Maximization of the second non-trivial Neumann eigenvalue

Dorin Bucur; Antoine Henrot

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arXiv · 1801.07435

Maximization of the second non-trivial Neumann eigenvalue

Dorin Bucur·Antoine Henrot

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Abstract

In this paper we prove that the second (non-trivial) Neumann eigenvalue of the Laplace operator on smooth domains of R N with prescribed measure m attains its maximum on the union of two disjoint balls of measure m 2. As a consequence, the P{\'o}lya 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. 2018-01-23. Maximization of the second non-trivial Neumann eigenvalue. https://arxiv.org/abs/1801.07435

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