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doi: 10.3934/cpaa.2022089
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Hopf's lemmas for parabolic fractional p-Laplacians

1. 

School of Mathematics and Statistics, Xinyang Normal University, Xinyang, Henan, China, 464000

2. 

Department of Mathematical Sciences, Yeshiva University, New York, NY, USA, 10033

*Corresponding author

Received  January 2022 Revised  April 2022 Early access May 2022

Fund Project: The first author was supported by the State China Scholarship(No. 201906290184) and the National Natural Science Foundation of China (No. 12101530). The second author was partially supported by National Natural Science Foundation of China (No.12071229)

In this paper, we first establish Hopf's lemmas for parabolic fractional $ p $-equations for $ p \geq 2 $. Then we derive an asymptotic Hopf's lemma for anti-symmetric solutions to parabolic fractional Laplacians. We believe that these Hopf's lemmas will become powerful tools in obtaining qualitative properties of solutions for nonlocal parabolic equations.

Citation: Pengyan Wang, Wenxiong Chen. Hopf's lemmas for parabolic fractional p-Laplacians. Communications on Pure and Applied Analysis, doi: 10.3934/cpaa.2022089
References:
[1]

M. BirknerJ. López-Mimbela and A. Wakolbinger, Comparison results and steady states for the Fujita equation with fractional Laplacian, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 83-97.  doi: 10.1016/J.ANIHPC.2004.05.002.

[2]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Commun. Partial Differ. Equ., 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.

[3]

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.

[4]

W. ChenC. Li and Y. Li, A drirect method of moving planes for the fractional Laplacian, Adv. Math., 308 (2017), 404-437.  doi: 10.1016/j.aim.2016.11.038.

[5]

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

[6]

W. ChenC. Li and S. Qi, A Hopf lemma and regularity for fractional $p$-Laplacians, Discrete Contin. Dyn. Syst., 40 (2020), 3235-3252.  doi: 10.3934/dcds.2020034.

[7]

W. ChenY. 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.

[8]

W. ChenC. Li and J. Zhu, Fractional equations with indefinite nonlinearities, Discrete Contin. Dyn. Syst., 39 (2019), 1257-1268.  doi: 10.3934/dcds.2019054.

[9]

W. Chen, P. Wang, Y. Niu and Y. Hu, Asymptotic method of moving planes for fractional parabolic equations, Adv. Math., 377 (2021), 47 pp. doi: 10.1016/j.aim.2020.107463.

[10]

W. Chen and L. Wu, Liouville theorems for fractional parabolic equations, Adv. Nonlinear Stud., 21 (2021), 939-958.  doi: 10.1515/ans-2021-2148.

[11]

W. DaiZ. Liu and G. Lu, Hardy-Sobolev type integral systems with Dirichlet boundary conditions in a half space, Commun. Pure Appl. Anal., 16 (2017), 1253-1264.  doi: 10.3934/cpaa.2017061.

[12]

W. Dai and G. Qin, Classification of nonnegative classical solutions to third-order equations, Adv. Math., 328 (2018), 822-857.  doi: 10.1016/j.aim.2018.02.016.

[13]

X. Fernández-Real and X. Ros-Oton, Regularity theory for general stable operators: parabolic equations, J. Funct. Anal., 272 (2017), 4165-4221.  doi: 10.1016/j.jfa.2017.02.015.

[14]

R. K. Getoor, First passage times for symmetric stable processes in space, Trans. Am. Math. Soc., 101 (1961), 75-90.  doi: 10.1090/S0002-9947-1961-0137148-5.

[15]

E. Hopf, Elementare Bemerkungen über die Lösungen partieller Differentialgleichngen zweiter Ordnung vom elliptischen Typus, Sitz. Ber. Preuss. Akad. Wissensch. Berlin, Math.-Phys. Kl, 19 (1927), 147–152.

[16]

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.

[17]

T. Jin and J. Xiong, A fractional Yamabe flow and some applications, J. Reine Angew. Math., 696 (2014), 187-223.  doi: 10.1515/crelle-2012-0110.

[18]

C. Li, Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on bounded domains, Commun. Partial Differ. Equ., 16 (1991), 491-526.  doi: 10.1080/03605309108820766.

[19]

Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres, Duke math. J., 80 (1995), 383-418.  doi: 10.1215/S0012-7094-95-08016-8.

[20]

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.

[21]

Z. Li and Q. Zhang, Sub-solutions and a point-wise Hopf's lemma for fractional $p$-Laplacian, Commun. Pure Appl. Anal., 20 (2021), 835-865.  doi: 10.3934/cpaa.2020293.

[22]

Z. Liu, Maximum principles and monotonicity of solutions for fractional $p$-equations in unbounded domains, J. Differ. Equ., 270 (2021), 1043-1078.  doi: 10.1016/j.jde.2020.09.001.

[23]

G. Lu and J. Zhu, An overdetermined problem in Riese-potential and fractional Laplacian, Nonlinear Anal., 75 (2012), 3036-3048.  doi: 10.1016/j.na.2011.11.036.

[24]

G. Lu and J. Zhu, The maximum principles and symmetry results for viscosity solutions of fully nonlinear equations, J. Differ. Equ., 258 (2015), 205-2079.  doi: 10.1016/j.jde.2014.11.022.

[25]

G. Lu and J. Zhu, Axial symmetry and regularity of solutions to an integral equation in a half-space, Pacific J. Math., 253 (2011), 455-473.  doi: 10.2140/pjm.2011.253.455.

[26]

P. Wang, Uniqueness and monotonicity of solutions for fractional equations with a gradient term, Electron. J. Qual. Theory Differ. Equ., 58 (2021), 1-19.  doi: 10.14232/ejqtde.2021.1.58.

[27]

L. Wu and W. Chen, Monotonicity of solutions for fractional equations with De Giorgi type nonlinearities, (in Chinese)., Sci. Sin. Math., 52 (2020), 1-22.  doi: 10.1360/SCM-2019-0668.

[28]

L. Wu and W. Chen, The sliding methods for the fractional $p$-Laplacian, Adv. Math., 361 (2020), 35 pp. doi: 10.1016/j.aim.2019.106933.

show all references

References:
[1]

M. BirknerJ. López-Mimbela and A. Wakolbinger, Comparison results and steady states for the Fujita equation with fractional Laplacian, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 83-97.  doi: 10.1016/J.ANIHPC.2004.05.002.

[2]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Commun. Partial Differ. Equ., 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.

[3]

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.

[4]

W. ChenC. Li and Y. Li, A drirect method of moving planes for the fractional Laplacian, Adv. Math., 308 (2017), 404-437.  doi: 10.1016/j.aim.2016.11.038.

[5]

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

[6]

W. ChenC. Li and S. Qi, A Hopf lemma and regularity for fractional $p$-Laplacians, Discrete Contin. Dyn. Syst., 40 (2020), 3235-3252.  doi: 10.3934/dcds.2020034.

[7]

W. ChenY. 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.

[8]

W. ChenC. Li and J. Zhu, Fractional equations with indefinite nonlinearities, Discrete Contin. Dyn. Syst., 39 (2019), 1257-1268.  doi: 10.3934/dcds.2019054.

[9]

W. Chen, P. Wang, Y. Niu and Y. Hu, Asymptotic method of moving planes for fractional parabolic equations, Adv. Math., 377 (2021), 47 pp. doi: 10.1016/j.aim.2020.107463.

[10]

W. Chen and L. Wu, Liouville theorems for fractional parabolic equations, Adv. Nonlinear Stud., 21 (2021), 939-958.  doi: 10.1515/ans-2021-2148.

[11]

W. DaiZ. Liu and G. Lu, Hardy-Sobolev type integral systems with Dirichlet boundary conditions in a half space, Commun. Pure Appl. Anal., 16 (2017), 1253-1264.  doi: 10.3934/cpaa.2017061.

[12]

W. Dai and G. Qin, Classification of nonnegative classical solutions to third-order equations, Adv. Math., 328 (2018), 822-857.  doi: 10.1016/j.aim.2018.02.016.

[13]

X. Fernández-Real and X. Ros-Oton, Regularity theory for general stable operators: parabolic equations, J. Funct. Anal., 272 (2017), 4165-4221.  doi: 10.1016/j.jfa.2017.02.015.

[14]

R. K. Getoor, First passage times for symmetric stable processes in space, Trans. Am. Math. Soc., 101 (1961), 75-90.  doi: 10.1090/S0002-9947-1961-0137148-5.

[15]

E. Hopf, Elementare Bemerkungen über die Lösungen partieller Differentialgleichngen zweiter Ordnung vom elliptischen Typus, Sitz. Ber. Preuss. Akad. Wissensch. Berlin, Math.-Phys. Kl, 19 (1927), 147–152.

[16]

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.

[17]

T. Jin and J. Xiong, A fractional Yamabe flow and some applications, J. Reine Angew. Math., 696 (2014), 187-223.  doi: 10.1515/crelle-2012-0110.

[18]

C. Li, Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on bounded domains, Commun. Partial Differ. Equ., 16 (1991), 491-526.  doi: 10.1080/03605309108820766.

[19]

Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres, Duke math. J., 80 (1995), 383-418.  doi: 10.1215/S0012-7094-95-08016-8.

[20]

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.

[21]

Z. Li and Q. Zhang, Sub-solutions and a point-wise Hopf's lemma for fractional $p$-Laplacian, Commun. Pure Appl. Anal., 20 (2021), 835-865.  doi: 10.3934/cpaa.2020293.

[22]

Z. Liu, Maximum principles and monotonicity of solutions for fractional $p$-equations in unbounded domains, J. Differ. Equ., 270 (2021), 1043-1078.  doi: 10.1016/j.jde.2020.09.001.

[23]

G. Lu and J. Zhu, An overdetermined problem in Riese-potential and fractional Laplacian, Nonlinear Anal., 75 (2012), 3036-3048.  doi: 10.1016/j.na.2011.11.036.

[24]

G. Lu and J. Zhu, The maximum principles and symmetry results for viscosity solutions of fully nonlinear equations, J. Differ. Equ., 258 (2015), 205-2079.  doi: 10.1016/j.jde.2014.11.022.

[25]

G. Lu and J. Zhu, Axial symmetry and regularity of solutions to an integral equation in a half-space, Pacific J. Math., 253 (2011), 455-473.  doi: 10.2140/pjm.2011.253.455.

[26]

P. Wang, Uniqueness and monotonicity of solutions for fractional equations with a gradient term, Electron. J. Qual. Theory Differ. Equ., 58 (2021), 1-19.  doi: 10.14232/ejqtde.2021.1.58.

[27]

L. Wu and W. Chen, Monotonicity of solutions for fractional equations with De Giorgi type nonlinearities, (in Chinese)., Sci. Sin. Math., 52 (2020), 1-22.  doi: 10.1360/SCM-2019-0668.

[28]

L. Wu and W. Chen, The sliding methods for the fractional $p$-Laplacian, Adv. Math., 361 (2020), 35 pp. doi: 10.1016/j.aim.2019.106933.

Figure 1.  Cylinder
Figure 2.  Subsolution
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