[Paper Review] Dirichlet eigenvalue sums on triangles are minimal for equilaterals

Richard S. Laugesen, Bartłomiej Siudeja|arXiv (Cornell University)|Aug 7, 2010

Spectral Theory in Mathematical Physics25 references3 citations

TL;DR

This paper proves that among all triangles of fixed diameter, the equilateral triangle minimizes the sum of the first $n$ Dirichlet eigenvalues for every $n \geq 1$, using a novel method of transplanting eigenfunctions from arbitrary triangles to known equilateral and right triangles. The key contribution is a geometrically sharp lower bound on eigenvalue sums, with equality only for equilateral triangles.

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.

Motivation & Objective

• To establish geometrically sharp lower bounds for the sum of the first $n$ Dirichlet eigenvalues on triangular domains.
• To resolve the conjecture that the equilateral triangle minimizes individual eigenvalues $\lambda_n$ and their sums under fixed diameter.
• To develop a new method for eigenvalue estimation that overcomes limitations of classical rearrangement techniques for higher eigenvalues.
• To support the broader conjecture that the disk minimizes eigenvalue sums and individual eigenvalues among all planar domains of fixed diameter.

Proposed method

• Introduce a novel 'Method of the Unknown Trial Function' that transplants eigenfunctions from an arbitrary triangle to known equilateral and right triangles.
• Use interpolation across multiple transplantations to eliminate derivative distortions in the Rayleigh quotient, enabling sharp lower bounds.
• Normalize eigenvalues by the square of the diameter $D^2$ to achieve scale invariance and enable comparison across triangles.
• Combine known upper bounds on $\lambda_1$ with new lower bounds on $\lambda_1 + \lambda_2$ to prove the critical inequality for $\lambda_2$.
• Leverage explicit formulas for eigenvalues of equilateral triangles (provided in Appendix A) to compute exact bounds.
• Apply the method to prove minimality of eigenvalue sums and individual eigenvalues under diameter normalization.

Experimental results

Research questions

• RQ1 Does the equilateral triangle minimize the sum of the first $n$ Dirichlet eigenvalues among all triangles of fixed diameter, for each $n \geq 1$?
• RQ2 Are the first, second, and third Dirichlet eigenvalues individually minimized for the equilateral triangle under diameter normalization?
• RQ3 Can the transplantation-based method overcome the failure of rearrangement techniques for higher eigenvalues ($n > 1$)?
• RQ4 Is the disk the minimizer of eigenvalue sums and individual eigenvalues among all planar domains of fixed diameter?
• RQ5 Does the minimality of eigenvalues for the equilateral triangle extend to all $n$, or are there exceptions?

Key findings

• For all $n \geq 1$, the sum $\left(\lambda_1 + \cdots + \lambda_n\right) D^2$ is minimized uniquely for the equilateral triangle among all triangles of given diameter.
• $\lambda_1 D^2 \geq \frac{3 \cdot 16\pi^2}{9}$, with equality if and only if the triangle is equilateral.
• $\lambda_2 D^2 \geq \frac{7 \cdot 16\pi^2}{9}$, and this bound is sharp, with equality only for equilateral triangles.
• $\lambda_3 D^2 \geq \frac{7 \cdot 16\pi^2}{9}$, following directly from the result for $\lambda_2$.
• The method provides the first geometrically sharp lower bounds for eigenvalue sums in this context, surpassing asymptotic bounds like Berezin–Li–Yau.
• The results support the conjecture that the disk minimizes $\left(\lambda_1 + \cdots + \lambda_n\right) D^2$ among all planar domains, with equality only for the disk.