Upper bound estimation for the ratio of the first two eigenvalues of Robin Laplacian

TL;DR

This work extends the Payne–Polya–Weinberger (PPW) paradigm to Robin Laplacians by proving that, in any dimension $N\ge2$ and for all boundary parameters $\sigma>0$, the ratio of the first two Robin eigenvalues $\mu_2/\mu_1$ is maximized by the ball under an interior sphere condition. The authors develop a Robin-specific Chiti-type comparison framework and establish strict monotonicity of a key trial-function pair $(w,B)$, enabling a sharp bound $\mu_2/\mu_1 \le k_{\nu+1,1}^2/k_{\nu,1}^2$ with equality only for balls; they also show the maximal bound is strictly decreasing in $\sigma$ and connects to the Dirichlet case as $\sigma\to\infty$. A secondary result provides a tunable upper bound for the eigenvalue ratio via $\sigma$ and clarifies the behavior of the spectral gap. Collectively, the results give a quantitative, parameter-dependent answer to whether an elastically supported drum is circular based on its first two eigenfrequencies and extend the AB inequality to Robin problems.

Abstract

The celebrated conjecture by Payne, Pólya and Weinberger (1956) states that for the fixed membrane problem, the ratio of the first two eigenvalues, $λ_2/λ_1$, is maximized by a disk. A more general dimensional version of this conjecture was later resolved by Ashbaugh and Benguria in the 1990s. For the Robin Laplacian, Payne and Schaefer (2001) formulated an analogous conjecture, positing that the ratio $μ_2/μ_1$ is also maximized by a disk for a range of the boundary parameter $σ$. This was later restated by Henrot in 2003. In this work, under some suitable conditions, we affirm this conjecture for all dimensions $N\geq2$ and for all $σ>0$. Furthermore, we prove that the maximum value of $μ_2/μ_1$ is strictly decreasing in $σ$ over the entire interval $(0,+\infty)$. Our result provides a positive answer to a variant of Yau's Problem 77: by measuring the ratio of the first two eigenfrequencies, one can determine whether an elastically supported drum is circular.

Figures (5)

• Figure 1: The schematic diagram of $k_{\nu+1,1}^2/k_{\nu,1}^2$.
• Figure 2: The schematic diagrams of $w(r)$ and $B(r)$.
• Figure 3: The schematic diagrams of $u_1^*(s)$ (solid line) and $z^*(s)$ (dotted line).
• Figure 4: The schematic diagrams of $u_1^*(s)$ (solid line) and $z^*(s)$ (dotted line).
• Figure 5: The schematic diagrams of Theorem 3.1.