On an estimate of the first eigenvalue of a self-adjoint elliptic operator
type: Item
instance of: scholarly article
title: On an estimate of the first eigenvalue of a self-adjoint elliptic operator
zbMATH Open document ID: 0553.35063
published in: Moscow University Mathematics Bulletin
publication date: 1983-01-01T00:00:00Z
review text: The authors consider the eigenvalue problem \(Lu=\lambda Q(x)u\) in the boundary domain \(\Omega \subset R^ n\) with Dirichlet conditions. L is a differential operator of order m, \((Lu,u)\geq c_ 0\sum_{| \alpha | \leq m}(D^{\alpha}u,D^{\alpha}u)\) and Q(x)\(\geq 0\), \(\int Q^{\alpha}dx\equiv 1.\) Let \(\lambda\) be the first eigenvalue. Theorem. If \(\alpha \geq \max (1,n/(2m)),\) \(n\neq 2m\), then \(\lambda \geq \lambda_ 0>0\) and \(\lambda\) can be as great as possible. If \(\alpha <1/(2m)\), \(\alpha\) \(\neq 0\), then \(\lambda \leq \lambda_ 0\), \(\lambda_ 0>0\), and \(\lambda\) can be any small positive number. If \(1/(2m)\leq \alpha <1\) then \(\lambda\) can be any small or any great positive number.
Mathematics Subject Classification ID: 35P15
Mathematics Subject Classification ID: 35J30
zbMATH DE Number: 3881171
zbMATH Keywords: first eigenvalue
zbMATH Keywords: self-adjoint elliptic operator
zbMATH Keywords: Dirichlet conditions
author: Yu V. Egorov
author: V. A. Kondrat'ev
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3881171