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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 |
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.
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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