带权Paneitz算子和带权圆盘振动问题的特征值不等式

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

本文研究光滑度量测度空间上带权Paneitz算子的闭特征值问题和带权圆盘振动问题,给出Euclid空间、单位球面、射影空间和一般Riemann流形的n维紧子流形的权重Paneitz算子和带权圆盘振动问题的前n个特征值上界估计.进一步地,本文给出带权Ricci曲率有界的紧致度量测度空间上带权圆盘振动问题的第一特征值的下界估计.

In this paper, we study the closed eigenvalue problem of the weighted Paneitz operator and weighted vibration problem for a clamped plate on 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 unit sphere, 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 space with the bounded weighted Ricci curvature.