Sufficient conditions yielding the Rayleigh Conjecture for the clamped plate

TL;DR

This paper establishes sufficient conditions under which the Rayleigh Conjecture for the clamped plate holds in any dimension, using an order reduction principle and shape optimization techniques.

Contribution

It introduces a new order reduction method and provides sufficient conditions for the Rayleigh Conjecture in higher dimensions, extending previous results.

Findings

The conjecture holds if the eigenfunction's mean value is minimal.

A second condition involves the Laplacian's constant normal derivative on the boundary.

Explicit eigenfunction computation for the ball is achieved.

Abstract

The Rayleigh Conjecture for the bilaplacian consists in showing that the clamped plate with least principal eigenvalue is the ball. The conjecture has been shown to hold in 1995 by Nadirashvili in dimension $2$ and by Ashbaugh and Benguria in dimension $3$. Since then, the conjecture remains open in dimension $d\geq 4$. In this paper, we contribute to answer this question, and show that the conjecture is true in any dimension as long as some special condition holds on the principal eigenfunction of an optimal shape. This condition regards the mean value of the eigenfunction, asking it to be in some sense minimal. This main result is based on an order reduction principle allowing to convert the initial fourth order linear problem into a second order affine problem, for which the classical machinery of shape optimization and elliptic theory is available. The order reduction principle turns out to be a general tool. In particular, it is used to derive another sufficient condition for the conjecture to hold, which is a second main result. This condition requires the Laplacian of the optimal eigenfunction to have constant normal derivative on the boundary. Besides our main two results, we detail shape derivation tools allowing to prove simplicity for the principal eigenvalue of an optimal shape and to derive optimality conditions. Eventually, because our first result involves the principal eigenfunction of a ball, we are led to compute it explicitly.