Maximization of the second non-trivial Neumann eigenvalue

Dorin Bucur Université Grenoble Alpes Antoine Henrot Université de Lorraine

Analysis of PDEs mathscidoc:1911.43032

Acta Mathematica, 222, (2), 337-361, 2019
In this paper we prove that the second (non-trivial) Neumann eigenvalue of the Laplace operator on smooth domains of RN with prescribed measure m attains its maximum on the union of two disjoint balls of measure m/2. As a consequence, the Pólya conjecture for the Neumann eigenvalues holds for the second eigenvalue and for arbitrary domains. We moreover prove that a relaxed form of the same inequality holds in the context of non-smooth domains and densities.
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@inproceedings{dorin2019maximization,
  title={Maximization of the second non-trivial Neumann eigenvalue},
  author={Dorin Bucur, and Antoine Henrot},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191128120533199921543},
  booktitle={Acta Mathematica},
  volume={222},
  number={2},
  pages={337-361},
  year={2019},
}
Dorin Bucur, and Antoine Henrot. Maximization of the second non-trivial Neumann eigenvalue. 2019. Vol. 222. In Acta Mathematica. pp.337-361. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191128120533199921543.
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