Open Access
November 2009 Maximization of the second positive Neumann eigenvalue for planar domains
Alexandre Girouard, Nikolai Nadirashvili, Iosif Polterovich
J. Differential Geom. 83(3): 637-662 (November 2009). DOI: 10.4310/jdg/1264601037


We prove that the second positive Neumann eigenvalue of a bounded simply-connected planar domain of a given area does not exceed the first positive Neumann eigenvalue on a disk of half this area. The estimate is sharp and attained by a sequence of domains degenerating to a union of two identical disks. In particular, this result implies the Pólya conjecture for the second Neumann eigenvalue. The proof is based on a combination of analytic and topological arguments. As a by-product of our method we obtain an upper bound on the second eigenvalue for conformally round metrics on odd-dimensional spheres.


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Alexandre Girouard. Nikolai Nadirashvili. Iosif Polterovich. "Maximization of the second positive Neumann eigenvalue for planar domains." J. Differential Geom. 83 (3) 637 - 662, November 2009.


Published: November 2009
First available in Project Euclid: 27 January 2010

zbMATH: 1186.35120
MathSciNet: MR2581359
Digital Object Identifier: 10.4310/jdg/1264601037

Rights: Copyright © 2009 Lehigh University

Vol.83 • No. 3 • November 2009
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