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A Hopf lemma and regularity for fractional $ p $-Laplacians
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 |
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 $.
References:
[1] |
G. Alberti and G. Bellettini,
A nonlocal anisotropicmodel for phase transitions Ⅰ: The optimal profile problem, Math. Ann., 310 (1998), 527-560.
doi: 10.1007/s002080050159. |
[2] |
C. Bjorland, L. Caffarelli and A. Figalli,
Non-local gradient dependent operators, Adv. Math., 230 (2012), 1859-1894.
doi: 10.1016/j.aim.2012.03.032. |
[3] |
C. Bjorland, L. Caffarelli and A. Figalli,
Nonlocal tug-of-war and the infinity fractional Laplacian, Comm. Pure Appl. Math., 65 (2012), 337-380.
doi: 10.1002/cpa.21379. |
[4] |
C. Brandle, E. Colorado, A. de Pablo and U. Sanchez,
A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 39-71.
doi: 10.1017/S0308210511000175. |
[5] |
L. Brasco, E. Lindgren and A. Schikorra,
Higher H$\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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