Maximization of the second positive Neumann eigenvalue for planar domains

Details Maximization of the second positive Neumann eigenvalue for planar domains Maximization of the second positive Neumann eigenvalue for planar domains

2008-01-14

arxiv.org/abs/0801.2142v2

J. Differential Geom. 83 (2009), no. 3, 637-661

Mathematics

Spectral Theory

24 pages, 2 figures; Conjecture 1.2.3 corrected

Scientific paper

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 a twice smaller area. This estimate is sharp and attained by a sequence of domains degenerating to a union of two identical disks. In particular, this result implies the Polya 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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