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

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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

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

Received September 2015 Revised March 2016 Published July 2016

Abstract
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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.
[2]	C. Gerhardt, Global regularity of the solutions to the capillary problem,, \emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci.}, 3 (1976), 157.
[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.
[4]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, 2$^{nd}$ edition, (2001).
[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.
[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.
[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.
[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.
[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.
[10]	G. M. Lieberman, Oblique Boundary Value Problems for Elliptic Equations,, World Scientific Publishing Co. Pte. Ltd., (2013). doi: 10.1142/8679.
[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.
[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.
[13]	J. Spruck, On the existence of a capillary surface with prescribed contact angle,, \emph{Comm. Pure Appl. Math.}, 28 (1975), 189.
[14]	N. S. Trudinger, The Dirichlet problem for the prescribed curvature equations,, \emph{Arch. Rational Mech. Anal.}, 111 (1990), 153. doi: 10.1007/BF00375406.
[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.
[16]	X. J. Wang, Interior gradient estimates for mean curvature equations,, \emph{Math.Z.}, 228 (1998), 73. 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,, \emph{Acta Math.}, 155 (1985), 261. doi: 10.1007/BF02392544.
[2]	C. Gerhardt, Global regularity of the solutions to the capillary problem,, \emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci.}, 3 (1976), 157.
[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.
[4]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, 2$^{nd}$ edition, (2001).
[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.
[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.
[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.
[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.
[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.
[10]	G. M. Lieberman, Oblique Boundary Value Problems for Elliptic Equations,, World Scientific Publishing Co. Pte. Ltd., (2013). doi: 10.1142/8679.
[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.
[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.
[13]	J. Spruck, On the existence of a capillary surface with prescribed contact angle,, \emph{Comm. Pure Appl. Math.}, 28 (1975), 189.
[14]	N. S. Trudinger, The Dirichlet problem for the prescribed curvature equations,, \emph{Arch. Rational Mech. Anal.}, 111 (1990), 153. doi: 10.1007/BF00375406.
[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.
[16]	X. J. Wang, Interior gradient estimates for mean curvature equations,, \emph{Math.Z.}, 228 (1998), 73. doi: 10.1007/PL00004604.

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