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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

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