A note on optimization of the second positive Neumann eigenvalue for parallelograms

TL;DR

This work addresses the problem of maximizing the third Neumann Laplacian eigenvalue $\mu_3$ under a fixed perimeter constraint within the class of parallelograms. It exploits a linear transformation to map parallelograms to a rectangle, enabling a variational setup where trial functions from rectangle eigenfunctions yield a tractable finite matrix bound. The authors derive explicit matrices $\mathsf L_{c,d}$ and $\mathsf M_{c,d}$ whose eigenvalues control $\mu_3(\Omega_{c,d})$, and combine this with Kröger’s universal bound to cover all parameter regions, ultimately proving that the maximum occurs at the rectangle with side ratio $2:1$, with strictness except in the limiting case. This establishes the conjectured optimizer within parallelograms and provides a sharp bound $|\partial\Omega|^2\mu_3(\Omega)\le 36\pi^2$ for this class. The approach integrates variational principles, explicit spectral computations, and region-wise analysis to achieve a complete proof in this domain.

Abstract

It has recently been conjectured by Bogosel, Henrot, and Michetti that the second positive eigenvalue of the Neumann Laplacian is maximized, among all planar convex domains of fixed perimeter, by the rectangle with one edge length equal to twice the other. In this note we prove that this conjecture is true within the class of parallelogram domains.

Figures (3)

• Figure 1: The parallelogram $\Omega_{c,d}$ with vertices having coordinates $(0,0)$, $(c,d)$, $(c+1,d)$, and $(1,0)$.
• Figure 2: The regions $R_1,R_2,R_3$, and $R_4$.
• Figure 3: Graph of the function $S$ on the interval $[\frac{1}{2},\frac{3}{4}]$

Theorems & Definitions (18)

• Lemma 2.1
• proof
• Corollary 2.2
• Proposition 3.1
• proof
• Lemma 4.1
• proof
• Theorem 4.2
• Theorem 4.3: Main result
• proof