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Some Properties of the Eigenfunctions of The Laplace-Operator on Riemannian Manifolds

Published online by Cambridge University Press:  20 November 2018

S. Minakshisundaram  and
Å. Pleijel
S. Minakshisundaram
Affiliation:
Andhra University, Waltair, South India
Å. Pleijel
Affiliation:
Andhra University, Waltair, South India
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Let V be a connected, compact, differentiable Riemannian manifold. If V is not closed we denote its boundary by S. In terms of local coordinates (xi), i = 1, 2, … Ν, the line-element dr is given by where gik (x1, x2, … xN) are the components of the metric tensor on V We denote by Δ the Beltrami-Laplace-Operator and we consider on V the differential equation (1) Δu + λu = 0.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 1 , Issue 3 , 01 June 1949 , pp. 242 - 256
Copyright
Copyright © Canadian Mathematical Society 1949

References

[1] Carleman, T., “Propriétés asymptotiques des fonctions fondamentales des membranes vibrantes,” Skand. Matent. Kongress (1934).Google Scholar
[2] Carleman, T., “Uber die asymptotische Verteilung der Eigenwerte partieller Differentialgleichungen,” Berichte Verhandl. Akad. Leipzig, vol. 88 (1936).Google Scholar
[3] Courant, R. and Hilbert, D., Methoden der Mathematischen Physik I (Berlin, 1924).Google Scholar
[4] Courant, R. and Hilbert, D., Methoden der Mathematischen Physik II(Berlin, 1937).Google Scholar
[5] Giraud, M. G., “Généralisation des problèmes sur les opérations du type elliptique,“ Bull. Sci. Math., Series 2, vol. 52 (1932).Google Scholar
[6] Hadamard, J., Le problème de Cauchy et les équations aux dérivées partielles linéaires hyperboliques (Paris, 1932).Google Scholar
[7] Minakshisundaram, S., “A Generalization of Epstein Zeta Functions,” will appear in this Journal.Google Scholar
[8] Minakshisundaram, S., “Zeta-functions on the Sphere,” will appear in J. Indian Math.Soc. Google Scholar
[9] Pleijel, Å., “Propriétés asymptotiques des fonctions et valeurs propres de certains problèmes de vibrations,” Arkiv Mat., Astr. o. Fys., vol. 27 A, no. 13 (1940).Google Scholar
[10] Pleijel, Å., “Asymptotic Properties of the Eigenfunctions of Certain Boundary-Value Problems of Polar Type,” will appear in Amer. J. Math. Google Scholar
[11] de Rham, G., “Sur la théorie des formes différentielles harmoniques,” Ann. Univ. Grenoble, Sec. Sci. Math. Phys., (N.S.) vol. 22 (1946).Google Scholar
[12] Watson, G. M., A Treatise on the Theory of Bessel Functions (Cambridge, 1944).Google Scholar
[13] Weyl, H., “Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung),” Math. Ann., vol. 71 (1911).Google Scholar
[14] Wiener, N., “Tauberian Theorems,” Ann. of Math., vol. 33 (1932).Google Scholar