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arXiv · 2610.13180

On the inequality $μ_3 < λ_1$ for the eigenvalues of the Laplace operator

N. Filonov
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Abstract

In 1952, Pólya proved the inequality $μ_2 < λ_1$ for the eigenvalues of Dirichlet and Neumann problems in domain $Ω\subset \mathbb{R}^d$, $d \ge 2$. We show that $μ_3 < λ_1$ if $d\ge 3$. In the two-dimensional case, in 2025, Rohleder proved that $μ_3 < λ_1$ for simply connected domains. We prove that this inequality holds also true for double connected domains symmetric with respect to the origin.

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N. Filonov. 2026-10-09. On the inequality $μ_3 < λ_1$ for the eigenvalues of the Laplace operator. https://arxiv.org/abs/2610.13180

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