Rayleigh's conjecture for vibrating plates: simpler planar proof and nonlinear extensions
Abstract
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 |Δu|q/|u|p among bounded open sets of prescribed area, for every 1≤ p,q≤∞. Equality is characterized for q>1, while exceptional behavior at q=1 is identified. In higher dimensions, we prove ball minimality for p=∞ and conjecture minimality for all p,q within the compact embedding range. We also establish the corresponding interval inequality in one dimension.
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BibTeX
@misc{bucur2026rayleighs,
title = {{Rayleigh's conjecture for vibrating plates: simpler planar proof and nonlinear extensions}},
author = {Dorin Bucur and Richard S. Laugesen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29840},
eprint = {2610.29840},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29840},
}
RIS
TY - UNPB
TI - Rayleigh's conjecture for vibrating plates: simpler planar proof and nonlinear extensions
AU - Dorin Bucur
AU - Richard S. Laugesen
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29840
UR - https://arxiv.org/abs/2610.29840
AB - 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 |Δu|_q/|u|_p among bounded open sets of prescribed area, for every 1≤ p,q≤∞. Equality is characterized for q>1, while exceptional behavior at q=1 is identified. In higher dimensions, we prove ball minimality for p=∞ and conjecture minimality for all p,q within the compact embedding range. We also establish the corresponding interval inequality in one dimension.
ER -