Isoperimetric inequalities for eigenvalues of triangles

TL;DR

This paper establishes lower bounds for the first eigenvalue of the Dirichlet Laplacian on triangles using symmetrization, and identifies the equilateral triangle as minimizing the spectral gap and eigenvalue ratio.

Contribution

It generalizes Pólya's isoperimetric bounds and proves that the equilateral triangle minimizes spectral gap and eigenvalue ratios among all triangles.

Findings

Lower bounds for the first eigenvalue on triangles

Equilateral triangle minimizes spectral gap

Equilateral triangle minimizes eigenvalue ratio under certain conditions

Abstract

Lower bounds estimates are proved for the first eigenvalue for the Dirichlet Laplacian on arbitrary triangles using various symmetrization techniques. These results can viewed as a generalization of P\'olya's isoperimetric bounds. It is also shown that amongst triangles, the equilateral triangle minimizes the spectral gap and (under additional assumption) the ratio of the first two eigenvalues. This last result resembles the Payne-P\'olya-Weinberger conjecture proved by Ashbaugh and Benguria.