Maximization of the second non-trivial Neumann eigenvalue

2018/01/23 by Bucur, Dorin, Henrot, Antoine · 3 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1801.07435

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 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.

Cited by

• Sharp shifted reciprocal sums of Neumann eigenvalues on space forms
• Improved Berezin-Li-Yau inequality and Kröger inequality and consequences
• Pólya's conjecture up to ε-loss and quantitative estimates for the remainder of Weyl's law

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