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

TL;DR

This paper proves that among parallelograms, the rectangle with one side twice the other maximizes the second positive Neumann Laplacian eigenvalue, supporting a broader conjecture about convex domains.

Contribution

It confirms the conjecture for parallelogram domains, establishing the optimal shape for the second positive Neumann eigenvalue within this class.

Findings

The second positive Neumann eigenvalue is maximized by a specific rectangle among parallelograms.

Supports the conjecture that the rectangle maximizes the eigenvalue among all convex domains of fixed perimeter.

Provides a partial proof for a broader conjecture in spectral geometry.

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.