The interior gradient estimate for some nonlinear curvature equations

Liangjun Weng

doi:10.3934/cpaa.2019076

Article Contents

2019, Volume 18, Issue 4: 1601-1612. Doi: 10.3934/cpaa.2019076

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The interior gradient estimate for some nonlinear curvature equations

Liangjun Weng^1,2,

1.
School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, Anhui, China
2.
University of Freiburg, Freiburg im Breisgau 79104, Germany

Received: November 2017

Revised: August 2018

Published: January 2019

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Abstract

In this paper, we obtain the interior gradient estimate of some nonlinear equations which arise naturally from prescribed curvature problem of graphs in hyperbolic space. The method depends on the maximum principle.

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Mathematics Subject Classification: Primary: 35J60. Secondary: 35B45.

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References

[1]	B. Andrews and J. Clutterbuck, Time-interior gradient estimates for quasilinear parabolic equations, Indiana Univ. Math. J., 58 (2009), 351-380. doi: 10.1512/iumj.2009.58.3756.
[2]	L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. Ⅲ. functions of eigenvalues of the Hessian, Acta Math., 155 (1985), 261-301. doi: 10.1007/BF02392544.
[3]	L. Caffarelli, L. Nirenberg and J. Spruck, Nonlinear second order elliptic equations. Ⅳ. Starshaped compact Weingarten hypersurfaces, Current Topics in Partial Differential Equation, 1–26, Kinokuniya, Tokyo, 1986.
[4]	L. Caffarelli, L. Nirenberg and J. Spruck, Nonlinear second order elliptic equations. Ⅴ. The Dirichlet problem for Weingarten hypersurfaces, Comm. Pure Appl. Math., 41 (1988), 47-70. doi: 10.1002/cpa.3160410105.
[5]	K. S. Chou and X. J. Wang, A variational theory of the Hessian equation, Comm. Pure Appl. Math., 54 (2001), 1029-1064. doi: 10.1002/cpa.1016.
[6]	J. Clutterbuck, Interior gradient estimates for anisotropic mean-curvature flow, Pacific J. Math., 229 (2007), 119-136. doi: 10.2140/pjm.2007.229.119.
[7]	Tobias H. Colding and William P. II Minicozzi, Sharp estimates for mean curvature flow of graphs, J. Reine Angew. Math., 574 (2004), 187-195. doi: 10.1515/crll.2004.069.
[8]	B. Guan, Gradient estimates for solutions of nonparametric curvature evolution with prescribed contact angle condition, Monge Ampère equation: applications to geometry and optimization, Contemp. Math., 226 (1999), 105-112, Amer. Math. Soc., Providence, RI. doi: 10.1090/conm/226/03237.
[9]	B. Guan and J. Spruck, Interior gradient estimates for solutions of prescribed curvature equations of parabolic type, Indiana Univ. Math. J., 40 (1991), 1471-1481. doi: 10.1512/iumj.1991.40.40066.
[10]	B. Guan and J. Spruck, Hypersurfaces of constant mean curvature in hyperbolic space with prescribed asymptotic boundary at infinity, Amer. J. Math., 122 (2000), 1039-1060.
[11]	B. Guan, J. Spruck and M. Szapiel, Hypersurfaces of constant curvature in hyperbolic space Ⅰ, J. Geom. Anal., 19 (2009), 772-795. doi: 10.1007/s12220-009-9086-7.
[12]	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.
[13]	O. A. Ladyzhenskaya and N. Uraltseva, Local estimates for gradients of non-uniformly elliptic and parabolic equations, Comm. Pure Appl. Math., 23 (1970), 677-703. doi: 10.1002/cpa.3160230409.
[14]	G. Lieberman, Second Order Parabolic Differential Equations, World Scientific, 1996. doi: 10.1142/3302.
[15]	Y. Y. Li, Interior gradient estimates for solutions of certain fully nonlinear elliptic equations, J. Diff. Equ., 90 (1991), 172-185. doi: 10.1016/0022-0396(91)90166-7.
[16]	L. Z. Lin and L. Xiao, Modified mean curvature flow of star-shaped hypersurfaces in hyperbolic space, Comm. Anal. Geom., 20 (2012), 1061-1096. doi: 10.4310/CAG.2012.v20.n5.a6.
[17]	M. Lin and N. S. Trudinger, On some inequalities for elementary symmetric functions, Bull. Aust. Math. Soc., 50 (1994), 317-326. doi: 10.1017/S0004972700013770.
[18]	D. Serre, Gradient estimate in terms of a Hilbert-like distance, for minimal surfaces and Chaplygin gas, Comm. Partial Diff. Equ., 41 (2016), 774-784. doi: 10.1080/03605302.2015.1127969.
[19]	N. S. Trudinger, The Dirichlet problem for the prescribed curvature equations, Arch. Rational Mech. Anal., 111 (1990), 153-179. doi: 10.1007/BF00375406.
[20]	X. J. Wang, Interior gradient estimates for mean curvature equations, Math. Z., 228 (1998), 73-81. doi: 10.1007/PL00004604.
[21]	X. J. Wang, The k-Hessian equation. Geometric analysis and PDEs, Lecture Notes in Math. 1977, (2009) 177–252, Springer, Dordrecht. doi: 10.1007/978-3-642-01674-5_5.