Lower bounds of Cheeger-Osserman type for the first eigenvalue of then-dimensional fixed membrane problem

  • Albert Avinyó
  • Xavier Mora
Brief Reports


In this note we derive new lower bounds for the first eigenvalue of the Laplacian in a boundedn-dimensional domain with Dirichlet boundary conditions. The lower bounds obtained are related to those of Cheeger (1970) [2] and Osserman (1977) [6], and they turn out to be sharper when the domain is not too far from being a ball.


Boundary Condition Lower Bound Mathematical Method Dirichlet Boundary Dirichlet Boundary Condition 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    L. A. Caffarelli and J. Spruck,Convexity properties of solutions to some classical variational problems, Comm. Partial Differential Equations 1337–1379 (1982).Google Scholar
  2. [2]
    J. Cheeger,A lower bound for the smallest eigenvalue of the Laplacian, Problems in Analysis (ed. R. C. Gunning, Princeton Univ. Press) 195–199 (1970).Google Scholar
  3. [3]
    C. B. Croke,The first eigenvalue of the Laplacian for plane domains, Proc. Amer. Math. Soc.81, 304–305 (1981).Google Scholar
  4. [4]
    M. T. Kohler-Jobin,Sur la première fonction propre d'une membrane: extension à N dimensions de l'inégalité isopérimétrique de Payne-Rayner, Z.A.M.P.28, 1137–1140 (1977).Google Scholar
  5. [5]
    N. J. Korevaar,Convex solutions to nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J.32, 603–614 (1983).Google Scholar
  6. [6]
    R. Osserman,A note on Hayman's theorem on the base note of a drum, Comm. Math. Helv.52, 545–555 (1977).Google Scholar
  7. [7]
    R. Osserman,The isoperimetric inequality, Bull. Amer. Math. Soc.84, 1182–1238 (1978).Google Scholar
  8. [8]
    R. Osserman,Bonnesen style isoperimetric inequalities, Amer. Math. Monthly86, 1–29 (1979).Google Scholar
  9. [9]
    L. Payne, M. Rayner,An isoperimetric inequality for the first eigenfunction in the fixed membrane problem, Z.A.M.P.23, 13–15 (1972).Google Scholar
  10. [10]
    S. T. Yau,Isoperimetric constants and the first eigenvalue of a compact Riemannian manifold, Ann. Sci. École Norm. Sup. (4)8, 487–507 (1975).Google Scholar

Copyright information

© Birkhäuser Verlag 1990

Authors and Affiliations

  • Albert Avinyó
    • 1
  • Xavier Mora
    • 2
  1. 1.Dept. Matemàtica Aplicada II. Fac. d'lnformàticaUniversitat Politècnica de CatalunyaBarcelona
  2. 2.Dept. de MatemàtiquesUniv. Autònoma de BarcelonaBellaterra, BarcelonaSpain

Personalised recommendations