Inequalities for eigenvalues of a clamped plate problem

Abstract:

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: \[ \left \{ \begin {aligned} &\Delta ^2 u =\lambda u, \ \text { in $D$,} &u|_{\partial D}=\left . \frac {\partial u}{\partial n}\right |_{\partial D}=0. \end {aligned} \right . \] 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: \[ \lambda _{k+1} \le \frac 1k\sum _{i=1}^k \lambda _i +\left [\frac {8(n+2)}{n^2} \right ]^{1/2} \frac 1k\sum _{i=1}^k \biggl [ \lambda _i(\lambda _{k+1} -\lambda _i) \biggl ]^{1/2}. \] Further, a more explicit inequality of eigenvalues is also obtained.