American Institute of Mathematical Sciences

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

 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

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.

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.

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.. [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.
 [1] A. M. Micheletti, Angela Pistoia. Multiple eigenvalues of the Laplace-Beltrami operator and deformation of the Riemannian metric. Discrete and Continuous Dynamical Systems, 1998, 4 (4) : 709-720. doi: 10.3934/dcds.1998.4.709 [2] Micol Amar, Roberto Gianni. Laplace-Beltrami operator for the heat conduction in polymer coating of electronic devices. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1739-1756. doi: 10.3934/dcdsb.2018078 [3] Andrea Bonito, Roland Glowinski. On the nodal set of the eigenfunctions of the Laplace-Beltrami operator for bounded surfaces in $R^3$: A computational approach. Communications on Pure and Applied Analysis, 2014, 13 (5) : 2115-2126. doi: 10.3934/cpaa.2014.13.2115 [4] Jan Bouwe van den Berg, Gabriel William Duchesne, Jean-Philippe Lessard. Rotation invariant patterns for a nonlinear Laplace-Beltrami equation: A Taylor-Chebyshev series approach. Journal of Computational Dynamics, 2022, 9 (2) : 253-278. doi: 10.3934/jcd.2022005 [5] Mikhail Karpukhin. Bounds between Laplace and Steklov eigenvalues on nonnegatively curved manifolds. Electronic Research Announcements, 2017, 24: 100-109. doi: 10.3934/era.2017.24.011 [6] Shingo Takeuchi. Partial flat core properties associated to the $p$-laplace operator. Conference Publications, 2007, 2007 (Special) : 965-973. doi: 10.3934/proc.2007.2007.965 [7] Eduardo Lara, Rodolfo Rodríguez, Pablo Venegas. Spectral approximation of the curl operator in multiply connected domains. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 235-253. doi: 10.3934/dcdss.2016.9.235 [8] Antonio Vitolo. $H^{1,p}$-eigenvalues and $L^\infty$-estimates in quasicylindrical domains. Communications on Pure and Applied Analysis, 2011, 10 (5) : 1315-1329. doi: 10.3934/cpaa.2011.10.1315 [9] George Baravdish, Yuanji Cheng, Olof Svensson, Freddie Åström. Generalizations of $p$-Laplace operator for image enhancement: Part 2. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3477-3500. doi: 10.3934/cpaa.2020152 [10] Guoqing Zhang, Jia-yu Shao, Sanyang Liu. Linking solutions for N-laplace elliptic equations with Hardy-Sobolev operator and indefinite weights. Communications on Pure and Applied Analysis, 2011, 10 (2) : 571-581. doi: 10.3934/cpaa.2011.10.571 [11] Umberto Biccari. Internal control for a non-local Schrödinger equation involving the fractional Laplace operator. Evolution Equations and Control Theory, 2022, 11 (1) : 301-324. doi: 10.3934/eect.2021014 [12] David Hoff. Pointwise bounds for the Green's function for the Neumann-Laplace operator in $\text{R}^3$. Kinetic and Related Models, 2022, 15 (4) : 535-550. doi: 10.3934/krm.2021037 [13] Dorina Mitrea, Marius Mitrea, Sylvie Monniaux. The Poisson problem for the exterior derivative operator with Dirichlet boundary condition in nonsmooth domains. Communications on Pure and Applied Analysis, 2008, 7 (6) : 1295-1333. doi: 10.3934/cpaa.2008.7.1295 [14] Mahamadi Warma. A fractional Dirichlet-to-Neumann operator on bounded Lipschitz domains. Communications on Pure and Applied Analysis, 2015, 14 (5) : 2043-2067. doi: 10.3934/cpaa.2015.14.2043 [15] Chao Zhang, Lihe Wang, Shulin Zhou, Yun-Ho Kim. Global gradient estimates for $p(x)$-Laplace equation in non-smooth domains. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2559-2587. doi: 10.3934/cpaa.2014.13.2559 [16] Martí Prats. Beltrami equations in the plane and Sobolev regularity. Communications on Pure and Applied Analysis, 2018, 17 (2) : 319-332. doi: 10.3934/cpaa.2018018 [17] Peter I. Kogut, Olha P. Kupenko. On optimal control problem for an ill-posed strongly nonlinear elliptic equation with $p$-Laplace operator and $L^1$-type of nonlinearity. Discrete and Continuous Dynamical Systems - B, 2019, 24 (3) : 1273-1295. doi: 10.3934/dcdsb.2019016 [18] Bin Guo, Wenjie Gao. Finite-time blow-up and extinction rates of solutions to an initial Neumann problem involving the $p(x,t)-Laplace$ operator and a non-local term. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 715-730. doi: 10.3934/dcds.2016.36.715 [19] Lok Ming Lui, Tsz Wai Wong, Wei Zeng, Xianfeng Gu, Paul M. Thompson, Tony F. Chan, Shing Tung Yau. Detection of shape deformities using Yamabe flow and Beltrami coefficients. Inverse Problems and Imaging, 2010, 4 (2) : 311-333. doi: 10.3934/ipi.2010.4.311 [20] Lingju Kong, Roger Nichols. On principal eigenvalues of biharmonic systems. Communications on Pure and Applied Analysis, 2021, 20 (1) : 1-15. doi: 10.3934/cpaa.2020254

2020 Impact Factor: 1.392