Given a compact Riemannian manifold M with boundary (possibly empty), we consider the eigenvalues of the biharmonic operator with weight on M, proving a general inequality involving them. Using this inequality, we consider these eigenvalues when M is a compact domain of one of the following three spaces: 1) a complex projective space, 2) a minimal submanifold of a Euclidean space and 3) a minimal submanifold of a unit sphere.
"Inequalities for eigenvalues of the biharmonic operator with weight on Riemannian manifolds." J. Math. Soc. Japan 62 (2) 597 - 622, April, 2010. https://doi.org/10.2969/jmsj/06220597