Estimates For Eigenvalues Of A Class Of Operators On Riemannian Manifolds

In recent years,there is increasing interest in the eigenvalue problem of elliptic operators in divergence form.In this paper,we study the eigenvalue problem of the operator about elliptic operators in divergence form on Riemann manifolds.We sort out some results of the eigenvalue problem of the biharmonic operator and the quadratic polynomial operator of the Laplacain on the Riemannian manifold.On this basis,we investigate the eigenvalue problem of the differ-ential operatorΔ~2-pdiv(A(?))+q on Riemannian manifolds admitting some special functions,and give some Yang-type inequalities.At the same time,we study the eigenvalue problem of the differential operatorΔ_f~2-pdiv_f(A(?))+q on smooth metric measure spaces,get a lower order eigenvalue inequality and generalize Shi’s result of the differential operatorΔ~2-div(A(?))+q on Riemannian manifolds.