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June 2019 Maximization of the second non-trivial Neumann eigenvalue
Dorin Bucur, Antoine Henrot
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Acta Math. 222(2): 337-361 (June 2019). DOI: 10.4310/ACTA.2019.v222.n2.a2

Abstract

In this paper we prove that the second (non-trivial) Neumann eigenvalue of the Laplace operator on smooth domains of $\mathbb{R}^N$ 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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Dorin Bucur. Antoine Henrot. "Maximization of the second non-trivial Neumann eigenvalue." Acta Math. 222 (2) 337 - 361, June 2019. https://doi.org/10.4310/ACTA.2019.v222.n2.a2

Information

Received: 22 January 2018; Published: June 2019
First available in Project Euclid: 16 April 2020

zbMATH: 1423.35271
MathSciNet: MR3974477
Digital Object Identifier: 10.4310/ACTA.2019.v222.n2.a2

Subjects:
Primary: 35P15 , 49Q10

Rights: Copyright © 2019 Institut Mittag-Leffler

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Vol.222 • No. 2 • June 2019
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