Advanced Search
Article Contents
Article Contents

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

Abstract Related Papers Cited by
  • 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.
    Mathematics Subject Classification: Primary: 35B45, 35B50; Secondary: 35J92.


    \begin{equation} \\ \end{equation}
  • [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.


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


    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.


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


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


    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.


    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.


    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.


    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.


    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.


    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.


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


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


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


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


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

  • 加载中

Article Metrics

HTML views() PDF downloads(175) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint