Estimates of the first eigenvalue for the fourth order elliptic equation
Let \(G\subset {\mathbb{R}}^ n\) (n\(\geq 2)\) be a bounded domain with the piecewise smooth boundary, \(G^*\) is a ball in \({\mathbb{R}}^ n\) having the same volume as G. The \(\lambda\) (G) be the first eigenvalue of the problem \(\Delta^ 2u+\lambda \Delta u=0\) in G, \(u|_{\partial G}=\partial u/\partial \nu |_{\partial G}=0\), and \(\lambda (G^*)\) be the first eigenvalue of the same problem in \(G^*\). The author proves that \(\lambda (G)\geq k(n)\lambda (G^*)\); the explicit expression for k(n) is given. A similar result for the equation \(\Delta^ 2u+\lambda u=0\) was earlier obtained by \textit{G. Talenti} [Ann. Mat. Pura Appl., IV. Ser. 129, 265-280 (1981; Zbl 0475.73050)].
scientific article; zbMATH DE number 4038360