Maximization of the second positive Neumann eigenvalue for planar domains
Spectral Theory
2012-02-24 v2
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.
Cite
@article{arxiv.0801.2142,
title = {Maximization of the second positive Neumann eigenvalue for planar domains},
author = {Alexandre Girouard and Nikolai Nadirashvili and Iosif Polterovich},
journal= {arXiv preprint arXiv:0801.2142},
year = {2012}
}
Comments
24 pages, 2 figures; Conjecture 1.2.3 corrected