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

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.

Figures (3)

• Figure 1: Holes in different situations. In each case, the open set $\omega$ is in dark grey, whereas its hole $T$, if it exists, is in light gray.
• Figure 2: Different configurations of the open sets $\omega_+$ and $\omega_-$. In each case, $\omega_+$ is in dark grey, and $\omega_-$ is in light gray. In situation \ref{['fig:trous/intersection des bords']}, the assumption $\partial\omega\cap\partial\omega_+\cap\partial\omega_-=\emptyset$ of Lemma \ref{['lemme:trou']} fails since $\partial\omega_+$ and $\partial\omega_-$ meet simultaneously $\partial\omega$ at the red points. In situation \ref{['fig:trous/omega non connexe']}, the assumption of connectedness of $\omega$ fails. In situation \ref{['fig:trous/omega_+ troué']}, all the assumptions of Lemma \ref{['lemme:trou']} hold, and hence $\omega_+$ admits a hole containing $\omega_-$.
• Figure 3: Qualitative properties of the function $F_\nu$ in the $(k,a)$-plane.

Theorems & Definitions (18)

• Conjecture
• proof
• Proposition 7
• proof
• proof
• proof
• proof : Proof of Theorem \ref{['thm:talenti modifie']}
• Proposition 20
• proof
• proof