Rayleigh's conjecture for vibrating plates: simpler planar proof and nonlinear extensions

We give a simpler proof of Nadirashvili’s theorem (Rayleigh’s conjecture) that among vibrating clamped plates of prescribed area in the plane, the disk has the lowest fundamental tone. The proof uses a one-shot Talenti type comparison for sign-changing solutions of the Poisson equation with mean-zero data, established in a companion paper. The comparison considers just two rearranged sources, rather than optimizing over all admissible sources as in the traditional two-ball technique. This ``paired comparison’’ simplification yields immediate nonlinear extensions: the disk minimizes the clamped quotient $\mathrm{\parallel }\mathrm{\Delta }u{\mathrm{\parallel }}_q\mathrm{/}\mathrm{\parallel }u{\mathrm{\parallel }}_p$ among bounded open sets of prescribed area, for every $1\le p,q\le \mathrm{\infty }$. Equality is characterized for $q>1$, while exceptional behavior at $q=1$ is identified. In higher dimensions, we prove ball minimality for $p=\mathrm{\infty }$ and conjecture minimality for all $p,q$ within the compact embedding range. We also establish the corresponding interval inequality in one dimension.