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arXiv · 0801.2142

Maximization of the second positive Neumann eigenvalue for planar domains

Alexandre Girouard·Nikolai Nadirashvili·Iosif Polterovich
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Abstract

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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Alexandre Girouard, Nikolai Nadirashvili, Iosif Polterovich. 2008-01-28. Maximization of the second positive Neumann eigenvalue for planar domains. https://arxiv.org/abs/0801.2142

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