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

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June 2020, 40(6): 3529-3559. doi: 10.3934/dcds.2020040

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

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

Received February 2019 Revised May 2019 Published October 2019

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

Abstract
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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.

Keywords: Eigenvalues, Laplace-Beltrami operator, zonal domains.

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

Citation: Yoshitsugu Kabeya. Eigenvalues of the Laplace-Beltrami operator under the homogeneous Neumann condition on a large zonal domain in the unit sphere. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3529-3559. doi: 10.3934/dcds.2020040

References:

[1]	C. Bandle, Y. Kabeya and H. Ninomiya, Bifurcating solutions of a nonlinear elliptic Neumann problem on large spherical caps, in Funk. Ekvac..
[2]	R. Beals and R. Wong, Special Functions: A Graduate Text, Cambridge Studies in Advanced Mathematics, 126. Cambridge University Press, Cambridge, 2010. doi: 10.1017/CBO9780511762543.
[3]	G. Courtois, Spectrum of manifolds with holes, J. Funct. Anal., 134 (1995), 194-221. doi: 10.1006/jfan.1995.1142.
[4]	E. W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics, Chelsea Publishing Company, New York, 1955.
[5]	Y. Kabeya, T. Kawakami, A. Kosaka and H. Ninomiya, Eigenvalues of the Laplace-Beltrami operator on a large spherical cap under the Robin problem, Kodai Math. J., 37 (2014), 620-645. doi: 10.2996/kmj/1414674613.
[6]	M. L. de Cristoforis, Simple Neumann eigenvalues for the Laplace operator in a domain with a small hole. A functional analytic approach, Rev. Mat. Complut., 25 (2012), 369-412. doi: 10.1007/s13163-011-0081-8.
[7]	H. M. Macdonald, Zeroes of the spherical harmonics $P^m_n(\mu)$ considered as a Function of $n$, Proc. London Math. Soc., 31 (1899), 264-278. doi: 10.1112/plms/s1-31.1.264.
[8]	C. Müller, Spherical Harmonics, Lecture Notes in Mathematics, 17. Springer-Verlag, Berlin-New York, 1966.
[9]	W.-M. Ni, The Mathematics of Diffusion, CBMS-NSF Regional Conference Series in Applied Mathematics, 82. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611971972.
[10]	W.-M. Ni and X. F. Wang, On the first positive Neumann eigenvalue, Discrete. Contin. Dyn. Syst., 17 (2007), 1-19. doi: 10.3934/dcds.2007.17.1.
[11]	S. Ozawa, Singular variations of domains and eigenvalues of the Laplacian, Duke Math. J., 48 (1981), 767-778. doi: 10.1215/S0012-7094-81-04842-0.
[12]	S. Ozawa, An asymptotic formula for the eigenvalues of the Laplacian in a three dimensional domain with a small hole, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 30 (1983), 243-257.
[13]	S. Ozawa, Asymptotic property of an eigenfunction of the Laplacian under singular variation of domains–the Neumann condition, Osaka J. Math., 22 (1985), 639-655.
[14]	N. Shimakura, Partial Differential Operators of Elliptic Type, Translations of Mathematical Monographs, 99. American Mathematical Society, Providence, RI, 1992.
[15]	E. C. Titchmarsh, Eigenfunction expansions associated with partial differential equations, V. Proc. London Math. Soc., 5 (1955), 1-21. doi: 10.1112/plms/s3-5.1.1.
[16]	E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations. Part I, Second Edition, Clarendon Press, Oxford, 1962.

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References:

[1]	C. Bandle, Y. Kabeya and H. Ninomiya, Bifurcating solutions of a nonlinear elliptic Neumann problem on large spherical caps, in Funk. Ekvac..
[2]	R. Beals and R. Wong, Special Functions: A Graduate Text, Cambridge Studies in Advanced Mathematics, 126. Cambridge University Press, Cambridge, 2010. doi: 10.1017/CBO9780511762543.
[3]	G. Courtois, Spectrum of manifolds with holes, J. Funct. Anal., 134 (1995), 194-221. doi: 10.1006/jfan.1995.1142.
[4]	E. W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics, Chelsea Publishing Company, New York, 1955.
[5]	Y. Kabeya, T. Kawakami, A. Kosaka and H. Ninomiya, Eigenvalues of the Laplace-Beltrami operator on a large spherical cap under the Robin problem, Kodai Math. J., 37 (2014), 620-645. doi: 10.2996/kmj/1414674613.
[6]	M. L. de Cristoforis, Simple Neumann eigenvalues for the Laplace operator in a domain with a small hole. A functional analytic approach, Rev. Mat. Complut., 25 (2012), 369-412. doi: 10.1007/s13163-011-0081-8.
[7]	H. M. Macdonald, Zeroes of the spherical harmonics $P^m_n(\mu)$ considered as a Function of $n$, Proc. London Math. Soc., 31 (1899), 264-278. doi: 10.1112/plms/s1-31.1.264.
[8]	C. Müller, Spherical Harmonics, Lecture Notes in Mathematics, 17. Springer-Verlag, Berlin-New York, 1966.
[9]	W.-M. Ni, The Mathematics of Diffusion, CBMS-NSF Regional Conference Series in Applied Mathematics, 82. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611971972.
[10]	W.-M. Ni and X. F. Wang, On the first positive Neumann eigenvalue, Discrete. Contin. Dyn. Syst., 17 (2007), 1-19. doi: 10.3934/dcds.2007.17.1.
[11]	S. Ozawa, Singular variations of domains and eigenvalues of the Laplacian, Duke Math. J., 48 (1981), 767-778. doi: 10.1215/S0012-7094-81-04842-0.
[12]	S. Ozawa, An asymptotic formula for the eigenvalues of the Laplacian in a three dimensional domain with a small hole, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 30 (1983), 243-257.
[13]	S. Ozawa, Asymptotic property of an eigenfunction of the Laplacian under singular variation of domains–the Neumann condition, Osaka J. Math., 22 (1985), 639-655.
[14]	N. Shimakura, Partial Differential Operators of Elliptic Type, Translations of Mathematical Monographs, 99. American Mathematical Society, Providence, RI, 1992.
[15]	E. C. Titchmarsh, Eigenfunction expansions associated with partial differential equations, V. Proc. London Math. Soc., 5 (1955), 1-21. doi: 10.1112/plms/s3-5.1.1.
[16]	E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations. Part I, Second Edition, Clarendon Press, Oxford, 1962.

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