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Maximization of the second positive neumann eigenvalue for planar domains

Alexandre Girouard Universit´e de Montr´eal Nikolai Nadirashvili Centre de Math´ematiques et Informatique Iosif Universit´e de Montr´eal

Differential Geometry mathscidoc:1609.10151

Journal of Differential Geometry, 83, (3), 637-661, 2009
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 P´olya 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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@inproceedings{alexandre2009maximization,
  title={MAXIMIZATION OF THE SECOND POSITIVE NEUMANN EIGENVALUE FOR PLANAR DOMAINS},
  author={Alexandre Girouard, Nikolai Nadirashvili, and Iosif},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160911203709397320808},
  booktitle={Journal of Differential Geometry},
  volume={83},
  number={3},
  pages={637-661},
  year={2009},
}
Alexandre Girouard, Nikolai Nadirashvili, and Iosif. MAXIMIZATION OF THE SECOND POSITIVE NEUMANN EIGENVALUE FOR PLANAR DOMAINS. 2009. Vol. 83. In Journal of Differential Geometry. pp.637-661. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160911203709397320808.
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