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

Guowei Dai; Yingxin Sun

Home/Archive/arXiv

arXiv · 2511.20988

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

Guowei Dai·Yingxin Sun

View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Guowei Dai, Yingxin Sun. 2025-12-20. Upper bound estimation for the ratio of the first two eigenvalues of Robin Laplacian. https://arxiv.org/abs/2511.20988

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Around the Quantum Lenard-Balescu equation

In the mean-field regime, a gas of quantum particles with Boltzmann statistics can be described by the Hartree-Fock equation. This dynamics becomes trivial if the initial distribution of particle is invariant by translation. However, the first correction is given on time of order $O(N)$ by the quantum Lenard--Balescu equation. In the first part of the present article, we justify this equation until time of order $O((\log N)^{1-δ})$ (for any $δ\in(0,1)$). A similar phenomenon exists in the classical setting (with a similar validity time obtained by Duerinckx \cite{Duerinckx}). In a second time, we prove the convergence for dimension $d\geq 2$ of the solutions of the quantum Lenard--Balescu equation to the solutions of its classical counterpart in the semi-classical limit. This problem can be interpreted as a grazing collision limit: the quantum Lenard--Balescu equation looks like a cut-off Boltzmann equation, when the classical one looks like the Landau equation.

math.AP↗

arXiv2026

Almost Periodic Solutions of The Cubic Defocusing Nonlinear Schrödinger Equation

This paper addresses the Cauchy problem for the cubic defocusing nonlinear Schrödinger equation (NLS) with almost periodic initial data. We prove that for small analytic quasiperiodic initial data satisfying Diophantine frequency conditions, the Cauchy problem admits a solution that is almost periodic in both space and time, and that this solution is unique among solutions locally bounded in a suitable sense. The analysis combines direct and inverse spectral theory. In the inverse spectral theory part, we prove existence, almost periodicity, and uniqueness for solutions with initial data whose associated Dirac operator has purely a.c.\ spectrum that is not too thin. This resolves novel challenges presented by the NLS hierarchy, such as an additional degree of freedom and an additional commuting flow. In the direct spectral theory part, for Dirac operators with small analytic quasiperiodic potentials with Diophantine frequency conditions, we prove pure a.c.\ spectrum, exponentially decaying spectral gaps, and spectral thickness conditions (homogeneity and Craig-type conditions).

math.AP↗

arXiv2026

Large-Amplitude Steady Solitary Water Waves with General Vorticity

We study two-dimensional steady solitary gravity water waves with general vorticity, allowing for overhanging free-surface profiles. The main challenges arise from the free boundary, the unbounded fluid domain, and the inherent complexity of the general vorticity setting. To address these, we introduce a conformal reformulation that reduces the problem to an equivalent system on a fixed strip, consisting of an overdetermined elliptic problem coupled with an elliptic boundary-value problem. This framework enables a local analysis without imposing restrictive assumptions on the vorticity. Using the center-manifold construction of Chen, Walsh and Wheeler \cite{chennonlinearity}, we first establish the existence of small-amplitude solitary waves for such non-trivial vorticity distributions. Subsequently, via a global analytic bifurcation argument, we prove the existence of continuous branches of large-amplitude solitary waves for analytic vorticity functions. Along these global solution curves, the free surfaces may develop overhanging profiles and need not remain graphs over the horizontal coordinate. Our results provide a new constructive framework for large-amplitude solitary waves with general vorticity, extending the existing theory beyond the classical settings of constant vorticity \cite{susannaarma} and non-overhanging profiles \cite{milesjma}.

math.AP↗