Eigenvalue inequalities for the clamped plate problem of L_(ν)^(2) operator  

作　　者：Lingzhong Zeng 

机构地区：[1]School of Mathematics and Statistics,Jiangxi Normal University,Nanchang,330022,China

出　　处：《Science China Mathematics》2022年第4期793-812,共20页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China(Grant Nos.11861036 and 11826213);the Natural Science Foundation of Jiangxi Province(Grant No.20171ACB21023).

摘　　要：The ■ operator is introduced by Xin(2015), which is an important extrinsic elliptic differential operator of divergence type and has profound geometric meaning. In this paper, we extend the ■ operator to a more general elliptic differential operator ■, and investigate the clamped plate problem of the bi-■ operator,which is denoted by ■ on the complete Riemannian manifolds. A general formula of eigenvalues for the ■ operator is established. Applying this formula, we estimate the eigenvalues on the Riemannian manifolds. As some further applications, we establish some eigenvalue inequalities for this operator on the translating solitons with respect to the mean curvature flows, submanifolds of the Euclidean spaces, unit spheres and projective spaces. In particular, for the case of translating solitons, all of the eigenvalue inequalities are universal.

关 键 词：mean curvature flows L^(2)_(ν)operator clamped plate problem EIGENVALUES Riemannian manifolds translating solitons 

分 类 号：O177[理学—数学] O178[理学—基础数学]