Some inequalities and asymptotic formulas for eigenvalues on Riemannian manifolds

In this paper, we establish sharp inequalities for four kinds of classical eigenvalues in bounded domains of Riemannian manifolds. We also obtain the Weyl-type asymptotic formulas for the eigenvalues of the buckling and clamped plate problems in bounded domains of Riemannian manifolds. In addition, we give a negative answer to the Payne conjecture for the one-dimensional case.