September  2016, 15(5): 1719-1742. doi: 10.3934/cpaa.2016010

A new proof of gradient estimates for mean curvature equations with oblique boundary conditions

1. 

University of Science and Technology of China, Hefei Anhui, 230026, China

Received  September 2015 Revised  March 2016 Published  July 2016

In this paper, we will use the maximum principle to give a new proof of the gradient estimates for mean curvature equations with some oblique derivative problems. In particular, we shall give a new proof for the capillary problem with zero gravity.
Citation: Jinju Xu. A new proof of gradient estimates for mean curvature equations with oblique boundary conditions. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1719-1742. doi: 10.3934/cpaa.2016010
References:
[1]

L. A. 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.

[2]

C. Gerhardt, Global regularity of the solutions to the capillary problem, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 3 (1976), 157-175.

[3]

P. F. Guan and X. N. Ma, The Christoffel-Minkowski problem I: Convexity of solutions of a Hessian equations, Invent. Math., 151 (2003), 553-577. doi: 10.1007/s00222-002-0259-2.

[4]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edition, Springer-Verlag Berlin, 2001,xiv+517 pp.

[5]

N. J. Korevaar, Maximum principle gradient estimates for the capillary problem, Comm. in Partial Differential Equations, 13 (1988), 1-31. doi: 10.1080/03605308808820536.

[6]

G. M. Lieberman, The conormal derivative problem for elliptic equations of variational type, J.Differential Equations, 49 (1983), 218-257. doi: 10.1016/0022-0396(83)90013-X.

[7]

G. M. Lieberman, The nonlinear oblique derivative problem for quasilinear elliptic equations, Nonlinear Analysis. Theory. Method $ & $ Applications, 8 (1984), 49-65. doi: 10.1016/0362-546X(84)90027-0.

[8]

G. M. Lieberman, Gradient bounds for solutions of nonuniformly elliptic oblique derivative problems, Nonlinear Anal., 11 (1987), 49-61. doi: 10.1016/0362-546X(87)90025-3.

[9]

G. M. Lieberman, Gradient estimates for capillary-type problems via the maximum principle, Commun. in Partial Differential Equations, 13 (1988), 33-59. doi: 10.1080/03605308808820537.

[10]

G. M. Lieberman, Oblique Boundary Value Problems for Elliptic Equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013. xvi+509 pp. doi: 10.1142/8679.

[11]

X. N. Ma and J. J. Xu, Gradient estimates of mean curvature equations with Neumann boundary condition, Advances in Mathematics, 290 (2016), 1010-1039. doi: 10.1016/j.aim.2015.10.031.

[12]

L. Simon and J. Spruck, Existence and regularity of a capillary surface with prescribed contact angle, Arch. Rational Mech. Anal., 61 (1976), 19-34.

[13]

J. Spruck, On the existence of a capillary surface with prescribed contact angle, Comm. Pure Appl. Math., 28 (1975), 189-200.

[14]

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

[15]

N. N. Ural'tseva, The solvability of the capillary problem, (Russian) Vestnik Leningrad. Univ. No. 19 Mat. Meh. Astronom.Vyp., 4 (1973), 54-64.

[16]

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

show all references

References:
[1]

L. A. 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.

[2]

C. Gerhardt, Global regularity of the solutions to the capillary problem, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 3 (1976), 157-175.

[3]

P. F. Guan and X. N. Ma, The Christoffel-Minkowski problem I: Convexity of solutions of a Hessian equations, Invent. Math., 151 (2003), 553-577. doi: 10.1007/s00222-002-0259-2.

[4]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edition, Springer-Verlag Berlin, 2001,xiv+517 pp.

[5]

N. J. Korevaar, Maximum principle gradient estimates for the capillary problem, Comm. in Partial Differential Equations, 13 (1988), 1-31. doi: 10.1080/03605308808820536.

[6]

G. M. Lieberman, The conormal derivative problem for elliptic equations of variational type, J.Differential Equations, 49 (1983), 218-257. doi: 10.1016/0022-0396(83)90013-X.

[7]

G. M. Lieberman, The nonlinear oblique derivative problem for quasilinear elliptic equations, Nonlinear Analysis. Theory. Method $ & $ Applications, 8 (1984), 49-65. doi: 10.1016/0362-546X(84)90027-0.

[8]

G. M. Lieberman, Gradient bounds for solutions of nonuniformly elliptic oblique derivative problems, Nonlinear Anal., 11 (1987), 49-61. doi: 10.1016/0362-546X(87)90025-3.

[9]

G. M. Lieberman, Gradient estimates for capillary-type problems via the maximum principle, Commun. in Partial Differential Equations, 13 (1988), 33-59. doi: 10.1080/03605308808820537.

[10]

G. M. Lieberman, Oblique Boundary Value Problems for Elliptic Equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013. xvi+509 pp. doi: 10.1142/8679.

[11]

X. N. Ma and J. J. Xu, Gradient estimates of mean curvature equations with Neumann boundary condition, Advances in Mathematics, 290 (2016), 1010-1039. doi: 10.1016/j.aim.2015.10.031.

[12]

L. Simon and J. Spruck, Existence and regularity of a capillary surface with prescribed contact angle, Arch. Rational Mech. Anal., 61 (1976), 19-34.

[13]

J. Spruck, On the existence of a capillary surface with prescribed contact angle, Comm. Pure Appl. Math., 28 (1975), 189-200.

[14]

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

[15]

N. N. Ural'tseva, The solvability of the capillary problem, (Russian) Vestnik Leningrad. Univ. No. 19 Mat. Meh. Astronom.Vyp., 4 (1973), 54-64.

[16]

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

[1]

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

[2]

Alberto Boscaggin, Francesca Colasuonno, Benedetta Noris. Positive radial solutions for the Minkowski-curvature equation with Neumann boundary conditions. Discrete and Continuous Dynamical Systems - S, 2020, 13 (7) : 1921-1933. doi: 10.3934/dcdss.2020150

[3]

Zhengchao Ji. Cylindrical estimates for mean curvature flow in hyperbolic spaces. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1199-1211. doi: 10.3934/cpaa.2021016

[4]

Tae Gab Ha. Global existence and general decay estimates for the viscoelastic equation with acoustic boundary conditions. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6899-6919. doi: 10.3934/dcds.2016100

[5]

Maria Francesca Betta, Rosaria Di Nardo, Anna Mercaldo, Adamaria Perrotta. Gradient estimates and comparison principle for some nonlinear elliptic equations. Communications on Pure and Applied Analysis, 2015, 14 (3) : 897-922. doi: 10.3934/cpaa.2015.14.897

[6]

Dejian Chang, Zhen Wu. Stochastic maximum principle for non-zero sum differential games of FBSDEs with impulse controls and its application to finance. Journal of Industrial and Management Optimization, 2015, 11 (1) : 27-40. doi: 10.3934/jimo.2015.11.27

[7]

Ha Tuan Dung, Nguyen Thac Dung, Jiayong Wu. Sharp gradient estimates on weighted manifolds with compact boundary. Communications on Pure and Applied Analysis, 2021, 20 (12) : 4127-4138. doi: 10.3934/cpaa.2021148

[8]

Hancheng Guo, Jie Xiong. A second-order stochastic maximum principle for generalized mean-field singular control problem. Mathematical Control and Related Fields, 2018, 8 (2) : 451-473. doi: 10.3934/mcrf.2018018

[9]

Tian Chen, Zhen Wu. A general maximum principle for partially observed mean-field stochastic system with random jumps in progressive structure. Mathematical Control and Related Fields, 2022  doi: 10.3934/mcrf.2022012

[10]

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

[11]

Chao Zhang, Xia Zhang, Shulin Zhou. Gradient estimates for the strong $p(x)$-Laplace equation. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 4109-4129. doi: 10.3934/dcds.2017175

[12]

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

[13]

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 and Applied Analysis, 2012, 11 (5) : 1983-2003. doi: 10.3934/cpaa.2012.11.1983

[14]

Zixiao Liu, Jiguang Bao. Asymptotic expansion of 2-dimensional gradient graph with vanishing mean curvature at infinity. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022081

[15]

Dimitra Antonopoulou, Georgia Karali. A nonlinear partial differential equation for the volume preserving mean curvature flow. Networks and Heterogeneous Media, 2013, 8 (1) : 9-22. doi: 10.3934/nhm.2013.8.9

[16]

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

[17]

Chiara Corsato, Colette De Coster, Pierpaolo Omari. Radially symmetric solutions of an anisotropic mean curvature equation modeling the corneal shape. Conference Publications, 2015, 2015 (special) : 297-303. doi: 10.3934/proc.2015.0297

[18]

Elias M. Guio, Ricardo Sa Earp. Existence and non-existence for a mean curvature equation in hyperbolic space. Communications on Pure and Applied Analysis, 2005, 4 (3) : 549-568. doi: 10.3934/cpaa.2005.4.549

[19]

Adel Chala, Dahbia Hafayed. On stochastic maximum principle for risk-sensitive of fully coupled forward-backward stochastic control of mean-field type with application. Evolution Equations and Control Theory, 2020, 9 (3) : 817-843. doi: 10.3934/eect.2020035

[20]

Eun Heui Kim. Boundary gradient estimates for subsonic solutions of compressible transonic potential flows. Conference Publications, 2007, 2007 (Special) : 573-579. doi: 10.3934/proc.2007.2007.573

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (156)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]