American Institute of Mathematical Sciences

Advanced
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

Jinju Xu 1,

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.
Keywords: mean curvature equation, Gradient estimates, zero gravity., maximum principle, oblique boundary conditions.
Mathematics Subject Classification: Primary: 35B45, 35B50; Secondary: 35J9.
Citation: Jinju Xu. A new proof of gradient estimates for mean curvature equations with oblique boundary conditions. Communications on Pure & 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,, \emph{Acta Math.}, 155 (1985), 261.  doi: 10.1007/BF02392544.  Google Scholar

[2]

C. Gerhardt, Global regularity of the solutions to the capillary problem,, \emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci.}, 3 (1976), 157.   Google Scholar

[3]

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

[4]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, 2$^{nd}$ edition, (2001).   Google Scholar

[5]

N. J. Korevaar, Maximum principle gradient estimates for the capillary problem,, \emph{Comm. in Partial Differential Equations}, 13 (1988), 1.  doi: 10.1080/03605308808820536.  Google Scholar

[6]

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

[7]

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

[8]

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

[9]

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

[10]

G. M. Lieberman, Oblique Boundary Value Problems for Elliptic Equations,, World Scientific Publishing Co. Pte. Ltd., (2013).  doi: 10.1142/8679.  Google Scholar

[11]

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

[12]

L. Simon and J. Spruck, Existence and regularity of a capillary surface with prescribed contact angle,, \emph{Arch. Rational Mech. Anal.}, 61 (1976), 19.   Google Scholar

[13]

J. Spruck, On the existence of a capillary surface with prescribed contact angle,, \emph{Comm. Pure Appl. Math.}, 28 (1975), 189.   Google Scholar

[14]

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

[15]

N. N. Ural'tseva, The solvability of the capillary problem,, \emph{(Russian) Vestnik Leningrad. Univ. No. 19 Mat. Meh. Astronom.Vyp.}, 4 (1973), 54.   Google Scholar

[16]

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

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,, \emph{Acta Math.}, 155 (1985), 261.  doi: 10.1007/BF02392544.  Google Scholar

[2]

C. Gerhardt, Global regularity of the solutions to the capillary problem,, \emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci.}, 3 (1976), 157.   Google Scholar

[3]

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

[4]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, 2$^{nd}$ edition, (2001).   Google Scholar

[5]

N. J. Korevaar, Maximum principle gradient estimates for the capillary problem,, \emph{Comm. in Partial Differential Equations}, 13 (1988), 1.  doi: 10.1080/03605308808820536.  Google Scholar

[6]

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

[7]

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

[8]

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

[9]

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

[10]

G. M. Lieberman, Oblique Boundary Value Problems for Elliptic Equations,, World Scientific Publishing Co. Pte. Ltd., (2013).  doi: 10.1142/8679.  Google Scholar

[11]

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

[12]

L. Simon and J. Spruck, Existence and regularity of a capillary surface with prescribed contact angle,, \emph{Arch. Rational Mech. Anal.}, 61 (1976), 19.   Google Scholar

[13]

J. Spruck, On the existence of a capillary surface with prescribed contact angle,, \emph{Comm. Pure Appl. Math.}, 28 (1975), 189.   Google Scholar

[14]

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

[15]

N. N. Ural'tseva, The solvability of the capillary problem,, \emph{(Russian) Vestnik Leningrad. Univ. No. 19 Mat. Meh. Astronom.Vyp.}, 4 (1973), 54.   Google Scholar

[16]

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

[1]

Vandana Sharma. Global existence and uniform estimates of solutions to reaction diffusion systems with mass transport type boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021001
[2]

Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048
[3]

Petr Pauš, Shigetoshi Yazaki. Segmentation of color images using mean curvature flow and parametric curves. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1123-1132. doi: 10.3934/dcdss.2020389
[4]

Huyuan Chen, Dong Ye, Feng Zhou. On gaussian curvature equation in $ \mathbb{R}^2 $ with prescribed nonpositive curvature. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3201-3214. doi: 10.3934/dcds.2020125
[5]

Gabrielle Nornberg, Delia Schiera, Boyan Sirakov. A priori estimates and multiplicity for systems of elliptic PDE with natural gradient growth. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3857-3881. doi: 10.3934/dcds.2020128
[6]

Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020110
[7]

Sihem Guerarra. Maximum and minimum ranks and inertias of the Hermitian parts of the least rank solution of the matrix equation AXB = C. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 75-86. doi: 10.3934/naco.2020016
[8]

Roland Schnaubelt, Martin Spitz. Local wellposedness of quasilinear Maxwell equations with absorbing boundary conditions. Evolution Equations & Control Theory, 2021, 10 (1) : 155-198. doi: 10.3934/eect.2020061
[9]

Kuntal Bhandari, Franck Boyer. Boundary null-controllability of coupled parabolic systems with Robin conditions. Evolution Equations & Control Theory, 2021, 10 (1) : 61-102. doi: 10.3934/eect.2020052
[10]

Qianqian Hou, Tai-Chia Lin, Zhi-An Wang. On a singularly perturbed semi-linear problem with Robin boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 401-414. doi: 10.3934/dcdsb.2020083
[11]

Wenrui Hao, King-Yeung Lam, Yuan Lou. Ecological and evolutionary dynamics in advective environments: Critical domain size and boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 367-400. doi: 10.3934/dcdsb.2020283
[12]

Franck Davhys Reval Langa, Morgan Pierre. A doubly splitting scheme for the Caginalp system with singular potentials and dynamic boundary conditions. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 653-676. doi: 10.3934/dcdss.2020353
[13]

Peter Frolkovič, Karol Mikula, Jooyoung Hahn, Dirk Martin, Branislav Basara. Flux balanced approximation with least-squares gradient for diffusion equation on polyhedral mesh. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 865-879. doi: 10.3934/dcdss.2020350
[14]

Md. Masum Murshed, Kouta Futai, Masato Kimura, Hirofumi Notsu. Theoretical and numerical studies for energy estimates of the shallow water equations with a transmission boundary condition. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1063-1078. doi: 10.3934/dcdss.2020230
[15]

Kazunori Matsui. Sharp consistency estimates for a pressure-Poisson problem with Stokes boundary value problems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1001-1015. doi: 10.3934/dcdss.2020380
[16]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247
[17]

Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, 2021, 20 (1) : 301-317. doi: 10.3934/cpaa.2020267
[18]

Amru Hussein, Martin Saal, Marc Wrona. Primitive equations with horizontal viscosity: The initial value and The time-periodic problem for physical boundary conditions. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020398
[19]

Fang Li, Bo You. On the dimension of global attractor for the Cahn-Hilliard-Brinkman system with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021024
[20]

Jan Březina, Eduard Feireisl, Antonín Novotný. On convergence to equilibria of flows of compressible viscous fluids under in/out–flux boundary conditions. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021009

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (55)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences