Sharp Dirichlet eigenvalue inequalities on triangles

TL;DR

This paper establishes sharp inequalities for the first Dirichlet eigenvalue of triangles, confirming a conjecture and introducing a new method based on shape derivatives and finite-element estimates.

Contribution

It proves a conjecture about the minimality of the equilateral triangle for a scale-invariant eigenvalue functional and develops a versatile new proof technique.

Findings

The equilateral triangle uniquely minimizes the eigenvalue functional.

Derived an optimal lower bound for the first Dirichlet eigenvalue based on area and perimeter.

Established a Cheeger-type inequality with an explicit best constant.

Abstract

We prove sharp Dirichlet eigenvalue inequalities for planar triangles. We settle a conjecture of Laugesen and Siudeja by showing that the equilateral triangle uniquely minimizes a scale-invariant functional of the first Dirichlet eigenvalue, area, and perimeter. Consequences include an optimal two-term lower bound for the first Dirichlet eigenvalue in terms of area and perimeter. We also prove a Cheeger-type inequality with an explicit best constant considered by Parini. To prove these conjectures we propose a new method for proving Dirichlet eigenvalue inequalities on triangles. Our method is based on a new computable lower bound for second-order directional shape derivatives under vertex perturbations. It also uses validated finite-element error estimates and recently developed analytic estimates for eigenvalues of nearly degenerate triangles. The method is not specific to the functionals considered in this paper and it can be used to prove various other eigenvalue inequalities on triangles.