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On the second eigenvalue of the Laplace operator on a spherical band


Author: Chung-Tsun Shieh
Journal: Proc. Amer. Math. Soc. 132 (2004), 157-164
MSC (2000): Primary 35P15
DOI: https://doi.org/10.1090/S0002-9939-03-07039-4
Published electronically: May 8, 2003
MathSciNet review: 2021258
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we prove that the second eigenvalue of the Laplacian for a spherical band on the unit sphere $S^2$ has multiplicity 2. We also show that among all spherical bands of given fixed area less than $2\pi$ the second eigenvalue is maximized at the band which is symmetrical with respect to the equator.


References [Enhancements On Off] (What's this?)

  • 1. C. Bandle, Isoperimetric Inequalities and Applications, Pitman, Landon, 1980. MR 81e:35095
  • 2. P. R. Beesack, Integral Inequalities of the Wirtinger Type, Duke. Math. J. vol. 25(1958), pp. 477-498. MR 20:3947
  • 3. R. Courant and D. Hilbert, Methods of Mathematical Physics, vol. I, Wiley-Interscience, New York-London, 1953. MR 16:426a
  • 4. G. H. Hardy, J. E. Littlewood and G. Polya. Inequalities, Cambridge University Press, 1952. MR 13:727e
  • 5. E. L. Ince, Ordinary Differential Equations, Dover, New York, 1956.
  • 6. J. R. Kuttler and V. G. Sigillito, Eigenvalues of the Laplacian in Two Dimensions, SIAM Review, vol 26(1984), no. 2 pp. 163-193. MR 85k:65086
  • 7. C.-L. Shen and C.-T. Shieh, Some properties of the first eigenvalue of the Laplace operator on the spherical bands in $S^2$, SIAM J. Math. Anal., vol. 23(1992), 1305-1308. MR 93f:35162
  • 8. G. P. Tolstov, Fourier Series, Dover, New York, 1962.

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Additional Information

Chung-Tsun Shieh
Affiliation: Department of Mathematics, Fu-Jen Catholic University, Taipei, Taiwan, Republic of China
Email: ctshieh@math.fju.edu.tw

DOI: https://doi.org/10.1090/S0002-9939-03-07039-4
Received by editor(s): April 27, 2001
Received by editor(s) in revised form: July 8, 2002, and August 21, 2002
Published electronically: May 8, 2003
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2003 American Mathematical Society

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