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 has multiplicity 2. We also show that among all spherical bands of given fixed area less than the second eigenvalue is maximized at the band which is symmetrical with respect to the equator.

**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**,*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