0199-x. Google Scholar
Imran H. Biswas, Indranil Chowdhury. On the differentiability of the solutions of non-local Isaacs equations involving $\frac{1}{2}$-Laplacian. Communications on Pure & Applied Analysis, 2016, 15 (3) : 907-927. doi: 10.3934/cpaa.2016.15.907
Massimiliano Ferrara, Giovanni Molica Bisci, Binlin Zhang. Existence of weak solutions for non-local fractional problems via Morse theory. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2483-2499. doi: 10.3934/dcdsb.2014.19.2483
Dariusz Idczak. A global implicit function theorem and its applications to functional equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2549-2556. doi: 10.3934/dcdsb.2014.19.2549
Pengyan Wang, Pengcheng Niu. Liouville's theorem for a fractional elliptic system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1545-1558. doi: 10.3934/dcds.2019067
Dorota Bors. Application of Mountain Pass Theorem to superlinear equations with fractional Laplacian controlled by distributed parameters and boundary data. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 29-43. doi: 10.3934/dcdsb.2018003
Keonhee Lee, Kazumine Moriyasu, Kazuhiro Sakai. $C^1$-stable shadowing diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 683-697. doi: 10.3934/dcds.2008.22.683
Hasib Khan, Cemil Tunc, Aziz Khan. Green function's properties and existence theorems for nonlinear singular-delay-fractional differential equations. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020139
Bassam Fayad, Zhiyuan Zhang. An effective version of Katok's horseshoe theorem for conservative C2 surface diffeomorphisms. Journal of Modern Dynamics, 2017, 11: 425-445. doi: 10.3934/jmd.2017017
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
Lizhi Zhang. Symmetry of solutions to semilinear equations involving the fractional laplacian. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2393-2409. doi: 10.3934/cpaa.2015.14.2393
Zhigang Wu, Hao Xu. Symmetry properties in systems of fractional Laplacian equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1559-1571. doi: 10.3934/dcds.2019068
María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Local pathwise solutions to stochastic evolution equations driven by fractional Brownian motions with Hurst parameters $H\in (1/3,1/2]$. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2553-2581. doi: 10.3934/dcdsb.2015.20.2553
Shaoming Guo. Oscillatory integrals related to Carleson's theorem: fractional monomials. Communications on Pure & Applied Analysis, 2016, 15 (3) : 929-946. doi: 10.3934/cpaa.2016.15.929
Lingyu Jin, Yan Li. A Hopf's lemma and the boundary regularity for the fractional p-Laplacian. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1477-1495. doi: 10.3934/dcds.2019063
Shu-Guang Shao, Shu Wang, Wen-Qing Xu, Yu-Li Ge. On the local C1, α solution of ideal magneto-hydrodynamical equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2103-2113. doi: 10.3934/dcds.2017090
Hongjie Dong, Doyoon Kim. Schauder estimates for a class of non-local elliptic equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2319-2347. doi: 10.3934/dcds.2013.33.2319
A. V. Bobylev, Vladimir Dorodnitsyn. Symmetries of evolution equations with non-local operators and applications to the Boltzmann equation. Discrete & Continuous Dynamical Systems - A, 2009, 24 (1) : 35-57. doi: 10.3934/dcds.2009.24.35
Yuanhong Wei, Xifeng Su. On a class of non-local elliptic equations with asymptotically linear term. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 6287-6304. doi: 10.3934/dcds.2018154
Fengping Yao, Shulin Zhou. Interior $C^{1,\alpha}$ regularity of weak solutions for a class of quasilinear elliptic equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1635-1649. doi: 10.3934/dcdsb.2016015
Kazuhisa Ichikawa, Mahemauti Rouzimaimaiti, Takashi Suzuki. Reaction diffusion equation with non-local term arises as a mean field limit of the master equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 115-126. doi: 10.3934/dcdss.2012.5.115
Matteo Cozzi