The eigenvalues inequalities of the weighted Paneitz operator and weighted vibration problem for a clamped plate

Abstract

In this paper, we study the closed eigenvalue problem of the weighted Paneitz operator and weighted vibration problem for a clamped plateon smooth metric measure spaces, and give the upper bounds for the first $n$ eigenvalues on $n$-dimensional compact submanifolds of a Euclidean space, or a unitsphere, or a projective space, or a general Riemannian manifold. In addition, we give the lower bounds of the first eigenvalue of the weighted vibration problem for a clamped plate on a compact smooth metric measure spacewith the bounded weighted Ricci curvature.