Inequalities for the first eigenvalues of the clamped plate and buckling problems

Abstract

In 1877 Rayleigh conjectured that the lowest frequency of vibration of a clamped plate of given area occurs when the plate is circular. This conjecture was finally proved by Nadirashvili in 1992, with important earlier contributions due to Szego and Talenti. An analogous conjecture concerning the critical buckling load of a clamped plate was made by P6lya and Szego around 1950. This conjecture remains open. This paper surveys the state of our knowledge of these and related problems, including their n-dimensional generalizations. In particular, we discuss our recent work proving Rayleigh's clamped plate conjecture for dimension 3 (and 2) and proving related, but presumably nonoptimal, inequalities for the clamped plate problem for dimension n 2: 4 and for the buckling problem. In the latter cases, the inequalities have the form of the conjectured lower bounds but contain an unwanted factor slightly less than 1. Our bounds for the clamped plate problem follow from detailed analysis involving Bessel functions, and compare favorably to earlier bounds of a similar form due to Talenti. Our bounds for the buckling problem follow by similar methods, and also from earlier bounds due to Payne and Krahn, as first noted by Bramble and Payne.