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June  2020, 40(6): 3235-3252. doi: 10.3934/dcds.2020034

A Hopf lemma and regularity for fractional $ p $-Laplacians

Wenxiong Chen 1, , Congming Li 2,, and  Shijie Qi 3,1,

1. 

Department of Mathematical Sciences, Yeshiva University, New York, NY 10033, USA
2. 

School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200234, China
3. 

Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China

* Corresponding author: Congming Li

Received  January 2019 Revised  May 2019 Published  October 2019

Fund Project: The first author was partially supported by the Simons Foundation Collaboration Grant for Mathematicians (245486), and the second author was partially supported by NSFC (11571233)

In this paper, we study qualitative properties of the fractional $ p $-Laplacian. Specifically, we establish a Hopf type lemma for positive weak super-solutions of the fractional $ p- $Laplacian equation with Dirichlet condition. Moreover, an optimal condition is obtained to ensure $ (-\triangle)_p^s u\in C^1(\mathbb{R}^n) $ for smooth functions $ u $.

Keywords: Fractional $ p $-Laplacian, Hopf type lemma, regularity.
Mathematics Subject Classification: Primary: 35R11, 35J91; Secondary: 35B65.
Citation: Wenxiong Chen, Congming Li, Shijie Qi. A Hopf lemma and regularity for fractional $ p $-Laplacians. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3235-3252. doi: 10.3934/dcds.2020034
References:
[1]

G. Alberti and G. Bellettini, A nonlocal anisotropicmodel for phase transitions Ⅰ: The optimal profile problem, Math. Ann., 310 (1998), 527-560.  doi: 10.1007/s002080050159.

[2]

C. Bjorland, L. Caffarelli and A. Figalli, Non-local gradient dependent operators, Adv. Math., 230 (2012), 1859-1894.  doi: 10.1016/j.aim.2012.03.032.

[3]

C. Bjorland, L. Caffarelli and A. Figalli, Nonlocal tug-of-war and the infinity fractional Laplacian, Comm. Pure Appl. Math., 65 (2012), 337-380.  doi: 10.1002/cpa.21379.

[4]

C. Brandle, E. Colorado, A. de Pablo and U. Sanchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 39-71.  doi: 10.1017/S0308210511000175.

[5]

L. Brasco, E. Lindgren and A. Schikorra, Higher H$\ddot{o}$der regularity for the fractional p-Laplacian in the superquadratic case, Adv. Math., 338 (2018), 782-846.  doi: 10.1016/j.aim.2018.09.009.

[6]

K. Bogdan, T. Grzywny and M. Ryznar, Heat kernel estimates for the fractional Laplacian with Dirichlet conditions, Ann. Probab., 38 (2010), 1901-1923.  doi: 10.1214/10-AOP532.

[7]

C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, 20, Springer, [Cham]; Unione Matematica Italiana, Bologna, 2016. doi: 10.1007/978-3-319-28739-3.

[8]

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.

[9]

W. Chen, Y. Fang and R. Yang, Liouville theorems involving the fractional Laplacian on a half space, Adv. Math., 274 (2015), 167-198.  doi: 10.1016/j.aim.2014.12.013.

[10]

W. Chen and C. Li, Maximum principles for the fractional $p$-Laplacian and symmetry of solutions, Adv. Math., 335 (2018), 735-758.  doi: 10.1016/j.aim.2018.07.016.

[11]

W. Chen, Y. Li and R. Zhang, A direct method of moving spheres on fractional order equations, J. Funct. Anal., 272 (2017), 4131-4157.  doi: 10.1016/j.jfa.2017.02.022.

[12]

W. Chen, C. Li and G. Li, Maximum principles for a fully nonlinear fractional order equation and symmetry of solutions, Calc. Var. Partial Differential Equations, 56 (2017), Art. 29, 18 pp. doi: 10.1007/s00526-017-1110-3.

[13]

W. Chen, C. Li and Y. Li, A direct method of moving planes for the fractional Laplacian, Adv. Math., 308 (2017), 404-437.  doi: 10.1016/j.aim.2016.11.038.

[14]

W. Chen, C. Li and Y. Li, A direct blowing-up and rescaling argument on nonlocal elliptic equations, Internat. J. Math., 27 (2016), 1650064, 20 pp. doi: 10.1142/S0129167X16500646.

[15]

W. Chen, Y. Li and P. Ma, The Fractional Laplacian, A book to be published by World Scientific Publishing C, 2019. doi: 10.1142/10550.

[16]

W. Chen and S. Qi, Direct methods on fractional equations, Discrete Contin. Dyn. Syst., 39 (2019), 1269-1310.  doi: 10.3934/dcds.2019055.

[17]

W. Chen and J. Zhu, Indefinite fractional elliptic problem and Liouville theorems, J. Differential Equations, 260 (2016), 4758-4785.  doi: 10.1016/j.jde.2015.11.029.

[18]

A. Di Castro, T. Kuusi and G. Palatucci, Local behavior of fractional p-minimizers, Ann. Inst. H. Poincare Anal. Non Lineaire, 33 (2016), 1279-1299.  doi: 10.1016/j.anihpc.2015.04.003.

[19]

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.

[20]

M. M. Fall and S. Jarohs, Overdetermined problems with fractional Laplacian, ESAIM Control Optim. Calc. Var., 21 (2015), 924-938.  doi: 10.1051/cocv/2014048.

[21]

A. Greco and R. Servadei, Hopf's lemma and constrained radial symmetry for the fractional Laplacian, Math. Res. Lett., 23 (2016), 863-885.  doi: 10.4310/MRL.2016.v23.n3.a14.

[22]

A. Iannizzotto, S. Mosconi and M. Squassina, Global H$\ddot{o}$der regularity for the fractional $p$-Laplacian, Rev. Mat. Iberoam., 32 (2016), 1353-1392.  doi: 10.4171/RMI/921.

[23]

H. Ishii and G. Nakamura, A class of integral equations and approximation of $p$-Laplace equations, Calc. Var. Partial Differential Equations, 37 (2010), 485-522.  doi: 10.1007/s00526-009-0274-x.

[24]

L. Jin and Y. Li, A Hopf's Lemma and the Boundary Regularity for the Fractional $p$-Laplacian, Discrete Contin. Dyn. Syst., 39 (2019), 1477-1495.  doi: 10.3934/dcds.2019063.

[25]

C. Li and W. Chen, A Hopf type lemma for fractional equations, Proc. Amer. Math. Soc., 147 (2019), 1565-1575.  doi: 10.1090/proc/14342.

[26]

S. Mosconi and M. Squassina, Recent progresses in the theory of nonlinear nonlocal problems, Bruno Pini Math. Analysis Sem., 7 (2016), 147-164. 

[27]

X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302.  doi: 10.1016/j.matpur.2013.06.003.

[28]

X. Ros-Oton and E. Valdinoci, The Dirichlet problem for nonlocal operators with singular kernels: convex and nonconvex domains, Adv. Math., 288 (2016), 732-790.  doi: 10.1016/j.aim.2015.11.001.

[29]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153.

[30]

Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: A geometric inequality and a symmetry result, J. Funct. Anal., 256 (2009), 1842-1864.  doi: 10.1016/j.jfa.2009.01.020.

[31]

Y. Sire and E. Valdinoci, Rigidity results for some boundary quasilinear phase transitions, Comm. Partial Differential Equations, 34 (2009), 765-784.  doi: 10.1080/03605300902892402.

[32]

E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl., 49 (2009), 33-44. 

[33]

R. Zhuo, W. Chen, X. Cui and Z. Yuan, Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1125-1141.  doi: 10.3934/dcds.2016.36.1125.

show all references

References:
[1]

G. Alberti and G. Bellettini, A nonlocal anisotropicmodel for phase transitions Ⅰ: The optimal profile problem, Math. Ann., 310 (1998), 527-560.  doi: 10.1007/s002080050159.

[2]

C. Bjorland, L. Caffarelli and A. Figalli, Non-local gradient dependent operators, Adv. Math., 230 (2012), 1859-1894.  doi: 10.1016/j.aim.2012.03.032.

[3]

C. Bjorland, L. Caffarelli and A. Figalli, Nonlocal tug-of-war and the infinity fractional Laplacian, Comm. Pure Appl. Math., 65 (2012), 337-380.  doi: 10.1002/cpa.21379.

[4]

C. Brandle, E. Colorado, A. de Pablo and U. Sanchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 39-71.  doi: 10.1017/S0308210511000175.

[5]

L. Brasco, E. Lindgren and A. Schikorra, Higher H$\ddot{o}$der regularity for the fractional p-Laplacian in the superquadratic case, Adv. Math., 338 (2018), 782-846.  doi: 10.1016/j.aim.2018.09.009.

[6]

K. Bogdan, T. Grzywny and M. Ryznar, Heat kernel estimates for the fractional Laplacian with Dirichlet conditions, Ann. Probab., 38 (2010), 1901-1923.  doi: 10.1214/10-AOP532.

[7]

C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, 20, Springer, [Cham]; Unione Matematica Italiana, Bologna, 2016. doi: 10.1007/978-3-319-28739-3.

[8]

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.

[9]

W. Chen, Y. Fang and R. Yang, Liouville theorems involving the fractional Laplacian on a half space, Adv. Math., 274 (2015), 167-198.  doi: 10.1016/j.aim.2014.12.013.

[10]

W. Chen and C. Li, Maximum principles for the fractional $p$-Laplacian and symmetry of solutions, Adv. Math., 335 (2018), 735-758.  doi: 10.1016/j.aim.2018.07.016.

[11]

W. Chen, Y. Li and R. Zhang, A direct method of moving spheres on fractional order equations, J. Funct. Anal., 272 (2017), 4131-4157.  doi: 10.1016/j.jfa.2017.02.022.

[12]

W. Chen, C. Li and G. Li, Maximum principles for a fully nonlinear fractional order equation and symmetry of solutions, Calc. Var. Partial Differential Equations, 56 (2017), Art. 29, 18 pp. doi: 10.1007/s00526-017-1110-3.

[13]

W. Chen, C. Li and Y. Li, A direct method of moving planes for the fractional Laplacian, Adv. Math., 308 (2017), 404-437.  doi: 10.1016/j.aim.2016.11.038.

[14]

W. Chen, C. Li and Y. Li, A direct blowing-up and rescaling argument on nonlocal elliptic equations, Internat. J. Math., 27 (2016), 1650064, 20 pp. doi: 10.1142/S0129167X16500646.

[15]

W. Chen, Y. Li and P. Ma, The Fractional Laplacian, A book to be published by World Scientific Publishing C, 2019. doi: 10.1142/10550.

[16]

W. Chen and S. Qi, Direct methods on fractional equations, Discrete Contin. Dyn. Syst., 39 (2019), 1269-1310.  doi: 10.3934/dcds.2019055.

[17]

W. Chen and J. Zhu, Indefinite fractional elliptic problem and Liouville theorems, J. Differential Equations, 260 (2016), 4758-4785.  doi: 10.1016/j.jde.2015.11.029.

[18]

A. Di Castro, T. Kuusi and G. Palatucci, Local behavior of fractional p-minimizers, Ann. Inst. H. Poincare Anal. Non Lineaire, 33 (2016), 1279-1299.  doi: 10.1016/j.anihpc.2015.04.003.

[19]

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.

[20]

M. M. Fall and S. Jarohs, Overdetermined problems with fractional Laplacian, ESAIM Control Optim. Calc. Var., 21 (2015), 924-938.  doi: 10.1051/cocv/2014048.

[21]

A. Greco and R. Servadei, Hopf's lemma and constrained radial symmetry for the fractional Laplacian, Math. Res. Lett., 23 (2016), 863-885.  doi: 10.4310/MRL.2016.v23.n3.a14.

[22]

A. Iannizzotto, S. Mosconi and M. Squassina, Global H$\ddot{o}$der regularity for the fractional $p$-Laplacian, Rev. Mat. Iberoam., 32 (2016), 1353-1392.  doi: 10.4171/RMI/921.

[23]

H. Ishii and G. Nakamura, A class of integral equations and approximation of $p$-Laplace equations, Calc. Var. Partial Differential Equations, 37 (2010), 485-522.  doi: 10.1007/s00526-009-0274-x.

[24]

L. Jin and Y. Li, A Hopf's Lemma and the Boundary Regularity for the Fractional $p$-Laplacian, Discrete Contin. Dyn. Syst., 39 (2019), 1477-1495.  doi: 10.3934/dcds.2019063.

[25]

C. Li and W. Chen, A Hopf type lemma for fractional equations, Proc. Amer. Math. Soc., 147 (2019), 1565-1575.  doi: 10.1090/proc/14342.

[26]

S. Mosconi and M. Squassina, Recent progresses in the theory of nonlinear nonlocal problems, Bruno Pini Math. Analysis Sem., 7 (2016), 147-164. 

[27]

X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302.  doi: 10.1016/j.matpur.2013.06.003.

[28]

X. Ros-Oton and E. Valdinoci, The Dirichlet problem for nonlocal operators with singular kernels: convex and nonconvex domains, Adv. Math., 288 (2016), 732-790.  doi: 10.1016/j.aim.2015.11.001.

[29]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153.

[30]

Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: A geometric inequality and a symmetry result, J. Funct. Anal., 256 (2009), 1842-1864.  doi: 10.1016/j.jfa.2009.01.020.

[31]

Y. Sire and E. Valdinoci, Rigidity results for some boundary quasilinear phase transitions, Comm. Partial Differential Equations, 34 (2009), 765-784.  doi: 10.1080/03605300902892402.

[32]

E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl., 49 (2009), 33-44. 

[33]

R. Zhuo, W. Chen, X. Cui and Z. Yuan, Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1125-1141.  doi: 10.3934/dcds.2016.36.1125.

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