Abstract
In this paper we give conditions sufficient to insure that the following two results hold for eigenfunctions of the fixed membrane problem defined over a bounded region of the plane:
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i)
The first eigenfunction has only one interior critical point.
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ii)
The second eigenfunction has no closed interior nodal curve.
Zusammenfassung
Gegeben werden hinreichende Bedingungen für die Gültigkeit der beiden folgenden Aussagen über Eigenfunktionen der eingespannten Membran auf einem beschränkten ebenen Gebiet:
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i)
Die erste Eigenfunktion hat einen einzigen kritischen Punkt;
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ii)
Die zweite Eigenfunktion besitzt keine innere geschlossene Knotenlinie.
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References
R. Courant andD. Hilbert,Methods in Mathematical Physics, vol. 1. Interscience, New York (1953).
E. Hopf.A Remark on Linear Elliptic Differential Equations of Second Order. Proc. Amer. Math. Soc.3, 291–293 (1952).
L. E. Payne,Isoperimetric Inequalities and their Applications, SIAM Review,9, 453–488 (1967).
L. E. Payne andI. Stakgold,On the Mean Value of the Fundamental Mode in the Fixed Membrane Problem, J. Appl. Anal., to appear.
L. E. Payne andH. F. Weinberger,A Faber-Krahn Inequality for Wedge Like Domains, J. Math. Phys.39, 182–188 (1960).
G. Pólya andG. Szegö,Isoperimetric Inequalities in Mathematical Physics. Annals of Math. Studies #27, Princeton Univ. Press, Princeton (1951).
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This research was supported in part by NSF GP-33031 and by the United States Army under Contract No. DA-31-124-ARO-D-462.
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Payne, L.E. On two conjectures in the fixed membrane eigenvalue problem. Journal of Applied Mathematics and Physics (ZAMP) 24, 721–729 (1973). https://doi.org/10.1007/BF01597076
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DOI: https://doi.org/10.1007/BF01597076