Inequalities for eigenvalues of a clamped plate problem

Let $D$ be a connected bounded domain in an $n$-dimensional Euclidean space $\mathbb{R}^n$. Assume that

$0 < \lambda_1 <\lambda_2 \le \cdots \le \lambda_k \le \cdots$

are eigenvalues of a clamped plate problem or an eigenvalue problem for the Dirichlet biharmonic operator:

$\left \{ \aligned&\Delta^2 u =\lambda u, \text{ in $D$,}\\ &u\ve... ...rac {\partial u}{\partial n}\right \vert _{\partial D}=0. \endaligned \right .$

Then, we give an upper bound of the $(k+1)$-th eigenvalue $\lambda_{k+1}$ in terms of the first $k$ eigenvalues, which is independent of the domain $D$, that is, we prove the following:

$\lambda_{k+1} \le \frac 1k\sum_{i=1}^k \lambda_i +\left [\frac {8... ...ac 1k\sum_{i=1}^k \biggl[ \lambda_i(\lambda_{k+1} -\lambda_i) \biggl ]^{1/2}.$

Further, a more explicit inequality of eigenvalues is also obtained.