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General Inequalities 7 pp 95–110Cite as

Birkhäuser

Inequalities for the first eigenvalues of the clamped plate and buckling problems

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Part of the book series: ISNM International Series of Numerical Mathematics ((ISNM,volume 123))

Abstract

In 1877 Rayleigh conjectured that the lowest frequency of vibration of a clamped plate of given area occurs when the plate is circular. This conjecture was finally proved by Nadirashvili in 1992, with important earlier contributions due to Szegő and Talenti. An analogous conjecture concerning the critical buckling load of a clamped plate was made by Pólya and Szegő around 1950. This conjecture remains open. This paper surveys the state of our knowledge of these and related problems, including their n-dimensional generalizations. In particular, we discuss our recent work proving Rayleigh’s clamped plate conjecture for dimension 3 (and 2) and proving related, but presumably nonoptimal, inequalities for the clamped plate problem for dimension n ≥ 4 and for the buckling problem. In the latter cases, the inequalities have the form of the conjectured lower bounds but contain an unwanted factor slightly less than 1. Our bounds for the clamped plate problem follow from detailed analysis involving Bessel functions, and compare favorably to earlier bounds of a similar form due to Talenti. Our bounds for the buckling problem follow by similar methods, and also from earlier bounds due to Payne and Krahn, as first noted by Bramble and Payne.

Partially supported by National Science Foundation (USA) grants DMS-9114162, INT-123481, and DMS-9500968.

Partially supported by Fondecyt (Chile) projects 193-0561 and 196-0462 and Conicyt (Chile).

Partially supported by National Science Foundation (USA) grants DMS-9414149 and DMS-9304580.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Missouri, Columbia, MO, 65211, USA

    M. S. Ashbaugh

  2. Facultad de Física, P. Universidad Católica de Chile, Casilla 306, Santiago, 22, Chile

    R. D. Benguria

  3. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109, USA

    R. S. Laugesen

Authors
  1. M. S. Ashbaugh
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  2. R. D. Benguria
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  3. R. S. Laugesen
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Editor information

Editors and Affiliations

  1. Institute of Mathematics, University of Basel, Rheinsprung 21, 4051, Basel, Switzerland

    Catherine Bandle

  2. School of Mathematics and Statistics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, England, UK

    William N. Everitt

  3. Department of Mathematics and Computer Science, Kuwait University, P.O. Box 5969, 13060, Safat, Kuwait

    Laszlo Losonczi

  4. Institue of Mathematics I, University of Karlsruhe, Kaiserstr. 12, 76128, Karlsruhe, Germany

    Wolfgang Walter

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© 1997 Springer Basel AG

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Ashbaugh, M.S., Benguria, R.D., Laugesen, R.S. (1997). Inequalities for the first eigenvalues of the clamped plate and buckling problems. In: Bandle, C., Everitt, W.N., Losonczi, L., Walter, W. (eds) General Inequalities 7. ISNM International Series of Numerical Mathematics, vol 123. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8942-1_9

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  • DOI: https://doi.org/10.1007/978-3-0348-8942-1_9

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9837-9

  • Online ISBN: 978-3-0348-8942-1

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