On Rayleigh's conjecture for the clamped plate and its generalization to three dimensions
The Rayleigh's conjecture for the clamped plate -- when the edges are clamped, the form of gravest pitch is doubtless the circle -- is considered. Namely, the eigenvalue problem \(\Delta^2 u= \lambda u\) in \(\Omega\subset \mathbb{R}^n\), \(u= |\nabla u|= 0\) on \(\partial\Omega\), where \(\Omega\) is any bounded domain in \(\mathbb{R}^n\) \((n= 2,3)\) with smooth boundary is investigated. Let \(\lambda(\Omega)\) be the principal eigenvalue. The main result is the following. For the eigenvalue problem mentioned above one has \(\lambda(\Omega)\geq \lambda(\Omega^*)\), where \(\Omega^*\) is the ball in \(\mathbb{R}^n\) having the same volume as the domain \(\Omega\) and equality holds if and only if \(\Omega\) is a ball.
scientific article; zbMATH DE number 784722
scientific article