The second Dirichlet eigenvalue is simple on every non-equilateral triangle

TL;DR

This work resolves the open question of whether the second Dirichlet eigenvalue $\lambda_2$ of the Laplacian on a triangle is simple for every non-equilateral triangle. By studying collapsing triangles, the authors introduce three related eigenvalue problems that connect to a one-dimensional Schrödinger operator with Airy-type asymptotics, and they derive rigorous two-sided bounds for the intermediate eigenvalues $\mu_k^t$, with $\mu_k^t$ linked to $\lambda_k^t$ via $\lambda_k^t = t^{-4/3}(\mu_k^t + \pi^2/t^2)$. A key result is the two-sided bound $\frac{\bar{\mu}_k}{1 + \frac{t_0^{2/3}}{3\pi^2}\,\bar{\mu}_k} \le \mu_k^t \le \hat{\mu}_k^{t_0}$, together with a projection-based lower bound, which implies $\mu_k^t \to \bar{\mu}_k$ as $t\to0+$ and yields the asymptotics $\lambda_k^t \sim t^{-2}(\pi^2 + \bar{\mu}_k t^{2/3})$. To complete the conjecture in the degenerate regime, a computer-assisted proof bounds $\hat{\mu}_2^{t_0}(s)$ and $\bar{\mu}_3(s)$ uniformly over a moduli space of nearly degenerate triangles, showing $\lambda_2(s,t) < \lambda_3(s,t)$ for all admissible $(s,t)$; this, together with prior results for less-degenerate cases, establishes the simplicity of $\lambda_2$ for all non-equilateral triangles. The approach blends analytic asymptotics with rigorous numerical verification (via interval arithmetic and Airy-function analysis), and the authors provide code publicly for reproducibility. This result completes the conjecture in Henrot's Shape Optimization and Spectral Theory and strengthens the connection between domain collapse, spectral stability, and Airy-type spectral problems.

Abstract

The Dirichlet eigenvalues of the Laplacian on a triangle that collapses into a line segment diverge to infinity. In this paper, to track the behavior of the eigenvalues during the collapsing process of a triangle, we establish a quantitative error estimate for the Dirichlet eigenvalues on collapsing triangles. As an application, we solve the open problem concerning the simplicity of the second Dirichlet eigenvalue for nearly degenerate triangles, offering a complete solution to Conjecture 6.47 posed by R. Laugesen and B. Siudeja in A. Henrot's book ``Shape Optimization and Spectral Theory".

Figures (4)

• Figure 1: Shape of triangle $T(s,t)$
• Figure 2: Shape of triangle $T^t$
• Figure 3: Moduli space of triangles
• Figure 4: Graph of $\hat{\mu}_2^{t_0}(s)$, $\bar{\mu}_3(s)$ and $\underline{\mu}_3(s)$

Theorems & Definitions (23)

• Conjecture 1.1: Conjecture 6.47 of henrot2017shape
• Theorem 1.1
• Lemma 1.2
• Lemma 2.1
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3