Speaker: 代国伟教授，大连理工大学

Inviter: 流体力学方程研究中心

Title: Rayleigh conjecture and the lower bound estimation of the first eigenvalue for vibrating clamped plate

Language: Chinese

Time & Venue: 2026.04.23 09:30-10:30 南楼N204

Abstract: In 1877, Lord Rayleigh conjectured that among all clamped plates ($\Omega$) of equal area, the circular plate minimizes the fundamental frequency. This conjecture was proven in two dimensions by Nadirashvili in 1995, a result soon extended to dimensions two and three by Ashbaugh and Benguria that same year. However, its validity in $\mathbb{R}^N$ for $N \geq 4$ remains an open problem. A key challenge in this setting arises from the fact that the eigenfunctions associated with the first eigenvalue of a clamped plate can change sign. This observation leads to a natural question: How does the size of nodal domains, or the degree of sign change, influence the first eigenvalue? Let $\Omega_+$ denote a positive nodal domain of such an eigenfunction. This work establishes lower bounds for the first eigenvalue in terms of the ratio $\left\vert\Omega_+\right\vert / \left\vert\Omega\right\vert$. Our results demonstrate that this lower bound increases monotonically with the size of the positive nodal domain, a finding that, in certain special cases, yields a positive answer to the Rayleigh conjecture.