New universal inequalities for eigenvalues of elliptic operators in weighted divergence form on smooth metric measure spaces

In this paper, we investigate the eigenvalue problem of elliptic operators in weighted divergence form on smooth metric measure spaces. Firstly, we give a general inequality for eigenvalues of the elliptic operators in weighted divergence form on a compact smooth metric measure space with boundary (possibly empty). Then, applying this general inequality, we get some new universal inequalities for the eigenvalues of fourth-order elliptic operators in weighted divergence form on smooth metric measure spaces. Also, using these general inequalities and three generalized Cheeger–Gromoll splitting theorems, we give some new universal inequalities for the eigenvalues of vibration problem for a clamped plate on the smooth metric measure spaces that satisfy some curvature conditions. Moreover, our result can reveal the relationship between the $(k + 1)$ -th eigenvalue and the first k eigenvalues in a relatively quick way.