Estimates on the Neumann and Steklov principal eigenvalues of collapsing domains

TL;DR

The paper investigates the relationship between the Neumann and Steklov principal eigenvalues on collapsing convex planar domains, recasting the problem in terms of thinning profiles $h$ within a 1D Sturm–Liouville framework. It identifies extremal thinning profiles: triangles $T_0$ and $T_1$ minimize the Steklov eigenvalue $\sigma_1(h)$, while the parabola $p(x)=6x(1-x)$ maximizes it, linking these to explicit eigenvalues involving Bessel functions via $\sigma_1(T_{x_0})$ and $(j'_{0,1})^2/2$. A key contribution is the construction of a transformation $\mathcal{G}$ that connects $\mu_1(h)$ and $\sigma_1(h)$, yielding sharp bounds on the ratio $\mu_1(h)/\sigma_1(h)$; specifically, $1\le \mu_1(h)/\sigma_1(h) \le 2$, with equality cases characterized by the extreme thinning shapes. Together, these results advance understanding of how domain thinning shapes influence Neumann–Steklov spectral ratios and provide tools for partially proving conjectured extremal behavior in convex thinning domains.

Abstract

We investigate the relationship between the Neumann and Steklov principal eigenvalues emerging from the study of collapsing convex domains in $\mathbb{R}^2$. Such a relationship allows us to give a partial proof of a conjecture concerning estimates of the ratio of the former to the latter: we show that thinning triangles maximize the ratio among convex thinning sets, while thinning rectangles minimize the ratio among convex thinning with some symmetry property.

Figures (2)

• Figure 1: Minimal width and diameter of a convex set.
• Figure 2: Function $w^\#$ when $w$ is a quasi-concave affine function

Theorems & Definitions (50)

• Conjecture 1: Henrot, Michetti HM22
• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 1
• Definition 2
• Definition 3
• Lemma 1
• Lemma 2
• Lemma 3