Reilly type inequalities for eigenvalues of the buckling problem on compact Riemannian manifolds

Abstract

In this paper, we firstly study the two types of buckling problem of the biharmonic operator and get some Reilly type lower bound estimates for the first nonzero eigenvalue of these eigenvalue problems on compact Riemannian manifolds with boundary. Moreover, using the orthogonalization of Gram and Schmidt, we give some Reilly type upper bound estimates for the sum of the first n eigenvalues of the buckling problem on compact Riemannian manifolds. Especially, interesting rigidity results can be obtained if sharp bounds are achieved.