Dirichlet eigenvalue sums on triangles are minimal for equilaterals

TL;DR

This paper proves that among triangles with a fixed diameter, the equilateral triangle minimizes the sum of the first n Dirichlet Laplacian eigenvalues, with specific eigenvalues also minimized individually.

Contribution

It establishes the minimality of the equilateral triangle for eigenvalue sums and individual eigenvalues among triangles of fixed diameter.

Findings

Equilateral triangle minimizes eigenvalue sums for all n ≥ 1 among triangles.

First, second, and third eigenvalues are individually minimized by the equilateral triangle.

Conjecture: the disk minimizes eigenvalues among all domains.

Abstract

Among all triangles of given diameter, the equilateral triangle is shown to minimize the sum of the first $n$ eigenvalues of the Dirichlet Laplacian, for each $n \geq 1$. In addition, the first, second and third eigenvalues are each proved to be minimal for the equilateral triangle. The disk is conjectured to be the minimizer among general domains.