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

  • S. Minakshisundaram (a1) and Å. Pleijel (a1)
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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 (x i ), i = 1, 2, … Ν, the line-element dr is given by where gik (x1, x2, … x N ) 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.

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References
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[1] Carleman, T., “Propriétés asymptotiques des fonctions fondamentales des membranes vibrantes,” Skand. Matent. Kongress (1934).
[2] Carleman, T., “Uber die asymptotische Verteilung der Eigenwerte partieller Differentialgleichungen,” Berichte Verhandl. Akad. Leipzig, vol. 88 (1936).
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[7] Minakshisundaram, S., “A Generalization of Epstein Zeta Functions,” will appear in this Journal.
[8] Minakshisundaram, S., “Zeta-functions on the Sphere,” will appear in J. Indian Math.Soc.
[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).
[10] Pleijel, Å., “Asymptotic Properties of the Eigenfunctions of Certain Boundary-Value Problems of Polar Type,” will appear in Amer. J. Math.
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[13] Weyl, H., “Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung),” Math. Ann., vol. 71 (1911).
[14] Wiener, N., “Tauberian Theorems,” Ann. of Math., vol. 33 (1932).
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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
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