Range of the first three eigenvalues of the planar Dirichlet Laplacian

TL;DR

The paper addresses the problem of determining the admissible region for the first three Dirichlet Laplacian eigenvalue ratios on planar domains, namely $\left(\frac{\lambda_2}{\lambda_1}, \frac{\lambda_3}{\lambda_1}\right)$. It combines a survey of universal and isoperimetric bounds with extensive numerical experiments across diverse domain classes and employs perturbation theory to infer structural properties of potential maximizers. A key contribution is the identification that the maximal ratio $\frac{\lambda_3}{\lambda_1}$ among tested domains is $Y^*\approx 3.202$, attained by a dumbbell-shaped domain with $\lambda_3\approx\lambda_4$, and that the rectangle $R_{\sqrt{\frac{8}{3}}}$ is not the global maximizer; the work also establishes that a local maximizer would have to satisfy $\lambda_3=\lambda_4$ under plausible assumptions. These results narrow the search for spectral-maximizers in planar domains and suggest that the optimal shapes are simply connected with smooth boundaries near dumbbell geometries, with implications for spectral-geometry and shape-optimization problems.

Abstract

We conduct extensive numerical experiments aimed at finding the admissible range of the ratios of the first three eigenvalues of a planar Dirichlet Laplacian. The results improve the previously known theoretical estimates of M Ashbaugh and R Benguria. We also prove some properties of a maximizer of the ratio $λ_3/λ_1$.

Figures (9)

• Figure 1: Admissible range (shaded) of $\left(\frac{\lambda_2}{\lambda_1},\frac{\lambda_3}{\lambda_1}\right)$ according to AshBen6. Shown for comparison are the maximum values of $\frac{\lambda_3}{\lambda_1}$ as functions of $\frac{\lambda_2}{\lambda_1}$ for rectangles and disjoint unions of circles.
• Figure 2: Parametrization of a quadrilateral.
• Figure 3: $y^\ast(x)$ for triangles, quadrilaterals and ellipses
• Figure 4: $y^\ast(x)$ for random sectors of annuli and pseudo-random polygons
• Figure 5: $y^\ast(x)$ for pseudo-random star-shaped domains

Theorems & Definitions (10)

• Remark 2.1
• Remark 2.2
• Remark 3.1
• Remark 3.2
• Theorem 4.1
• Theorem 4.2
• Conjecture 4.3
• Proposition 4.4
• Proposition 4.5
• Remark 4.6