Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-24T08:58:09.206Z Has data issue: false hasContentIssue false
  • English
  • Français

A Phragmén–Lindelöf principle for the equation of a surface of constant mean curvature

Published online by Cambridge University Press:  14 November 2011

R. J. Knops  and
L. E. Payne
R. J. Knops
Affiliation:
Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, U.K.
L. E. Payne
Affiliation:
Department of Mathematics, Cornell University, Ithaca, New York 14853, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper studies the surface of constant mean curvature on a semi-infinite strip, and shows by means of a first-order differential inequality that the solution in a given measure either becomes asymptotically unbounded at least to polynomial order, or decays at most exponentially to the solution of an associated one-dimensional problem. A proof is also presented for uniqueness in the class of functions having bounded gradient and subject to specified growth conditions for large values of the longitudinal distance. Extensions of these results to the whole strip and to more general types of equations are also described.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 124 , Issue 1 , 1994 , pp. 105 - 119
Copyright
Copyright © Royal Society of Edinburgh 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Bernstein, S.. Über ein geometrisches Theorem und seine Anwendung auf die partiellen Differentialgleichungen vom elliptischen Typus. Math. Zeit. 26 (1927), 551–558.Google Scholar
2Chen, X.-Y., Matano, H. and Veron, L.. Anisotropic singularities of solutions of nonlinear elliptic equations in ℝ2. J. Differential Fund. Anal. 83 (1989), 50–97.CrossRefGoogle Scholar
3Flavin, J. N., Knops, R. J. and Payne, L. E.. Decay estimates for the constrained elastic cylinder of variable cross section. Quart. Appl. Math. 47 (1989), 325–350.Google Scholar
4Holmes, P. J. and Mielke, A.. Spatially complex equilibria of buckled rods. Arch. Rational Mech. Anal. 101 (1988), 309–348.Google Scholar
5Horgan, C. O. and Payne, L. E.. Decay estimates for second-order quasilinear partial differential equations. Adv. Appl. Math. 5 (1984), 309–332.CrossRefGoogle Scholar
6Horgan, C. O. and Payne, L. E.. Decay estimates for a class of second-order quasilinear equations in three dimensions. Arch. Rational Mech. Anal. 86 (1984), 279–289.CrossRefGoogle Scholar
7Horgan, C. O. and Payne, L. E.. Decay estimates for a class of nonlinear boundary value problems in two dimensions. SIAM J. Math. Anal. 20 (1989), 782–788.CrossRefGoogle Scholar
8Horgan, C. O. and Payne, L. E.. On the asymptotic behaviour of solutions of inhomogeneous secondorder quasilinear partial differential equations. Quart. Appl. Math. 47 (1989), 753–771.CrossRefGoogle Scholar
9Horgan, C. O. and Payne, L. E.. Exponential decay estimates for capillary surfaces and extensible films. S.A.A.C.M. 1 (1991), 261–282.Google Scholar
10Horgan, C. O. and Payne, L. E.. On Saint-Venant's principle in finite anti-plane shear: an energy approach. Arch. Rational Mech. Anal. 109 (1990), 107–137.CrossRefGoogle Scholar
11Mielke, A.. On Saint-Venant's problem for an elastic strip. Proc. Roy. Soc. Edinburgh Sect. A 110 (1988), 161–181.Google Scholar
12Mielke, A.. Normal hyperbolicity of center manifolds and Saint-Venant's principle. Arch. Rational Mech. Anal. 110 (1990), 353–372.Google Scholar
13Payne, L. E. and Webb, J. R. L.. Spatial decay estimates for second-order partial differential equations. Nonlinear Anal. 18 (1992), 143–156.CrossRefGoogle Scholar
14Payne, L. E. and Weinberger, H. F.. Note on a lemma of Finn and Gilbarg. Acta Math. 98 (1951), 297–299.Google Scholar
15Serrin, J. B.. The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables. Philos. Trans. Roy. Soc. London 164 (1969), 413–496.Google Scholar
16Serrin, J. B.. Nonlinear elliptic equations of second order. Lectures Symposium on Partial Differential Equations, Berkeley (mimeographical notes) (1971).Google Scholar
17Shiffman, M.. Differentiability and analyticity of solutions of double integral variational problems. Ann. of Math. 48 (1948), 174–284.Google Scholar
18Collin, P. and Krust, R.. Le probleme de Dirichlet pour l'équation des surfaces minimales sur des domaines non bornés. Bull. Soc. Math. France 119 (1991), 443–462.CrossRefGoogle Scholar