American Institute of Mathematical Sciences

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March  2019, 39(3): 1477-1495. doi: 10.3934/dcds.2019063

A Hopf's lemma and the boundary regularity for the fractional p-Laplacian

Lingyu Jin 1, and  Yan Li 2,,

1. 

Department of Mathematics, South China Agricultural University, Guangzhou 510642, China
2. 

Department of Mathematics, Baylor University, Waco, TX 76706, USA

* Corresponding author: Yan Li

Received  January 2018 Revised  April 2018 Published  December 2018

Fund Project: The first author is partially supported by the Natural Science Foundation of China (11101160, 11271141) and the China Scholarship Council (201508440330).

We begin the paper with a Hopf's lemma for a fractional p-Laplacian problem on a half-space. Specifically speaking, we show that the derivative of the solution along the outward normal vector is strictly positive on the boundary of the half-space. Next we show that positive solutions for a fractional p-Laplacian equation possess certain Hölder continuity up to the boundary.

Keywords: Fractional p-Laplacian, Dirichlet problem, Hopf's lemma, boundary regularity.
Mathematics Subject Classification: Primary: 35S05, 35B50, 35B65; Secondary: 35B09.
Citation: Lingyu Jin, Yan Li. A Hopf's lemma and the boundary regularity for the fractional p-Laplacian. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1477-1495. doi: 10.3934/dcds.2019063
References:
[1]

G. Antonio and S. Raffaella, 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.  Google Scholar

[2]

D. Applebaum, Lévy processes - from probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2014), 1336-1347.   Google Scholar

[3]

M. Barlow, R. Bass, Z. Chen and M. Kassmann, Non-local Dirichlet formsand symmetric jump processes, Trans. Amer. Math. Soc., 361 (2009), 1963-1999.  doi: 10.1090/S0002-9947-08-04544-3.  Google Scholar

[4]

J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, 121 Cambridge University Press, Cambridge, 1996.  Google Scholar

[5]

K. Bogdan, The boundary Harnack principle for the fractional Laplacian, Studia Mathematica, 123 (1997), 43-80.  doi: 10.4064/sm-123-1-43-80.  Google Scholar

[6]

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

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J. Bouchard and A. Georges, Anomalous diffusion in disordered media: Statistical mechanics, models and physical applications, Physics reports, 195 (1990), 127-293.  doi: 10.1016/0370-1573(90)90099-N.  Google Scholar

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L. Caffarelli, J. M. Roquejoffre and Y. Sire, Variational problems with free boundaries for the fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1151-1179.  doi: 10.4171/JEMS/226.  Google Scholar

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L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Part. Diff. Eqs., 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

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A. Cesaroni, M. Cirant, S. Dipierro, M. Novaga and E. Valdinoci, On stationary fractional mean field games, J. Math. Pures Appl., (2017).  doi: 10.1016/j.matpur.2017.10.013.  Google Scholar

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W. Chen and C. Li, A Hopf type lemma for fractional equations, arXiv:1705.04889. Google Scholar

[12]

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.  Google Scholar

[13]

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

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W. Chen, Y. Li and P. Ma, The Fractional Laplacian, accepted for publication by World Scientific Publishing Co. Pte. Ltd.. Google Scholar

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W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure. Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.  Google Scholar

[16]

W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. Cont. Dyn. Sys., 12 (2005), 347-354.  doi: 10.3934/dcds.2005.12.347.  Google Scholar

[17]

Z. Chen, P. Kim and T. Kumagai, Global heat kernel estimates for symmetric jump processes, Trans. Amer. Math. Soc., 363 (2011), 5021-5055.  doi: 10.1090/S0002-9947-2011-05408-5.  Google Scholar

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S. Holm, S. P. Näsholm, F. Prieur and R. Sinkus, Deriving fractional acoustic wave equations from mechanical and thermal constitutive equations, Comput. Math. Appl., 66 (2013), 621-629.  doi: 10.1016/j.camwa.2013.02.024.  Google Scholar

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E. Hopf, Selected works of Eberhard Hopf with commentaries, C. Morawetz, J. Serrin, Y. Sinai eds., 2002, Providence, RI: American Mathematical Society, ISBN 0-8218-2077-X.  Google Scholar

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E. Hopf, Elementare Bemerkungenüber die Lösungen partieller Differentialgleichungen zweiter Ordnung vomelliptische n Typus, Sitzungsberichte Preussiche Akademie Wissenschaften, Berlin, 1927,147-152. Google Scholar

[22]

A. Iannizzotto, S. Mosconi and M. Squassina, Global Hölder regularity for the fractional p-Laplacian, arXiv:1411.2956. Google Scholar

[23]

A. Iannizzotto, S. Liu, K. Perera and M. Squassina, Existence results for fractional p-Laplacian problems via Morse theory, Adv. Calc. Var., 9 (2016), 101-125.  doi: 10.1515/acv-2014-0024.  Google Scholar

[24]

S. Kim and K. Lee, Geometric property of the ground state eigenfunction for Cauchy process, arXiv:1105.3283v1. Google Scholar

[25]

Y. Li and P. Ma, Symmetry of solutions for a fractional system, Sci. China Ser. A, 10 (2017), 1805-1824.   Google Scholar

[26]

E. Lindgren and P. Lindqvist, Fractional eigenvalues, Calc. Var. Partial Differential Equations, 49 (2014), 795-826.  doi: 10.1007/s00526-013-0600-1.  Google Scholar

[27]

B. Mathieu, P. Melchior, A. Oustaloup and Ch. Ceyral, Fractional differentiation for edge detection, Singal Processing, 83 (2003), 2421-2432.   Google Scholar

[28]

L. Pezzo and A. Quaas, A Hopf's lemma and a strong minimum principle for the fractional p-Laplacian, J. Differential Equations, 263 (2017), 765-778.  doi: 10.1016/j.jde.2017.02.051.  Google Scholar

[29]

P. Pucci and J. Serrin, The strong maximum principle revisited, J. Differential Equations, 196 (2004), 1-66.  doi: 10.1016/j.jde.2003.05.001.  Google Scholar

[30]

X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, Journal De Mathématiques Pures Et Appliquées, 101 (2014), 275-302.  doi: 10.1016/j.matpur.2013.06.003.  Google Scholar

[31]

K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge Studies in Advanced Mathematics, 68. Cambridge University Press, Cambridge, 1999.  Google Scholar

[32]

J. L. Vázquez, Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators, Disc. Cont. Dyn. Sys.-S., 7 (2014), 857-885.  doi: 10.3934/dcdss.2014.7.857.  Google Scholar

[33]

R. Xavier and R. Xavier, Boundary regularity for the fractional heat equation, Revista De La Real Academia De Ciencias Exactas Físicas Y Naturales.serie A.matemáticas, 110 (2016), 49-64.  doi: 10.1007/s13398-015-0218-6.  Google Scholar

[34]

R. Yang, Optimal Regularity and Nondegeneracy of a Free Boundary Problem Related to the Fractional Laplacian, Arch. Ration. Mech. Anal., 208 (2013), 693-723.  doi: 10.1007/s00205-013-0619-7.  Google Scholar

[35] G. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, 2008.   Google Scholar

show all references

References:
[1]

G. Antonio and S. Raffaella, 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.  Google Scholar

[2]

D. Applebaum, Lévy processes - from probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2014), 1336-1347.   Google Scholar

[3]

M. Barlow, R. Bass, Z. Chen and M. Kassmann, Non-local Dirichlet formsand symmetric jump processes, Trans. Amer. Math. Soc., 361 (2009), 1963-1999.  doi: 10.1090/S0002-9947-08-04544-3.  Google Scholar

[4]

J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, 121 Cambridge University Press, Cambridge, 1996.  Google Scholar

[5]

K. Bogdan, The boundary Harnack principle for the fractional Laplacian, Studia Mathematica, 123 (1997), 43-80.  doi: 10.4064/sm-123-1-43-80.  Google Scholar

[6]

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

[7]

J. Bouchard and A. Georges, Anomalous diffusion in disordered media: Statistical mechanics, models and physical applications, Physics reports, 195 (1990), 127-293.  doi: 10.1016/0370-1573(90)90099-N.  Google Scholar

[8]

L. Caffarelli, J. M. Roquejoffre and Y. Sire, Variational problems with free boundaries for the fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1151-1179.  doi: 10.4171/JEMS/226.  Google Scholar

[9]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Part. Diff. Eqs., 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

[10]

A. Cesaroni, M. Cirant, S. Dipierro, M. Novaga and E. Valdinoci, On stationary fractional mean field games, J. Math. Pures Appl., (2017).  doi: 10.1016/j.matpur.2017.10.013.  Google Scholar

[11]

W. Chen and C. Li, A Hopf type lemma for fractional equations, arXiv:1705.04889. Google Scholar

[12]

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.  Google Scholar

[13]

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

[14]

W. Chen, Y. Li and P. Ma, The Fractional Laplacian, accepted for publication by World Scientific Publishing Co. Pte. Ltd.. Google Scholar

[15]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure. Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.  Google Scholar

[16]

W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. Cont. Dyn. Sys., 12 (2005), 347-354.  doi: 10.3934/dcds.2005.12.347.  Google Scholar

[17]

Z. Chen, P. Kim and T. Kumagai, Global heat kernel estimates for symmetric jump processes, Trans. Amer. Math. Soc., 363 (2011), 5021-5055.  doi: 10.1090/S0002-9947-2011-05408-5.  Google Scholar

[18]

R. Cont and P. Tankov, Financial Modeling with Jump Processes, Chapman & Hall/CRC, Boca Raton, FL, 2004.  Google Scholar

[19]

S. Holm, S. P. Näsholm, F. Prieur and R. Sinkus, Deriving fractional acoustic wave equations from mechanical and thermal constitutive equations, Comput. Math. Appl., 66 (2013), 621-629.  doi: 10.1016/j.camwa.2013.02.024.  Google Scholar

[20]

E. Hopf, Selected works of Eberhard Hopf with commentaries, C. Morawetz, J. Serrin, Y. Sinai eds., 2002, Providence, RI: American Mathematical Society, ISBN 0-8218-2077-X.  Google Scholar

[21]

E. Hopf, Elementare Bemerkungenüber die Lösungen partieller Differentialgleichungen zweiter Ordnung vomelliptische n Typus, Sitzungsberichte Preussiche Akademie Wissenschaften, Berlin, 1927,147-152. Google Scholar

[22]

A. Iannizzotto, S. Mosconi and M. Squassina, Global Hölder regularity for the fractional p-Laplacian, arXiv:1411.2956. Google Scholar

[23]

A. Iannizzotto, S. Liu, K. Perera and M. Squassina, Existence results for fractional p-Laplacian problems via Morse theory, Adv. Calc. Var., 9 (2016), 101-125.  doi: 10.1515/acv-2014-0024.  Google Scholar

[24]

S. Kim and K. Lee, Geometric property of the ground state eigenfunction for Cauchy process, arXiv:1105.3283v1. Google Scholar

[25]

Y. Li and P. Ma, Symmetry of solutions for a fractional system, Sci. China Ser. A, 10 (2017), 1805-1824.   Google Scholar

[26]

E. Lindgren and P. Lindqvist, Fractional eigenvalues, Calc. Var. Partial Differential Equations, 49 (2014), 795-826.  doi: 10.1007/s00526-013-0600-1.  Google Scholar

[27]

B. Mathieu, P. Melchior, A. Oustaloup and Ch. Ceyral, Fractional differentiation for edge detection, Singal Processing, 83 (2003), 2421-2432.   Google Scholar

[28]

L. Pezzo and A. Quaas, A Hopf's lemma and a strong minimum principle for the fractional p-Laplacian, J. Differential Equations, 263 (2017), 765-778.  doi: 10.1016/j.jde.2017.02.051.  Google Scholar

[29]

P. Pucci and J. Serrin, The strong maximum principle revisited, J. Differential Equations, 196 (2004), 1-66.  doi: 10.1016/j.jde.2003.05.001.  Google Scholar

[30]

X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, Journal De Mathématiques Pures Et Appliquées, 101 (2014), 275-302.  doi: 10.1016/j.matpur.2013.06.003.  Google Scholar

[31]

K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge Studies in Advanced Mathematics, 68. Cambridge University Press, Cambridge, 1999.  Google Scholar

[32]

J. L. Vázquez, Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators, Disc. Cont. Dyn. Sys.-S., 7 (2014), 857-885.  doi: 10.3934/dcdss.2014.7.857.  Google Scholar

[33]

R. Xavier and R. Xavier, Boundary regularity for the fractional heat equation, Revista De La Real Academia De Ciencias Exactas Físicas Y Naturales.serie A.matemáticas, 110 (2016), 49-64.  doi: 10.1007/s13398-015-0218-6.  Google Scholar

[34]

R. Yang, Optimal Regularity and Nondegeneracy of a Free Boundary Problem Related to the Fractional Laplacian, Arch. Ration. Mech. Anal., 208 (2013), 693-723.  doi: 10.1007/s00205-013-0619-7.  Google Scholar

[35] G. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, 2008.   Google Scholar
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