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On the first eigenfunction of the fixed membrane: Some extensions of results of Payne and Stakgold

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Résumé

A l'aide des deux principes de maximum de E. Hopf [3], [4], on obtient des bornes pour grad2 w en fonction du quotientw=u/h, oùu est la première fonction propre d'une membrane vibranteD fixée sur son contour, et oùh est une solution positive de l'équation de Helmholz dansD. Dans le cas particulierh ≡ 1, on retrouve les résultats de L. E. Payne et I. Stakgold [7].

Zusammenfassung

Mit Hilfe der zwei Maximumprinzipien von E. Hopf [3], [4] werden Schranken für grad2 w als Funktion des Quotientenw=u/h konstruiert. Dabei istu die erste Eigenfunktion einer am Rande eingespannten schwingenden MembranD, undh ist eine positive Lösung der Helmholz'schen Gleichung inD. Der Spezialfallh ≡ 1 führt auf die Resultate von L. E. Payne und I. Stakgold [7] zurück.

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References

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Authors and Affiliations

  1. Dept. of Mathematics, Cornell University, Ithaca, New York, USA

    Gérard A. Philippin

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  1. Gérard A. Philippin
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This research was supported by the Swiss Nationalfonds.

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Philippin, G.A. On the first eigenfunction of the fixed membrane: Some extensions of results of Payne and Stakgold. Journal of Applied Mathematics and Physics (ZAMP) 28, 151–159 (1977). https://doi.org/10.1007/BF01590715

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