Extrinsic Estimates For Eigenvalues Of The Biharmonic Operator With Weight

Now the research of eigenvalues of Riemannian manifolds has been an important fields in analysis of manifolds. It has many applications in mathematics, physics and so on. Let Ω be a bounded domain in an n-dimensional Riemannian manifold Mn andâ†'n be the unit outward normal vector field of OM. The well-known eigenvalues problem is called a biharmonic operator with weight problem. Hereâ–³is the Laplacian of Mn, V is the positive and bounded function on Mn, and p is a weight function.In this paper, we discuss the eigenvalues of the biharmonic operator with a weight on compact Riemannian manifolds with boundary (?)M (possibly empty), and prove a general inequality for them. As its application, we obtain several universal inequalities for such biharmonic operators.