Inequalities for Eigenvalues of a General Clamped Plate Problem

Ghanbari, K. ; Shekarbeigi, B.

Canadian mathematical bulletin, Tome 55 (2012) no. 1, pp. 88-97

Résumé

Let $D$ be a connected bounded domain in ${{\mathbb{R}}^{n}}$ . Let $0\,<\,{{\mu }_{1}}\,\le \,{{\mu }_{2}}\,\le \,\cdots \,\le \,{{\mu }_{k}}\,\le \,\cdots $ be the eigenvalues of the following Dirichlet problem: $$\left\{ \begin{align}& {{\Delta }^{2}}u(x)\,+\,V(x)u(x)\,=\,\mu \rho (x)u(x),x\in \,D \\& u{{|}_{\partial D}}\,=\,\frac{\partial u}{\partial n}\,{{|}_{\partial D}}\,=\,0, \\ \end{align} \right.$$ where $V(x)$ is a nonnegative potential, and $\rho (x)\,\in \,C(\overset{-}{\mathop{D}}\,)$ is positive. We prove the following inequalities: $$\begin{align}& {{\mu }_{k+1}}\le \frac{1}{k}\sum\limits_{i=1}^{k}{\mu i+}{{[\frac{8(n+2)}{{{n}^{2}}}{{\left( \frac{\rho \max }{\rho \min } \right)}^{2}}]}^{1/2}}\times \frac{1}{k}{{\sum\limits_{i=1}^{k}{[{{\mu }_{i}}({{\mu }_{k+1}}-{{\mu }_{i}})]}}^{1/2}}, \\& \frac{{{n}^{2}}{{k}^{2}}}{8(n+2)}\le {{\left( \frac{\rho \max }{\rho \min } \right)}^{2}}[\sum\limits_{i=1}^{k}{\frac{\mu _{i}^{1/2}}{{{\mu }_{k+1}}-{{\mu }_{i}}}}]\times \sum\limits_{i=1}^{k}{\mu _{i}^{1/2}}. \\ \end{align}$$

Détail

DOI : 10.4153/CMB-2011-031-x

Mots-clés : 35P15, biharmonic operator, eigenvalue, eigenvector, inequality