Vol. 42, No. 3, 1972

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ISSN: 0030-8730
Extensions of an inequality by Pólya and Schiffer for vibrating membranes

Catherine Bandle

Vol. 42 (1972), No. 3, 543–555
Abstract

The inequality by Pólya and Schiffer considered in this paper is concerned with the sume of the n first reciprocal eigenvalues of the problem Δu + λu = 0 in G,u = 0 on ∂G. First we extend this inequality to the problem of an inhomogeneous membrane Δu + λρu = 0 in G,u = 0 on ∂G. Then we prove a sharper form of it for a class of homogeneous membranes with partially free boundary. The proofs are based on a variational characterization for the eigenvalues and use conformal mapping and transplantation arguments.

Mathematical Subject Classification 2000
Primary: 49G05
Secondary: 52A40, 53A99, 49F10
Milestones
Received: 8 June 1971
Published: 1 September 1972
Authors
Catherine Bandle