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Sufficient conditions yielding the Rayleigh Conjecture for the clamped plate

Roméo Leylekian

TL;DR

The paper addresses the Rayleigh Conjecture for the bilaplacian in the clamped-plate setting and extends understanding to arbitrary dimensions by proposing two sufficient conditions that force the optimising domain to be a ball. It introduces an order reduction principle that converts the fourth-order bilaplacian problem into a second-order affine framework, enabling shape-derivative and symmetrization techniques, as well as a Serrin-type overdetermined argument. The main contributions are (i) a Faber–Krahn-type result under a minimal-mean condition on the principal eigenfunction, valid in any dimension, and (ii) a separate condition where a constant normal derivative of $\Delta u$ on the boundary implies the domain is a ball; the principal eigenvalue is shown to be simple, and the ball eigenfunction is computed explicitly. Collectively, these results provide dimension-free sufficient criteria for the Rayleigh Conjecture, introduce a general reduction tool for higher-order operators, and supply computable benchmarks (via the ball) to guide future verification efforts.

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.

Sufficient conditions yielding the Rayleigh Conjecture for the clamped plate

TL;DR

The paper addresses the Rayleigh Conjecture for the bilaplacian in the clamped-plate setting and extends understanding to arbitrary dimensions by proposing two sufficient conditions that force the optimising domain to be a ball. It introduces an order reduction principle that converts the fourth-order bilaplacian problem into a second-order affine framework, enabling shape-derivative and symmetrization techniques, as well as a Serrin-type overdetermined argument. The main contributions are (i) a Faber–Krahn-type result under a minimal-mean condition on the principal eigenfunction, valid in any dimension, and (ii) a separate condition where a constant normal derivative of on the boundary implies the domain is a ball; the principal eigenvalue is shown to be simple, and the ball eigenfunction is computed explicitly. Collectively, these results provide dimension-free sufficient criteria for the Rayleigh Conjecture, introduce a general reduction tool for higher-order operators, and supply computable benchmarks (via the ball) to guide future verification efforts.

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 and by Ashbaugh and Benguria in dimension . Since then, the conjecture remains open in dimension . 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.
Paper Structure (7 sections, 3 theorems, 63 equations, 1 figure, 1 table)

This paper contains 7 sections, 3 theorems, 63 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Order reduction principle
  3. Shape derivatives
  4. Simplicity of the eigenvalue and optimality conditions
  5. Proofs of Theorem \ref{['thm:faber-krahn moyenne']} and Theorem \ref{['thm:surdetermine']}
  6. Consequences of Theorem \ref{['thm:faber-krahn moyenne']}
  7. Appendix

Key Result

Proposition 3

Let $\Omega$ be a $C^4$ bounded open set, and $u\in H_0^2(\Omega)$ an eigenfunction of the bilaplacian in $\Omega$ associated with an eigenvalue $\mu$, so that $\Delta u$ has trace in $H^{\frac{3}{2}}(\partial\Omega)$. Finally, let $g_u$ satisfy Then, the function $z_u:=\frac{\Delta}{\sqrt{\mu}}u+u-g_u$ solves the equation In particular, $z_u$ solves the following problem, the value of which is

Figures (1)

  • Figure 1: Analytic branches near a multiple eigenvalue $\Gamma$ on domains of the form $(\text{id}+tV)\Omega$ for a given set $\Omega$ and vector field $V$. The blue, the green, and the red lines are respectively the graphs of $t\mapsto\Gamma_k^\Omega(tV)$, $t\mapsto\Gamma_{k+1}^\Omega(tV)$, and $t\mapsto\Gamma_{k+2}^\Omega(tV)$. The segment $\partial^-$ represents the tangent generated by the left partial derivative of $\Gamma_k^\Omega$ at $0$ in the direction of $V$. The segment $\partial^+$ represents the tangent generated by the right partial derivative of $\Gamma_k^\Omega$.

Theorems & Definitions (15)

  • Conjecture
  • Proposition 3
  • proof
  • proof
  • proof
  • Proposition 8
  • proof
  • proof : Proof of Theorem \ref{["thm:condition d'optimalité"]}
  • proof
  • proof
  • ...and 5 more