Abstract
We study upper and lower bounds for the lowest eigenvalueλ of the Laplace operator on a spherical capC θ inm-dimensional space (m ≥ 3).
We prove that these bounds are sharp by finding asymptotic expressions forλ asθ → π and asθ → 0.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Debiard A, Gaveau B, Mazet E (1975) Théoremes de comparison en géometrie Riemannienne. Comptes Rendus Acad Sciences Paris, Ser A, 281:455–458
Erdelyi A, Magnus W, Oberhettinger F, Tricomi F (1953) Higher transcendental functions, vol 1. McGraw-Hill, New York
Friedland S, Hayman WK (1976) “Eigenvalue inequalities for the Dirichlet problem on spheres and the growth of subharmonic functions. Comment Math Helvetici 51:133–161
Hobson EW (1931) The theory of spherical and ellipsoidal harmonics. Cambridge Univ. Press
Pinsky MA (1981) The first eigenvalue of a spherical cap. Appl Math Optim 7:137–139
Sperner E (1973) Zur Symmetrisierung von Funktionen auf spharen. Math Z 134:317–327
Author information
Authors and Affiliations
Additional information
Communicated by A. K. Balakrishnan
Rights and permissions
About this article
Cite this article
Betz, C., Cámera, G.A. & Gzyl, H. Bounds for the First eigenvalue of a spherical cap. Appl Math Optim 10, 193–202 (1983). https://doi.org/10.1007/BF01448386
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01448386