Consistent With The Upper Bound For Eigenvalues Of The Biharmonic Operator On A Class Of Submanifolds

Eigenvalue problem is an important subject on differential geometry and geometry analysis. It is also a hot issue and has been studied extensively by mathematicians at domestic and abroad. This thesis is mainly to study eigenvalues of the biharmonic oper-ator in some submanifolds in a Euclidean, and obtain the estimate about the consistent with the upper bound.In the first part, we briefly introduce the background of eigenvalue problem.In the second part, we briefly introduce some basic definitions, the basic properties and research of eigenvalues.The third part is the main content. Firstly, we briefly introduce eigenvalues of the Dirichlet biharmonic operator on compact Riemannn manifolds with boundary and prove a general inequality for them. Then, we give the consistent with the upper bound on the (k+1)th eigenvalue on such objects in terms of the first k eigenvalues indepen-dent of the domains.