Skip to main content
SpringerLink
Log in

The first eigenvalue of a spherical cap

  • Published:
Applied Mathematics and Optimization Submit manuscript

Abstract

We find upper and lower bounds for the first eigenvalue of the Laplacian on the two-sphere from which a disk has been removed, with Dirichlet conditions imposed on the resulting boundary. When the radius of the disk tends to zero our lower bound is sharper than that obtained by Del Grosso, Gerardi, and Marchetti in the preceding paper.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. G. del Grosso, A. Gerardi, and F. Marchetti, A diffusion model for patch formation on cellular surfaces,Appl. Math Optim., this issue, 125–136.

  2. E. B. Dynkin,Markov Processes, Chapter 2, Springer-Verlag, 1965.

  3. A. Debiard, B. Gaveau, and E. Mazet, Théoremes de comparison en géometrie riemannienne,Comptes Rendus Acad. Sciences Paris, Serie A, 281, 455–458 (1975).

    Google Scholar 

  4. I. Chavel, and D. Feldman, Spectra of domains in compact manifolds,Jour. Functional Analysis, 30, 198–222 (1978).

    Google Scholar 

  5. A. Levy-Bruhl, Courbure riemannienne et developpements infinitesimaux du Laplacien,Comptes Rendus Acad. Sciences Paris, Serie A, 279, 197–200 (1974).

    Google Scholar 

  6. A. Friedman,Stochastic Differential Equations, Academic Press, New York, 1976.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Northwestern University, 60201, Evanston, IL, USA

    Mark A. Pinsky

Authors
  1. Mark A. Pinsky
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by W. Fleming

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pinsky, M.A. The first eigenvalue of a spherical cap. Appl Math Optim 7, 137–139 (1981). https://doi.org/10.1007/BF01442111

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01442111

Keywords

Search

Navigation