On Clamped Plates with Log-Convex Density

TL;DR

This paper proves a lower bound for the lowest eigenvalue of the bi-Laplace operator with drift on domains with log-convex density, extending Rayleigh's conjecture and employing comparison theorems and hypergeometric functions.

Contribution

It establishes a bound relating eigenvalues of clamped plates with log-convex density to those of centered balls, generalizing classical spectral inequalities.

Findings

Lower bound for eigenvalues in weighted domains

Constant C(V,n) close to 1 in low dimensions

Analysis involving confluent hypergeometric functions

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.