Universal bounds for eigenvalues of a buckling problem. II (Q2844743)

The aim of the paper is an universal inequality for eigenvalues of the buckling problem NEWLINE\[NEWLINE\begin{cases} \Delta^2 u = - \Lambda \Delta u\text{ in }\Omega, \\ u|_{\partial \Omega} = \dfrac{\partial u}{\partial \nu} = 0, \end{cases}NEWLINE\]NEWLINE where \(\Omega\) is a bounded domain in a Euclidean space \(\mathbb{R}^n\) and \(\nu\) is the outward unit normal vector. The following inequality was proven by \textit{Q.-M. Cheng} and \textit{H. Yang} [Commun. Math. Phys. 262, No. 3, 663--675 (2006; Zbl 1170.35379)]: NEWLINE\[NEWLINE \sum_{i=1}^k(\Lambda_{k+1}-\Lambda_i)^2 \leq \frac{4(n+2)}{n^2} \sum_{i=1}^k(\Lambda_{k+1} - \Lambda_i)\Lambda_i . NEWLINE\]NEWLINE The authors obtain a sharper form of this inequality where \(\frac{4(n+2)}{n^2} \) is replaced by \(\frac{4(n+4/3)}{n^2}\) and conjecture that the inequality holds with \(\frac{4}{n}\), which would be sharp in the sense of the order of \(k\). For the case that \(\Omega\) is a domain in the unit sphere, similar results are given by \textit{Q. Wang} and \textit{C. Xia} [ibid. 270, No. 3, 759--775 (2007; Zbl 1112.74017)]. The paper contains also an improvement on this inequality.