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

Wenxiong Chen; Congming Li; Shijie Qi

doi:10.3934/dcds.2020034

Previous Article
Boundary spike of the singular limit of an energy minimizing problem
DCDS Home
This Issue
Next Article
Nondegeneracy of harmonic maps from $ {{\mathbb{R}}^{2}} $ to $ {{\mathbb{S}}^{2}} $

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 & Continuous Dynamical Systems - A, 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[16]	W. Chen and S. Qi, Direct methods on fractional equations, Discrete Contin. Dyn. Syst., 39 (2019), 1269-1310. doi: 10.3934/dcds.2019055. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[26]	S. Mosconi and M. Squassina, Recent progresses in the theory of nonlinear nonlocal problems, Bruno Pini Math. Analysis Sem., 7 (2016), 147-164. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[32]	E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl., 49 (2009), 33-44. Google Scholar
[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. Google Scholar

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[16]	W. Chen and S. Qi, Direct methods on fractional equations, Discrete Contin. Dyn. Syst., 39 (2019), 1269-1310. doi: 10.3934/dcds.2019055. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[26]	S. Mosconi and M. Squassina, Recent progresses in the theory of nonlinear nonlocal problems, Bruno Pini Math. Analysis Sem., 7 (2016), 147-164. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[32]	E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl., 49 (2009), 33-44. Google Scholar
[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. Google Scholar

[1]	Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213
[2]	Z. Reichstein and B. Youssin. Parusinski's "Key Lemma" via algebraic geometry. Electronic Research Announcements, 1999, 5: 136-145.
[3]	Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261
[4]	María J. Garrido-Atienza, Bohdan Maslowski, Jana Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088
[5]	Martial Agueh, Reinhard Illner, Ashlin Richardson. Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type. Kinetic & Related Models, 2011, 4 (1) : 1-16. doi: 10.3934/krm.2011.4.1
[6]	Andrea Cianchi, Adele Ferone. Improving sharp Sobolev type inequalities by optimal remainder gradient norms. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1363-1386. doi: 10.3934/cpaa.2012.11.1363
[7]	Hakan Özadam, Ferruh Özbudak. A note on negacyclic and cyclic codes of length $p^s$ over a finite field of characteristic $p$. Advances in Mathematics of Communications, 2009, 3 (3) : 265-271. doi: 10.3934/amc.2009.3.265
[8]	Alexandre B. Simas, Fábio J. Valentim. $W$-Sobolev spaces: Higher order and regularity. Communications on Pure & Applied Analysis, 2015, 14 (2) : 597-607. doi: 10.3934/cpaa.2015.14.597
[9]	Philippe G. Lefloch, Cristinel Mardare, Sorin Mardare. Isometric immersions into the Minkowski spacetime for Lorentzian manifolds with limited regularity. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 341-365. doi: 10.3934/dcds.2009.23.341
[10]	Shihu Li, Wei Liu, Yingchao Xie. Large deviations for stochastic 3D Leray-$ \alpha $ model with fractional dissipation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2491-2509. doi: 10.3934/cpaa.2019113
[11]	A. Aghajani, S. F. Mottaghi. Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (3) : 887-898. doi: 10.3934/cpaa.2018044
[12]	Hyeong-Ohk Bae, Hyoungsuk So, Yeonghun Youn. Interior regularity to the steady incompressible shear thinning fluids with non-Standard growth. Networks & Heterogeneous Media, 2018, 13 (3) : 479-491. doi: 10.3934/nhm.2018021
[13]	Hirofumi Notsu, Masato Kimura. Symmetry and positive definiteness of the tensor-valued spring constant derived from P1-FEM for the equations of linear elasticity. Networks & Heterogeneous Media, 2014, 9 (4) : 617-634. doi: 10.3934/nhm.2014.9.617
[14]	Luigi C. Berselli, Jishan Fan. Logarithmic and improved regularity criteria for the 3D nematic liquid crystals models, Boussinesq system, and MHD equations in a bounded domain. Communications on Pure & Applied Analysis, 2015, 14 (2) : 637-655. doi: 10.3934/cpaa.2015.14.637
[15]	Lei Liu, Li Wu. Multiplicity of closed characteristics on $ P $-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020378

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors