Wolf-Keller theorem for Neumann eigenvalues

Details Wolf-Keller theorem for Neumann eigenvalues Wolf-Keller theorem for Neumann eigenvalues

2010-07-27

arxiv.org/abs/1007.4771v1

Mathematics

Spectral Theory

9 pages, 1 figure

Scientific paper

The classical Szego-Weinberger inequality states that among bounded planar domains of given area, the first nonzero Neumann eigenvalue is maximized by a disk. Recently, it was shown by Girouard, Nadirashvili and Polterovich that, for simply connected planar domains of given area, the second nonzero Neumann eigenvalue is maximized in the limit by a sequence of domains degenerating to a disjoint union of two identical disks. We prove that Neumann eigenvalues of planar domains of fixed area are not always maximized by a disjoint union of arbitrary disks. This is an analogue of a result by Wolf and Keller proved earlier for Dirichlet eigenvalues.

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