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Isoperimetric inequalities for eigenvalue ratios. (English) Zbl 0814.35081

Alvino, Angelo (ed.) et al., Partial differential equations of elliptic type. Proceedings of the conference held October 12-16, 1992 in Cortona, Italy. Cambridge: University Press. Symp. Math. 35, 1-36 (1994).
This paper is divided into three parts. The first contains a survey of classical inequalities for different eigenvalues of the membrane problem. The list is fairly complete, starting from the pioneering work of Pólya and Szegö till recent results. Emphasis is put on the inequalities of Payne, Pólya and Weinberger and Hile and Protter to which the authors have done major contributions.
The second part deals with extensions to domains on the spheres. It is well-known that the Rayleigh-Faber-Krahn inequality holds for any domain on the sphere. However for other inequalities there seems to be a restriction.
The third part contains open problems, some of them go back to Pólya. This survey is well-written and can be of great use. It also contains a large list of references.
For the entire collection see [Zbl 0799.00025].
Reviewer: C.Bandle (Basel)

MSC:

35P15 Estimates of eigenvalues in context of PDEs
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