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Eigenvalues of the Laplace-Beltrami operator under the homogeneous Neumann condition on a large zonal domain in the unit sphere
Department of Mathematical Sciences, Osaka Prefecture University, Gakuencho, Sakai, 599-8531, Japan |
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.
References:
[1] |
C. Bandle, Y. Kabeya and H. Ninomiya, Bifurcating solutions of a nonlinear elliptic Neumann problem on large spherical caps, in Funk. Ekvac.. Google Scholar |
[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. |
show all references
References:
[1] |
C. Bandle, Y. Kabeya and H. Ninomiya, Bifurcating solutions of a nonlinear elliptic Neumann problem on large spherical caps, in Funk. Ekvac.. Google Scholar |
[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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