On Clamped Plates with Log-Convex Density

Abstract

We consider the analogue of Rayleigh's conjecture for the clamped plate in Euclidean space weighted by a log-convex density. We show that the lowest eigenvalue of the bi-Laplace operator with drift in a given domain is bounded below by a constant C(V, n) times the lowest eigenvalue of a centered ball of the same volume; the constant depends on the volume V of the domain and the dimension n of the ambient space. Our result is driven by a comparison theorem in the spirit of Talenti, and the constant C(V, n) is defined in terms of a minimization problem following the work of Ashbaugh and Benguria. When the density is an "anti-Gaussian," we estimate C(V, n) using a delicate analysis that involves confluent hypergeometric functions, and we illustrate numerically that C(V, n) is close to 1 for low dimensions.