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

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

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.

Citation: Lingyu Jin, Yan Li. A Hopf's lemma and the boundary regularity for the fractional p-Laplacian. Discrete and 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.

[2]

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

[3]

M. BarlowR. BassZ. 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.

[4]

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

[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.

[6]

K. BogdanT. 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.

[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.

[8]

L. CaffarelliJ. 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.

[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.

[10]

A. CesaroniM. CirantS. DipierroM. Novaga and E. Valdinoci, On stationary fractional mean field games, J. Math. Pures Appl., (2017).  doi: 10.1016/j.matpur.2017.10.013.

[11]

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

[12]

W. ChenC. 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.

[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.

[14]

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

[15]

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

[16]

W. ChenC. 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.

[17]

Z. ChenP. 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.

[18]

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

[19]

S. HolmS. P. NäsholmF. 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.

[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.

[21]

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

[22]

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

[23]

A. IannizzottoS. LiuK. 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.

[24]

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

[25]

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

[26]

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

[27]

B. MathieuP. MelchiorA. Oustaloup and Ch. Ceyral, Fractional differentiation for edge detection, Singal Processing, 83 (2003), 2421-2432. 

[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.

[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.

[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.

[31]

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

[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.

[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.

[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.

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

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.

[2]

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

[3]

M. BarlowR. BassZ. 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.

[4]

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

[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.

[6]

K. BogdanT. 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.

[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.

[8]

L. CaffarelliJ. 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.

[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.

[10]

A. CesaroniM. CirantS. DipierroM. Novaga and E. Valdinoci, On stationary fractional mean field games, J. Math. Pures Appl., (2017).  doi: 10.1016/j.matpur.2017.10.013.

[11]

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

[12]

W. ChenC. 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.

[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.

[14]

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

[15]

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

[16]

W. ChenC. 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.

[17]

Z. ChenP. 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.

[18]

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

[19]

S. HolmS. P. NäsholmF. 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.

[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.

[21]

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

[22]

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

[23]

A. IannizzottoS. LiuK. 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.

[24]

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

[25]

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

[26]

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

[27]

B. MathieuP. MelchiorA. Oustaloup and Ch. Ceyral, Fractional differentiation for edge detection, Singal Processing, 83 (2003), 2421-2432. 

[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.

[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.

[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.

[31]

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

[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.

[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.

[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.

[35] G. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, 2008. 
Figure 1.  Subregions
[1]

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

[2]

Zaizheng Li, Qidi Zhang. Sub-solutions and a point-wise Hopf's lemma for fractional $ p $-Laplacian. Communications on Pure and Applied Analysis, 2021, 20 (2) : 835-865. doi: 10.3934/cpaa.2020293

[3]

Patrizia Pucci, Mingqi Xiang, Binlin Zhang. A diffusion problem of Kirchhoff type involving the nonlocal fractional p-Laplacian. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 4035-4051. doi: 10.3934/dcds.2017171

[4]

Agnid Banerjee, Nicola Garofalo. On the Dirichlet boundary value problem for the normalized $p$-laplacian evolution. Communications on Pure and Applied Analysis, 2015, 14 (1) : 1-21. doi: 10.3934/cpaa.2015.14.1

[5]

Wen Tan. The regularity of pullback attractor for a non-autonomous p-Laplacian equation with dynamical boundary condition. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 529-546. doi: 10.3934/dcdsb.2018194

[6]

Yansheng Zhong, Yongqing Li. On a p-Laplacian eigenvalue problem with supercritical exponent. Communications on Pure and Applied Analysis, 2019, 18 (1) : 227-236. doi: 10.3934/cpaa.2019012

[7]

Francesca Colasuonno, Benedetta Noris. A p-Laplacian supercritical Neumann problem. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3025-3057. doi: 10.3934/dcds.2017130

[8]

Dimitri Mugnai, Kanishka Perera, Edoardo Proietti Lippi. A priori estimates for the Fractional p-Laplacian with nonlocal Neumann boundary conditions and applications. Communications on Pure and Applied Analysis, 2022, 21 (1) : 275-292. doi: 10.3934/cpaa.2021177

[9]

Dimitri Mugnai. Bounce on a p-Laplacian. Communications on Pure and Applied Analysis, 2003, 2 (3) : 371-379. doi: 10.3934/cpaa.2003.2.371

[10]

CÉSAR E. TORRES LEDESMA. Existence and symmetry result for fractional p-Laplacian in $\mathbb{R}^{n}$. Communications on Pure and Applied Analysis, 2017, 16 (1) : 99-114. doi: 10.3934/cpaa.2017004

[11]

Leyun Wu, Pengcheng Niu. Symmetry and nonexistence of positive solutions to fractional p-Laplacian equations. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1573-1583. doi: 10.3934/dcds.2019069

[12]

Everaldo S. de Medeiros, Jianfu Yang. Asymptotic behavior of solutions to a perturbed p-Laplacian problem with Neumann condition. Discrete and Continuous Dynamical Systems, 2005, 12 (4) : 595-606. doi: 10.3934/dcds.2005.12.595

[13]

Stanislav Nikolaevich Antontsev, Serik Ersultanovich Aitzhanov, Guzel Rashitkhuzhakyzy Ashurova. An inverse problem for the pseudo-parabolic equation with p-Laplacian. Evolution Equations and Control Theory, 2022, 11 (2) : 399-414. doi: 10.3934/eect.2021005

[14]

Bernd Kawohl, Jiří Horák. On the geometry of the p-Laplacian operator. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 799-813. doi: 10.3934/dcdss.2017040

[15]

Pengyan Wang, Wenxiong Chen. Hopf's lemmas for parabolic fractional p-Laplacians. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022089

[16]

Vincenzo Ambrosio, Teresa Isernia. Multiplicity and concentration results for some nonlinear Schrödinger equations with the fractional p-Laplacian. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5835-5881. doi: 10.3934/dcds.2018254

[17]

Leszek Gasiński. Positive solutions for resonant boundary value problems with the scalar p-Laplacian and nonsmooth potential. Discrete and Continuous Dynamical Systems, 2007, 17 (1) : 143-158. doi: 10.3934/dcds.2007.17.143

[18]

Eun Kyoung Lee, R. Shivaji, Inbo Sim, Byungjae Son. Analysis of positive solutions for a class of semipositone p-Laplacian problems with nonlinear boundary conditions. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1139-1154. doi: 10.3934/cpaa.2019055

[19]

Genni Fragnelli, Dimitri Mugnai, Nikolaos S. Papageorgiou. Robin problems for the p-Laplacian with gradient dependence. Discrete and Continuous Dynamical Systems - S, 2019, 12 (2) : 287-295. doi: 10.3934/dcdss.2019020

[20]

Salvatore A. Marano, Nikolaos S. Papageorgiou. Positive solutions to a Dirichlet problem with $p$-Laplacian and concave-convex nonlinearity depending on a parameter. Communications on Pure and Applied Analysis, 2013, 12 (2) : 815-829. doi: 10.3934/cpaa.2013.12.815

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (388)
  • HTML views (128)
  • Cited by (2)

Other articles
by authors

[Back to Top]