Maximising Neumann eigenvalues on rectangles

TL;DR

This paper investigates the optimization of Neumann eigenvalues on rectangles, demonstrating convergence of maximizers to the unit square and identifying unique maximizers under perimeter constraints.

Contribution

It provides new results on spectral optimization of Neumann eigenvalues on rectangles, including asymptotic behavior and maximizer uniqueness under perimeter constraints.

Findings

Maximizers converge to the unit square as eigenvalue index increases

Unique maximizer of Neumann eigenvalues identified for fixed perimeter

Results applicable to spectral optimization problems in geometric analysis

Abstract

We obtain results for the spectral optimisation of Neumann eigenvalues on rectangles in $\mathbb{R}^2$ with a measure or perimeter constraint. We show that the rectangle with measure $1$ which maximises the $k$'th Neumann eigenvalue converges to the unit square in the Hausdorff metric as $k\rightarrow \infty$. Furthermore, we determine the unique maximiser of the $k$'th Neumann eigenvalue on a rectangle with given perimeter.