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.