Estimates of the first eigenvalue for the fourth order elliptic equation
type: Item
instance of: scholarly article
title: Estimates of the first eigenvalue for the fourth order elliptic equation
zbMATH Open document ID: 0637.35061
author: R. Yu. Nabokov
published in: Moscow University Mathematics Bulletin
publication date: 1986-01-01T00:00:00Z
review text: 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)].
Mathematics Subject Classification ID: 35P15
Mathematics Subject Classification ID: 35J40
zbMATH DE Number: 4038360
zbMATH Keywords: bounded domain
zbMATH Keywords: piecewise smooth boundary
zbMATH Keywords: first eigenvalue
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4038360