Taiwanese Journal of Mathematics

Boundary Continuity of Nonparametric Prescribed Mean Curvature Surfaces

Mozhgan Nora Entekhabi and Kirk E. Lancaster

Full-text: Open access


We investigate the boundary behavior of variational solutions of Dirichlet problems for prescribed mean curvature equations at smooth boundary points where certain boundary curvature conditions are satisfied (which preclude the existence of local barrier functions). We prove that if the Dirichlet boundary data $\phi$ is continuous at such a point (and possibly nowhere else), then the solution of the variational problem is continuous at this point.

Article information

Taiwanese J. Math., Volume 24, Number 2 (2020), 483-499.

Received: 19 December 2018
Revised: 28 April 2019
Accepted: 8 May 2019
First available in Project Euclid: 16 May 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J67: Boundary values of solutions to elliptic equations
Secondary: 35J93: Quasilinear elliptic equations with mean curvature operator 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42]

prescribed mean curvature Dirichlet problem boundary continuity


Entekhabi, Mozhgan Nora; Lancaster, Kirk E. Boundary Continuity of Nonparametric Prescribed Mean Curvature Surfaces. Taiwanese J. Math. 24 (2020), no. 2, 483--499. doi:10.11650/tjm/190504. https://projecteuclid.org/euclid.twjm/1557972013

Export citation


  • M. Bergner, On the Dirichlet problem for the prescribed mean curvature equation over general domains, Differential Geom. Appl. 27 (2009), no. 3, 335–343.
  • T. Bourni, $C^{1,\alpha}$ theory for the prescribed mean curvature equation with Dirichlet data, J. Geom. Anal. 21 (2011), no. 4, 982–1035.
  • P. Concus and R. Finn, On capillary free surfaces in the absence of gravity, Acta Math. 132 (1974), 177–198.
  • R. Courant, Dirichlet's Principle, Conformal Mapping, and Minimal Surfaces, Interscience, New York, 1950.
  • A. R. Elcrat and K. E. Lancaster, Boundary behavior of a nonparametric surface of prescribed mean curvature near a reentrant corner, Trans. Amer. Math. Soc. 297 (1986), no. 2, 645–650.
  • ––––, Bernstein functions and the Dirichlet problem, SIAM J. Math. Anal. 20 (1989), no. 5, 1055–1068.
  • M. Entekhabi and K. Lancaster, Radial limits of bounded nonparametric prescribed mean curvature surfaces, Pacific J. Math. 283 (2016), no. 2, 341–351.
  • R. Finn, Equilibrium Capillary Surfaces, Grundlehren der Mathematischen Wissenschaften 284, Springer-Verlag, New York, 1986.
  • ––––, Moon surfaces, and boundary behaviour of capillary surfaces for perfect wetting and non-wetting, Proc. Lond. Math. Soc. s3-57 (1988), no. 3, 542–576.
  • D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, (revised second edition), Grundlehren der Mathematischen Wissenschaften 224, Springer-Verlag, New York, 1998.
  • K. Hayasida and M. Nakatani, On the Dirichlet problem of prescribed mean curvature equations without $H$-convexity condition, Nagoya Math. J 157 (2000), 177–209.
  • 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, Theorems of Phragmén-Lindelöf type for quasilinear elliptic equations, J. Reine Angew. Math. 514 (1999), 165–197.
  • A. Korn, Über Minimalflächen, deren Randkurven wenig von ebenen Kurven abweichen [On minimal surfaces whose boundary curves deviate little from plane curves], Abh. Königl. Preuss. Akad. Wiss., Berlin, Phys.-Math. Cl. II (1909), 1–37.
  • K. E. Lancaster, Existence and nonexistence of radial limits of minimal surfaces, Proc. Amer. Math. Soc. 106 (1989), no. 3, 757–762.
  • K. E. Lancaster and D. Siegel, Existence and behavior of the radial limits of a bounded capillary surface at a corner, Pacific J. Math. 176 (1996), no. 1, 165–194. Correction to: “Existence and behavior of the radial limits of a bounded capillary surface at a corner", Pacific J. Math. 179 (1997), no. 2, 397–402.
  • C. P. Lau, Quasilinear elliptic equations with small boundary data, Manuscripta Math. 53 (1985), no. 1-2, 77–99.
  • F.-T. Liang, An absolute gradient bound for nonparametric surfaces of constant mean curvature, Indiana Univ. Math. J. 41 (1992), no. 3, 569–604.
  • ––––, Moon hypersurfaces and some related existence results of capillary hypersurfaces without gravity and of rotational symmetry, Pacific J. Math. 172 (1996), no. 2, 433–460.
  • F.-H. Lin, Behaviour of nonparametric solutions and free boundary regularity, in: Miniconference on Geometry and Partial Differential Equations 2 (Canberra, 1986), 96–116, Proc. Centre Math. Aanl. Austral. Nat. Univ. 12, Austral. Nat. Univ., Canberra, 1987.
  • J. C. C. Nitsche, Lectures on Minimal Surfaces I, Cambridge University Press, Cambridge, 1989.
  • J. Ripoll and F. Tomi, On solutions to the exterior Dirichlet problem for the minimal surface equation with catenoidal ends, Adv. Calc. Var. 7 (2014), no. 2, 205–226.
  • J. Serrin, The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables, Philos. Trans. Roy. Soc. London Ser. A 264 (1969), 413–496.
  • L. Simon, Boundary regularity for solutions of the non-parametric least area problem, Ann. of Math. (2) 103 (1976), no. 3, 429–455.
  • G. H. Williams, The Dirichlet problem for the minimal surface equation with Lipschitz continuous boundary data, J. Reine Angew. Math. 354 (1984), 123–140.
  • ––––, The Dirichlet problem for the minimal surface equation, in: Miniconference on Nonlinear Analysis (Canberra, 1984), 233–239, Proc. Centre Math. Anal. Austral. Nat. Univ. 8, Austral. Nat. Univ., Canberra, 1984.