Nagoya Mathematical Journal

On the Dirichlet problem of prescribed mean curvature equations without $H$-convexity condition

Kazuya Hayasida and Masao Nakatani

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The Dirichlet problem of prescribed mean curvature equations is well posed, if the boundery is H-convex. In this article we eliminate the H-convexity condition from a portion $\Gamma$ of the boundary and prove the existence theorem, where the boundary condition is satisfied on $\Gamma$ in the weak sense.

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Nagoya Math. J., Volume 157 (2000), 177-209.

First available in Project Euclid: 27 April 2005

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Zentralblatt MATH identifier

Primary: 35J60: Nonlinear elliptic equations
Secondary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42] 58E15: Application to extremal problems in several variables; Yang-Mills functionals [See also 81T13], etc.


Hayasida, Kazuya; Nakatani, Masao. On the Dirichlet problem of prescribed mean curvature equations without $H$-convexity condition. Nagoya Math. J. 157 (2000), 177--209.

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  • C. Gerhardt, Existence, regularity, and boundary behaviour of generalized surfaces of prescribed mean curvature , Math. Z., 139 (1974), 173–198.
  • M. Giaquinta, On the Dirichlet problem for surfaces of prescribed mean curvature , Manuscripta Math., 12 (1974), 73–86.
  • D. Gilberg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, New York (1983).
  • A. Giusti, Boundary value problems for non-parametric surfaces of prescribed mean curvature , Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 3 (1976), 501–548.
  • G. Gregori, Generalized solutions for a class of non-uniformly elliptic equations in divergence form , Comm. Partial Differential Equations, 22 (1997), 581–617.
  • N. M. Ivochkina, The Dirichlet problem for the equations of curvature of order $m$ , Leningrad Math. J., 2 (1991), 631–654.
  • H. Jenkins and J. Serrin, The Dirichlet problem for the minimal surface equation in higher dimensions , J. Reine Angew. Math., 229 (1968), 170–187.
  • Z. Jin and K. Lancaster, Behavior of solutions for some Dirichlet problems near reentrant corners , Indiana Univ. Math. J., 46 (1997), 827–862.
  • N. J. Korevaar and L. Simon, Continuity estimates for solutions to the prescribed- curvature Dirichlet problem , Math. Z., 197 (1987), 457–464.
  • O. A. Ladyzhenskaya and N. N. Ural'ceva, Local estimates for gradients of solutions of non-uniformly elliptic and parabolic equations , Comm. Pure Appl. Math., 23 (1970), 677–703.
  • ––––, Local estimates of the gradients of solution to a simplest regularization for some class of non uniformly elliptic equations , Zap. Nauchn. Sem. S.-Peterburg Otdel Mat. Inst. Steklov, 213 (1994), 75–92, (Russian).
  • K. E. Lancaster, Nonexistence of some nonparametric surfaces of prescribed mean curvature , Proc. Amer. Math. Soc., 96 (1986), 187–188.
  • C. P. Lau and F. H. Lin, The best Hölder exponent for solutions of the non-parametric least area problem , Indiana Univ. Math. J., 34 (1985), 809–813.
  • C. P. Lau, Quasilinear elliptic equations with small boundary data , Manuscripta Math., 53 (1985), 77–99.
  • A. Lichnewsky, Sur le comportement au bord des solutions généralisées du probl$\grave e$me non param$\acute e$trique des surfaces minimals , J. Math. Pures Appl., 53 (1974), 397–476.
  • F. Schultz and G. H. Williams, Barriers and existence results for a class of equations of mean curvature type , Analysis, 7 (1987), 359–374.
  • J. Serrin, The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables , Phil. Trans. Royal Soc. London, A 264 (1969), 413–496.
  • L. Simon, Interior gradient bounds for non-uniformly elliptic equations , Indiana Univ. Math. J., 25 (1976), 821–855.
  • ––––, Boundary behaviour of solutions of the non-parametric least area problem , Bull. Austral. Math. Soc., 26 (1982), 17–27.
  • R. Teman, Applications de l'analyse convexe au calcul des variations, Lecture Notes in Math., 543 , Springer (1975).
  • A. S. Tersenov, On quasilinear non-uniformly elliptic equations in some non-convex domains , Comm. Partial Differential Equations, 23 (1998), 2165–2185.
  • G. H. Williams, The Dirichlet problem for the minimal surface equation with Lipschitz continuous boundary data , J. Reine Anegew. Math., 354 (1984), 123–140.
  • ––––, Solutions of the minimal surface equation-continuous and discontinuous at the boundary , Comm. Partial Differential Equations, 11 (1986), 1439–1457.