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A Rayleigh-Faber-Krahn Inequality and Some Monotonicity Properties for Eigenvalue Problems with Mixed Boundary Conditions

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Inequalities and Applications

Part of the book series: International Series of Numerical Mathematics ((ISNM,volume 157))

Abstract

An eigenvalue problem is considered whose eigenvalues appear in the interior and on the boundary. It has been shown in [1] that there exists an infinite sequence of positive and an infinite sequence of negative eigenvalues. The lowest positive and the largest negative eigenvalue λ 1, resp. λ −1 can be characterised by means of a Rayleigh principle. It turns out that among all domains of given volume the ball has the smallest λ 1. A partial result in this direction is established for λ −1. The proof uses the isoperimetric inequality of Krahn-Bossel-Daners. Some monotonicity properties similar to those for the elastically supported membrane are included.

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References

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Author information

Authors and Affiliations

  1. Mathematisches Institut, Universität Basel, Rheinsprung 21, CH-4051, Basel, Switzerland

    Catherine Bandle

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  1. Catherine Bandle
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Editor information

Editors and Affiliations

  1. Institut Mathematik, Universität Basel, Rheinsprung 21, 4051, Basel, Switzerland

    Catherine Bandle

  2. Department Economic Analysis & Information Technology for Business, University of Debrecen, Kassai út.26, 4028, Debrecen, Hungary

    László Losonczi

  3. Department of Analysis Institute Mathematics & Informatics, University of Debrecen, Pf. 12, 4010, Debrecen, Hungary

    Attila Gilányi  & Zsolt Páles  & 

  4. Institut für Analysis, Universität Karlsruhe, Kaiserstrasse 12, 76128, Karlsruhe, Germany

    Michael Plum

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© 2008 Birkhäuser Verlag Basel/Switzerland

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Bandle, C. (2008). A Rayleigh-Faber-Krahn Inequality and Some Monotonicity Properties for Eigenvalue Problems with Mixed Boundary Conditions. In: Bandle, C., Losonczi, L., Gilányi, A., Páles, Z., Plum, M. (eds) Inequalities and Applications. International Series of Numerical Mathematics, vol 157. Birkhäuser, Basel. https://doi.org/10.1007/978-3-7643-8773-0_1

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