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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

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  March 2021 Early access  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 and 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.

[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.

[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.

[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.

[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.

[6]

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

[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.

[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.

[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. 

[10]

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

[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.

[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.

[13]

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

[14]

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

[15]

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

[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.

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.

[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.

[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.

[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.

[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.

[6]

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

[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.

[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.

[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. 

[10]

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

[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.

[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.

[13]

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

[14]

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

[15]

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

[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.

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