Eigenvalues of the Laplace-Beltrami operator under the homogeneous Neumann condition on a large zonal domain in the unit sphere

Yoshitsugu Kabeya

doi:10.3934/dcds.2020040

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2020, Volume 40, Issue 6: 3529-3559. Doi: 10.3934/dcds.2020040 open access

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Eigenvalues of the Laplace-Beltrami operator under the homogeneous Neumann condition on a large zonal domain in the unit sphere

Yoshitsugu Kabeya^,

Department of Mathematical Sciences, Osaka Prefecture University, Gakuencho, Sakai, 599-8531, Japan

^* Corresponding author: Yoshitsugu Kabeya
^* Corresponding author: Yoshitsugu Kabeya

Dedicated to Professor Wei-Ming Ni on the occasion of his seventieth birthday

Received: January 31, 2019

Revised: April 30, 2019

Published: March 2020

The author is supported in part by JSPS KAKENHI Grant Numbers 15K04965 and 15H03631

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Abstract

We consider the eigenvalues of the Laplace-Beltrami operator under the homogeneous Neumann condition on a spherical domain. Especially, we investigate the case when the domain is a large zonal one and letting the zone larger so that the zone covers the whole sphere as a limit. We discuss the behavior of eigenvalues according to the rate of expansion of the zone.

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Mathematics Subject Classification: Primary: 35J15, 35P15; Secondary: 35J25, 33C55.

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References

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