March  2021, 20(3): 1187-1198. doi: 10.3934/cpaa.2021012

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

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.

Citation: Xinqun Mei, Jundong Zhou. The interior gradient estimate of prescribed Hessian quotient curvature equation in the hyperbolic space. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1187-1198. 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. CaffarelliL. 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. ChenL. 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. GuanJ. 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. CaffarelliL. 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. ChenL. 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. GuanJ. 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

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