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Classification for positive solutions of degenerate elliptic system
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 |
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.
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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[11] |
W. Chen and C. Li, A Hopf type lemma for fractional equations, arXiv:1705.04889. |
[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. |
[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. 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. |
[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. |
[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. |
[18] |
R. Cont and P. Tankov,
Financial Modeling with Jump Processes, Chapman & Hall/CRC, Boca Raton, FL, 2004. |
[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. |
[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. 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. |
[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. Mathieu, P. Melchior, A. 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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[11] |
W. Chen and C. Li, A Hopf type lemma for fractional equations, arXiv:1705.04889. |
[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. |
[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. 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. |
[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. |
[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. |
[18] |
R. Cont and P. Tankov,
Financial Modeling with Jump Processes, Chapman & Hall/CRC, Boca Raton, FL, 2004. |
[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. |
[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. 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. |
[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. Mathieu, P. Melchior, A. 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.
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