Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Upper and lower bounds for eigenvalues of the Laplacian on a spherical cap


Author: Frank E. Baginski
Journal: Quart. Appl. Math. 48 (1990), 569-573
MSC: Primary 35P15; Secondary 35J05, 73H05, 73K15
DOI: https://doi.org/10.1090/qam/1074972
MathSciNet review: MR1074972
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Abstract: In the following, we derive upper and lower bounds for eigenvalues of the Laplacian on a domain that is a spherical cap whose angular width $ 2{\vartheta _0}$ is less than $ \pi $. While previous work of this nature seems to focus on the principle eigenvalue, our results apply to any eigenvalue when $ 0 < {\vartheta _0} < \pi /2$. In addition, some of our results also apply to spherical caps for which $ 0 < {\vartheta _0} < \pi $. When our estimates for the principle eigenvalue are compared to the results of [4, 8], we find that our upper and lower bounds are sharper.


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DOI: https://doi.org/10.1090/qam/1074972
Article copyright: © Copyright 1990 American Mathematical Society


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