American Institute of Mathematical Sciences

Advanced
doi: 10.3934/cpaa.2021012

The interior gradient estimate of prescribed Hessian quotient curvature equation in the hyperbolic space

Xinqun Mei 1,, and  Jundong Zhou 1,2,

1. 

School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, Anhui Province, China
2. 

School of Mathematics and Statistics, Fuyang Normal University, Fuyang 236037, Anhui Province, China

* Corresponding author

Received  August 2020 Revised  December 2020 Published  February 2021

In this paper, we obtain the interior gradient estimate of the Hessian quotient curvature equation in the hyperbolic space. The method depends on the maximum principle.

Keywords: Interior gradient estimate, Hessian quotient curvature equation, maximum principle.
Mathematics Subject Classification: Primary:35J60;Secondary:35B45.
Citation: Xinqun Mei, Jundong Zhou. The interior gradient estimate of prescribed Hessian quotient curvature equation in the hyperbolic space. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021012
References:
[1]

L. A. Caffarelli, L. Nirenberg and J. Spruck, Nonlinear second order elliptic equations IV: Starshaped compact Weigarten hypersurfaces, in Current topics in partial differential equations, Kinokunize, Tokyo, 1985.  Google Scholar

[2]

L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second order elliptic equations III, Functions of the eigenvalues of the Hessian, Acta Math., 155 (1985), 261-301.  doi: 10.1007/BF02392544.  Google Scholar

[3]

C. Q. Chen, The interior gradient estimate of Hessian quotient equations, J. Differ. Equ., 259 (2015), 1014-1023.  doi: 10.1016/j.jde.2015.02.035.  Google Scholar

[4]

C. Q. Chen, L. Xu and D. k. Zhang, The interior gradient estimate of prescribed Hessian quotient curvature equations, manuscripta mathematica, 153 (2016), 1-13.  doi: 10.1007/s00229-016-0877-4.  Google Scholar

[5]

K. S. Chou and X. J. Wang, A variation theory of the Hessian equation., Commun. Pure Appl. Math., 54 (2001), 1029-1064.  doi: 10.1002/cpa.1016.  Google Scholar

[6]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[7]

B. Guan and J. Spruck, Hypersurfaces of constant mean curvature in hyperbolic space with prescribed asymptotic boundary at infinity, Am. J. Math., 122 (2000), 1039–1060.  Google Scholar

[8]

B. Guan, J. Spruck and M. Szapiel, Hypersurfaces of constant curvature in hyperbolic space, J. Geom. Anal., 19 (2009), 772-795.  doi: 10.1007/s12220-009-9086-7.  Google Scholar

[9]

N. J. Korevaar, A priori interior gradient bounds for solutions to elliptic Weingarten equations, Ann. Inst. H. Poincaré, Anal. Non linéaire, 4 (1987), 405-421.   Google Scholar

[10]

G. Lieberman, Second order parabolic differential equations, World Scientific, 1996. doi: 10.1142/3302.  Google Scholar

[11]

Y. Y. Li, Interior gradient estimates for solutions of certain fully nonlinear elliptic equations, J. Differ. Equ., 90 (1991), 172-185.  doi: 10.1016/0022-0396(91)90166-7.  Google Scholar

[12]

M. Lin and N. Trudinger, On some inequalities for elementary symmetric functions, Bull. Austral. Math. Soc., 50 (1994), 317-326.  doi: 10.1017/S0004972700013770.  Google Scholar

[13]

J. Spruck, Geometric aspects of the theory of fully nonlinear elliptic equations, Clay Mathematics Proceedings, 2 (2005), 283-309.   Google Scholar

[14]

N. S. Trudinger, The Dirichlet problem for the precribed curvature equations, Arch. Ration. Mech. Anal., 111 (1990), 152-179.  doi: 10.1007/BF00375406.  Google Scholar

[15]

X. J. Wang, Interior gradient estimates for mean curvature equations, Math. Z., 228 (1998), 73-81.  doi: 10.1007/PL00004604.  Google Scholar

[16]

L. Weng, The interior gradient estimate for some nonlinear curvature equations, Commun. Pure Appl. Anal., 18 (2019), 1601-1612.  doi: 10.3934/cpaa.2019076.  Google Scholar

show all references

References:
[1]

L. A. Caffarelli, L. Nirenberg and J. Spruck, Nonlinear second order elliptic equations IV: Starshaped compact Weigarten hypersurfaces, in Current topics in partial differential equations, Kinokunize, Tokyo, 1985.  Google Scholar

[2]

L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second order elliptic equations III, Functions of the eigenvalues of the Hessian, Acta Math., 155 (1985), 261-301.  doi: 10.1007/BF02392544.  Google Scholar

[3]

C. Q. Chen, The interior gradient estimate of Hessian quotient equations, J. Differ. Equ., 259 (2015), 1014-1023.  doi: 10.1016/j.jde.2015.02.035.  Google Scholar

[4]

C. Q. Chen, L. Xu and D. k. Zhang, The interior gradient estimate of prescribed Hessian quotient curvature equations, manuscripta mathematica, 153 (2016), 1-13.  doi: 10.1007/s00229-016-0877-4.  Google Scholar

[5]

K. S. Chou and X. J. Wang, A variation theory of the Hessian equation., Commun. Pure Appl. Math., 54 (2001), 1029-1064.  doi: 10.1002/cpa.1016.  Google Scholar

[6]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[7]

B. Guan and J. Spruck, Hypersurfaces of constant mean curvature in hyperbolic space with prescribed asymptotic boundary at infinity, Am. J. Math., 122 (2000), 1039–1060.  Google Scholar

[8]

B. Guan, J. Spruck and M. Szapiel, Hypersurfaces of constant curvature in hyperbolic space, J. Geom. Anal., 19 (2009), 772-795.  doi: 10.1007/s12220-009-9086-7.  Google Scholar

[9]

N. J. Korevaar, A priori interior gradient bounds for solutions to elliptic Weingarten equations, Ann. Inst. H. Poincaré, Anal. Non linéaire, 4 (1987), 405-421.   Google Scholar

[10]

G. Lieberman, Second order parabolic differential equations, World Scientific, 1996. doi: 10.1142/3302.  Google Scholar

[11]

Y. Y. Li, Interior gradient estimates for solutions of certain fully nonlinear elliptic equations, J. Differ. Equ., 90 (1991), 172-185.  doi: 10.1016/0022-0396(91)90166-7.  Google Scholar

[12]

M. Lin and N. Trudinger, On some inequalities for elementary symmetric functions, Bull. Austral. Math. Soc., 50 (1994), 317-326.  doi: 10.1017/S0004972700013770.  Google Scholar

[13]

J. Spruck, Geometric aspects of the theory of fully nonlinear elliptic equations, Clay Mathematics Proceedings, 2 (2005), 283-309.   Google Scholar

[14]

N. S. Trudinger, The Dirichlet problem for the precribed curvature equations, Arch. Ration. Mech. Anal., 111 (1990), 152-179.  doi: 10.1007/BF00375406.  Google Scholar

[15]

X. J. Wang, Interior gradient estimates for mean curvature equations, Math. Z., 228 (1998), 73-81.  doi: 10.1007/PL00004604.  Google Scholar

[16]

L. Weng, The interior gradient estimate for some nonlinear curvature equations, Commun. Pure Appl. Anal., 18 (2019), 1601-1612.  doi: 10.3934/cpaa.2019076.  Google Scholar

[1]

Ka Luen Cheung, Man Chun Leung. Asymptotic behavior of positive solutions of the equation $ \Delta u + K u^{\frac{n+2}{n-2}} = 0$ in $IR^n$ and positive scalar curvature. Conference Publications, 2001, 2001 (Special) : 109-120. doi: 10.3934/proc.2001.2001.109
[2]

Olena Naboka. On synchronization of oscillations of two coupled Berger plates with nonlinear interior damping. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1933-1956. doi: 10.3934/cpaa.2009.8.1933
[3]

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
[4]

Tuan Hiep Pham, Jérôme Laverne, Jean-Jacques Marigo. Stress gradient effects on the nucleation and propagation of cohesive cracks. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 557-584. doi: 10.3934/dcdss.2016012
[5]

Matthias Erbar, Jan Maas. Gradient flow structures for discrete porous medium equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1355-1374. doi: 10.3934/dcds.2014.34.1355
[6]

Feng Luo. A combinatorial curvature flow for compact 3-manifolds with boundary. Electronic Research Announcements, 2005, 11: 12-20.

[7]

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
[8]

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
[9]

Guido De Philippis, Antonio De Rosa, Jonas Hirsch. The area blow up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7031-7056. doi: 10.3934/dcds.2019243
[10]

Hong Seng Sim, Wah June Leong, Chuei Yee Chen, Siti Nur Iqmal Ibrahim. Multi-step spectral gradient methods with modified weak secant relation for large scale unconstrained optimization. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 377-387. doi: 10.3934/naco.2018024
[11]

Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068
[12]

Daoyin He, Ingo Witt, Huicheng Yin. On the strauss index of semilinear tricomi equation. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4817-4838. doi: 10.3934/cpaa.2020213
[13]

Min Li. A three term Polak-Ribière-Polyak conjugate gradient method close to the memoryless BFGS quasi-Newton method. Journal of Industrial & Management Optimization, 2020, 16 (1) : 245-260. doi: 10.3934/jimo.2018149
[14]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437
[15]

Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109
[16]

Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25
[17]

Jumpei Inoue, Kousuke Kuto. On the unboundedness of the ratio of species and resources for the diffusive logistic equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2441-2450. doi: 10.3934/dcdsb.2020186
[18]

Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309
[19]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258
[20]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1649-1672. doi: 10.3934/dcdss.2020448

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (17)
  • HTML views (40)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences