Towards the optimality of the ball for the Rayleigh Conjecture concerning the clamped plate

TL;DR

This paper investigates the Rayleigh Conjecture for the first eigenvalue of the bilaplacian with clamped boundary conditions, providing a conditional proof in higher dimensions based on the eigenfunction's critical points.

Contribution

It offers a new conditional proof of the Rayleigh Conjecture in dimensions greater than three, assuming the eigenfunction's critical points are only at its extrema.

Findings

Conditional proof of the Rayleigh Conjecture in higher dimensions

Improved Talenti's comparison principle for eigenfunction analysis

Analysis of eigenfunction nodal domain geometry

Abstract

In 1995, Nadirashvili and subsequently Ashbaugh and Benguria proved the Rayleigh Conjecture concerning the first eigenvalue of the bilaplacian with clamped boundary conditions in dimension $2$ and $3$. Since then, the conjecture has remained open in dimension $d>3$. In this document, we contribute in answering the conjecture under a particular assumption regarding the critical values of the optimal eigenfunction. More precisely, we prove that if the optimal eigenfunction has no critical value except its minimum and maximum, then the conjecture holds. This is performed thanks to an improvement of Talenti's comparison principle, made possible after a fine study of the geometry of the eigenfunction's nodal domains.