DOI: 10.4153/cjm-1949-021-5. Some Properties of the Eigenfunctions of The Laplace-Operator on Riemannian Manifolds

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

10.4153/cjm-1949-021-5

Crossref journal-article

Canadian Mathematical Society

Canadian Journal of Mathematics (2643)

Abstract

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.

Bibliography

Minakshisundaram, S., & Pleijel, Ã. (1949). Some Properties of the Eigenfunctions of The Laplace-Operator on Riemannian Manifolds. Canadian Journal of Mathematics, 1(3), 242â256.

Authors 2

S. Minakshisundaram (first)
Å. Pleijel (additional)

References 14 Referenced 454

Pleijel Å. , “Asymptotic Properties of the Eigenfunctions of Certain Boundary-Value Problems of Polar Type,” will appear in Amer. J. Math.
{'key': 'S0008414X00037093_ref009', 'article-title': 'Arkiv Mat., Astr. o. Fys.', 'volume': '27', 'author': 'Pleijel', 'year': '1940', 'journal-title': '“Propriétés asymptotiques des fonctions et valeurs propres de certains problèmes de vibrations,”'} / “Propriétés asymptotiques des fonctions et valeurs propres de certains problèmes de vibrations,” / Arkiv Mat., Astr. o. Fys. by Pleijel (1940)
Minakshisundaram S. , “Zeta-functions on the Sphere,” will appear in J. Indian Math.Soc.
Minakshisundaram S. , “A Generalization of Epstein Zeta Functions,” will appear in this Journal.
{'key': 'S0008414X00037093_ref006', 'author': 'Hadamard', 'year': '1932'} by Hadamard (1932)
{'key': 'S0008414X00037093_ref011', 'article-title': 'Ann. Univ. Grenoble, Sec. Sci. Math. Phys.', 'volume': '22', 'author': 'de Rham', 'year': '1946', 'journal-title': '“Sur la théorie des formes différentielles harmoniques,”'} / “Sur la théorie des formes différentielles harmoniques,” / Ann. Univ. Grenoble, Sec. Sci. Math. Phys. by de Rham (1946)
{'key': 'S0008414X00037093_ref005', 'article-title': 'Series 2', 'volume': '52', 'author': 'Giraud', 'year': '1932', 'journal-title': '“Généralisation des problèmes sur les opérations du type elliptique,“ Bull. Sci. Math.'} / “Généralisation des problèmes sur les opérations du type elliptique,“ Bull. Sci. Math. / Series 2 by Giraud (1932)
{'key': 'S0008414X00037093_ref003', 'author': 'Courant', 'year': '1924'} by Courant (1924)
{'key': 'S0008414X00037093_ref002', 'volume-title': '“Uber die asymptotische Verteilung der Eigenwerte partieller Differentialgleichungen,”', 'volume': '88', 'author': 'Carleman', 'year': '1936'} / “Uber die asymptotische Verteilung der Eigenwerte partieller Differentialgleichungen,” by Carleman (1936)
{'key': 'S0008414X00037093_ref012', 'author': 'Watson', 'year': '1944'} by Watson (1944)
{'key': 'S0008414X00037093_ref014', 'article-title': 'Ann. of Math.', 'volume': '33', 'author': 'Wiener', 'year': '1932', 'journal-title': '“Tauberian Theorems,”'} / “Tauberian Theorems,” / Ann. of Math. by Wiener (1932)
Carleman T. , “Propriétés asymptotiques des fonctions fondamentales des membranes vibrantes,” Skand. Matent. Kongress (1934).
{'key': 'S0008414X00037093_ref013', 'article-title': 'Math. Ann.', 'volume': '71', 'author': 'Weyl', 'year': '1911', 'journal-title': '“Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung),”'} / “Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung),” / Math. Ann. by Weyl (1911)
{'key': 'S0008414X00037093_ref004', 'author': 'Courant', 'year': '1937'} by Courant (1937)

Dates

Type	When
Created	14 years, 8 months ago (Dec. 7, 2010, 1:24 p.m.)
Deposited	6 years, 2 months ago (June 8, 2019, 2:31 a.m.)
Indexed	2 weeks, 4 days ago (Aug. 7, 2025, 5:03 p.m.)
Issued	76 years, 2 months ago (June 1, 1949)
Published	76 years, 2 months ago (June 1, 1949)
Published Online	6 years, 9 months ago (Nov. 20, 2018)
Published Print	76 years, 2 months ago (June 1, 1949)

Funders 0

None

BibTeX

@article{Minakshisundaram_1949, title={Some Properties of the Eigenfunctions of The Laplace-Operator on Riemannian Manifolds}, volume={1}, ISSN={1496-4279}, url={http://dx.doi.org/10.4153/cjm-1949-021-5}, DOI={10.4153/cjm-1949-021-5}, number={3}, journal={Canadian Journal of Mathematics}, publisher={Canadian Mathematical Society}, author={Minakshisundaram, S. and Pleijel, Å.}, year={1949}, month=jun, pages={242–256} }

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  "abstract": "<jats:p>Let V be a connected, compact, differentiable Riemannian manifold. If <jats:italic>V</jats:italic> is not closed we denote its boundary by <jats:italic>S</jats:italic>. In terms of local coordinates (<jats:italic>x</jats:italic><jats:sup><jats:italic>i</jats:italic></jats:sup>), <jats:italic>i</jats:italic> = 1, 2, \u2026 \u039d, the line-element <jats:italic>dr</jats:italic> is given by <jats:disp-formula><jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0008414X00037093_eqn01\" /></jats:disp-formula> where gik (x<jats:sup>1</jats:sup>, x<jats:sup>2</jats:sup>, \u2026 x<jats:sup><jats:italic>N</jats:italic></jats:sup>) are the components of the metric tensor on <jats:italic>V</jats:italic> We denote by \u0394 the Beltrami-Laplace-Operator <jats:disp-formula><jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0008414X00037093_eqn02\" /></jats:disp-formula>and we consider on V the differential equation (1) \u0394u + \u03bbu = 0.</jats:p>",
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      "article-title": "Arkiv Mat., Astr. o. Fys.",
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      "author": "Pleijel",
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      "journal-title": "\u201cPropri\u00e9t\u00e9s asymptotiques des fonctions et valeurs propres de certains probl\u00e8mes de vibrations,\u201d"
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      "article-title": "Ann. Univ. Grenoble, Sec. Sci. Math. Phys.",
      "volume": "22",
      "author": "de Rham",
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      "journal-title": "\u201cSur la th\u00e9orie des formes diff\u00e9rentielles harmoniques,\u201d"
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