Sharp ratios for low-index Neumann eigenvalues on convex domains

Summary · qwen2.5:32b

Researchers prove sharp bounds for low-index Neumann eigenvalues on convex domains, specifically $\mu_2(\Omega)\le 4\mu_1(\Omega)$ and $\mu_3(\Omega)\le 9\mu_1(\Omega)$, resolving a problem attributed to Henrot and confirming predictions from the one-dimensional model with constants optimal in every dimension.

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Discovering sharp ratios for low-index Neumann eigenvalues on convex domains could lead to advancements in real-time rendering of complex shapes in graphics.

Excerpt

arXiv:2607.05388v1 Announce Type: new Abstract: Let $\Omega\subset\mathbb{R}^N$ be a bounded open convex set, and let $0=\mu_0(\Omega)<\mu_1(\Omega)\le \mu_2(\Omega)\le\cdots$ be the Neumann eigenvalues of the Laplacian, repeated according to multiplicity. We prove the sharp bounds $$ \mu_2(\Omega)\le 4\mu_1(\Omega),\qquad \mu_3(\Omega)\le 9\mu_1(\Omega). $$ The first estimate resolves a problem attributed to Henrot, while the second gives the next sharp case predicted by the one-dimensional model. The constants are optimal in every dimension.