• Previous Article
    On the existence of solutions and causality for relativistic viscous conformal fluids
  • CPAA Home
  • This Issue
  • Next Article
    Effects of localized spatial variations on the uniform persistence and spreading speeds of time periodic two species competition systems
July  2019, 18(4): 1601-1612. doi: 10.3934/cpaa.2019076

The interior gradient estimate for some nonlinear curvature equations

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

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.

Citation: Liangjun Weng. The interior gradient estimate for some nonlinear curvature equations. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1601-1612. doi: 10.3934/cpaa.2019076
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. Google Scholar

[2]

L. CaffarelliL. 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. Google Scholar

[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. Google Scholar

[4]

L. CaffarelliL. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[11]

B. GuanJ. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

G. Lieberman, Second Order Parabolic Differential Equations, World Scientific, 1996. doi: 10.1142/3302. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

N. S. Trudinger, The Dirichlet problem for the prescribed curvature equations, Arch. Rational Mech. Anal., 111 (1990), 153-179. doi: 10.1007/BF00375406. Google Scholar

[20]

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

[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. Google Scholar

show all references

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. Google Scholar

[2]

L. CaffarelliL. 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. Google Scholar

[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. Google Scholar

[4]

L. CaffarelliL. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[11]

B. GuanJ. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

G. Lieberman, Second Order Parabolic Differential Equations, World Scientific, 1996. doi: 10.1142/3302. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

N. S. Trudinger, The Dirichlet problem for the prescribed curvature equations, Arch. Rational Mech. Anal., 111 (1990), 153-179. doi: 10.1007/BF00375406. Google Scholar

[20]

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

[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. Google Scholar

[1]

Diego Castellaneta, Alberto Farina, Enrico Valdinoci. A pointwise gradient estimate for solutions of singular and degenerate pde's in possibly unbounded domains with nonnegative mean curvature. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1983-2003. doi: 10.3934/cpaa.2012.11.1983

[2]

H. O. Fattorini. The maximum principle in infinite dimension. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 557-574. doi: 10.3934/dcds.2000.6.557

[3]

Brian D. O. Anderson, Shaoshuai Mou, A. Stephen Morse, Uwe Helmke. Decentralized gradient algorithm for solution of a linear equation. Numerical Algebra, Control & Optimization, 2016, 6 (3) : 319-328. doi: 10.3934/naco.2016014

[4]

Carlo Orrieri. A stochastic maximum principle with dissipativity conditions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5499-5519. doi: 10.3934/dcds.2015.35.5499

[5]

Chiara Corsato, Franco Obersnel, Pierpaolo Omari, Sabrina Rivetti. On the lower and upper solution method for the prescribed mean curvature equation in Minkowski space. Conference Publications, 2013, 2013 (special) : 159-169. doi: 10.3934/proc.2013.2013.159

[6]

Torsten Lindström. Discrete models and Fisher's maximum principle in ecology. Conference Publications, 2003, 2003 (Special) : 571-579. doi: 10.3934/proc.2003.2003.571

[7]

Dan Mangoubi. A gradient estimate for harmonic functions sharing the same zeros. Electronic Research Announcements, 2014, 21: 62-71. doi: 10.3934/era.2014.21.62

[8]

Futoshi Takahashi. On the number of maximum points of least energy solution to a two-dimensional Hénon equation with large exponent. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1237-1241. doi: 10.3934/cpaa.2013.12.1237

[9]

N. A. Chernyavskaya, L. A. Shuster. Spaces admissible for the Sturm-Liouville equation. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1023-1052. doi: 10.3934/cpaa.2018050

[10]

Mikko Kaasalainen. Multimodal inverse problems: Maximum compatibility estimate and shape reconstruction. Inverse Problems & Imaging, 2011, 5 (1) : 37-57. doi: 10.3934/ipi.2011.5.37

[11]

Timothy Blass, Rafael De La Llave, Enrico Valdinoci. A comparison principle for a Sobolev gradient semi-flow. Communications on Pure & Applied Analysis, 2011, 10 (1) : 69-91. doi: 10.3934/cpaa.2011.10.69

[12]

Frédéric Abergel, Jean-Michel Rakotoson. Gradient blow-up in Zygmund spaces for the very weak solution of a linear elliptic equation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1809-1818. doi: 10.3934/dcds.2013.33.1809

[13]

John Sylvester. An estimate for the free Helmholtz equation that scales. Inverse Problems & Imaging, 2009, 3 (2) : 333-351. doi: 10.3934/ipi.2009.3.333

[14]

Chiun-Chuan Chen, Li-Chang Hung, Hsiao-Feng Liu. N-barrier maximum principle for degenerate elliptic systems and its application. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 791-821. doi: 10.3934/dcds.2018034

[15]

Yunkyong Hyon, Do Young Kwak, Chun Liu. Energetic variational approach in complex fluids: Maximum dissipation principle. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1291-1304. doi: 10.3934/dcds.2010.26.1291

[16]

Zaidong Zhan, Shuping Chen, Wei Wei. A unified theory of maximum principle for continuous and discrete time optimal control problems. Mathematical Control & Related Fields, 2012, 2 (2) : 195-215. doi: 10.3934/mcrf.2012.2.195

[17]

Yan Wang, Yanxiang Zhao, Lei Wang, Aimin Song, Yanping Ma. Stochastic maximum principle for partial information optimal investment and dividend problem of an insurer. Journal of Industrial & Management Optimization, 2018, 14 (2) : 653-671. doi: 10.3934/jimo.2017067

[18]

Shanjian Tang. A second-order maximum principle for singular optimal stochastic controls. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1581-1599. doi: 10.3934/dcdsb.2010.14.1581

[19]

H. O. Fattorini. The maximum principle for linear infinite dimensional control systems with state constraints. Discrete & Continuous Dynamical Systems - A, 1995, 1 (1) : 77-101. doi: 10.3934/dcds.1995.1.77

[20]

Isabeau Birindelli, Francoise Demengel. Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators. Communications on Pure & Applied Analysis, 2007, 6 (2) : 335-366. doi: 10.3934/cpaa.2007.6.335

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (80)
  • HTML views (164)
  • Cited by (0)

Other articles
by authors

[Back to Top]