Global and Local Bounds on the Fundamental Ratio of Triangles and Quadrilaterals

TL;DR

This paper provides a computer-assisted proof that equilateral triangles uniquely maximize the ratio of the first two Dirichlet-Laplacian eigenvalues among all triangles, and shows squares are local optimizers among quadrilaterals.

Contribution

It offers a new, rigorous computational method combining perturbative estimates and continuity arguments to establish extremal properties of geometric shapes for eigenvalue ratios.

Findings

Equilateral triangles uniquely maximize the fundamental eigenvalue ratio among triangles.

Squares are strict local optimizers of the fundamental ratio among quadrilaterals.

The method confirms conjectures through computer-assisted, rigorous proofs.

Abstract

We present a new, computer-assisted, proof that for all triangles in the plane, the equilateral triangle uniquely maximizes the ratio of the first two Dirichlet-Laplacian eigenvalues. This proves an independent proof the triangular Ashbaugh-Benguria-Payne-Polya-Weinberger conjecture first proved in arXiv:0707.3631 [math.SP] and arXiv:2009.00927 [math.SP]. Inspired by arXiv:1109.4117 [math.SP], the primary method is to use a perturbative estimate to determine a local optimum, and to then use a continuity estimate for the fundamental ratio to perform a rigorous computational search of parameter space. We repeat a portion of this proof to show that the square is a strict local optimizer of the fundamental ratio among quadrilaterals in the plane