Towards the Optimality of the Ball for the Rayleigh Conjecture Concerning the Clamped Plate

Abstract

The first eigenvalue of the Dirichlet bilaplacian shall be interpreted as the principal frequency of a vibrating plate with clamped boundary. In 1894, Rayleigh conjectured that, upon prescribing the area, the vibrating clamped plate with least principal frequency is circular. In 1995, Nadirashvili proved the Rayleigh Conjecture. Subsequently, Ashbaugh and Benguria proved the analogue of the conjecture in dimension 3. Since then, the conjecture has remained open in dimension \(d>3\). In this document, we contribute in answering the conjecture in high dimension 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.