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1. CMB 2011 (vol 55 pp. 88)
Inequalities for Eigenvalues of a General Clamped Plate Problem
Let $D$ be a
connected bounded domain in $\mathbb{R}^n$. Let
$0<\mu_1\leq\mu_2\leq\dots\leq\mu_k\leq\cdots$ be the eigenvalues
of the following Dirichlet
problem:
$$
\begin{cases}\Delta^2u(x)+V(x)u(x)=\mu\rho(x)u(x),\quad x\in
D
u|_{\partial D}=\frac{\partial u}{\partial n}|_{\partial
D}=0,
\end{cases}
$$
where $V(x)$ is a nonnegative potential,
and $\rho(x)\in C(\bar{D})$ is positive.
We prove the following inequalities:
$$\mu_{k+1}\leq\frac{1}{k}\sum_{i=1}^k\mu_i+\Bigl[\frac{8(n+2)}{n^2}\Bigl(\frac{\rho_{\max}}
{\rho_{\min}}\Bigr)^2\Bigr]^{1/2}\times
\frac{1}{k}\sum_{i=1}^k[\mu_i(\mu_{k+1}-\mu_i)]^{1/2},
$$
$$\frac{n^2k^2}{8(n+2)}\leq
\Bigl(\frac{\rho_{\max}}{\rho_{\min}}\Bigr)^2\Bigl[\sum_{i=1}^k\frac{\mu_i^{1/2}}{\mu_{k+1}-\mu_i}\Bigr]
\times\sum_{i=1}^k\mu_i^{1/2}.
$$
Keywords:biharmonic operator, eigenvalue, eigenvector, inequality Category:35P15 |