Extremal eigenvalues of the Dirichlet biharmonic operator on rectangles

TL;DR

This paper investigates the extremal eigenvalues of the Dirichlet biharmonic operator on rectangles, showing the minimal principal eigenvalue occurs within a specific aspect ratio range and that the sequence of minimizers converges to a square as eigenvalue index increases.

Contribution

It establishes bounds on the aspect ratio for minimal principal eigenvalues and proves the convergence of minimising rectangles to a square for higher eigenvalues.

Findings

Principal eigenvalue minimized for rectangles with aspect ratio ≤ 1.066459

Sequence of minimal eigenvalues' rectangles converges to a square

Provides bounds and convergence results for extremal eigenvalues

Abstract

We study the behaviour of extremal eigenvalues of the Dirichlet biharmonic operator over rectangles with a given fixed area. We begin by proving that the principal eigenvalue is minimal for a rectangle for which the ratio between the longest and the shortest side lengths does not exceed $1.066459$. We then consider the sequence formed by the minimal $k^{\rm th}$ eigenvalue and show that the corresponding sequence of minimising rectangles converges to the square as $k$ goes to infinity.