Some bounds for eigenfunctions of an elliptic operator
type: Item
instance of: scholarly article
title: Some bounds for eigenfunctions of an elliptic operator
zbMATH Open document ID: 0588.35070
published in: Moscow University Mathematics Bulletin
publication date: 1985-01-01T00:00:00Z
review text: The authors study an elliptic problem \[ Lu=\lambda^{2m}u\quad (in \Omega),\quad B_ ju|_{\partial \Omega}=0\quad (j=1,...,m);\quad \| u\|_{L^ 2(\Omega)}=1, \] here \(\Omega \subset {\mathbb{R}}^ n\), \(\partial \Omega\) and the coefficients are smooth. The estimate (1) \(\max_{x\in {\bar \Omega}}| u(x)|^ 2\leq c| \lambda^ n|\) is proved; the constant \(c>0\) does not depend on \(\lambda\). In the case of Dirichlet problem for the equation with constant coefficients under certain conditions the estimate \(\max_{x\in {\bar \Omega}}| u(x)|^ 2\leq c| \lambda^{n-1}| \ell n| \lambda |\) is proved. The estimate (1) is proved also for the second order equation with non-smooth coefficients.
Mathematics Subject Classification ID: 35P20
Mathematics Subject Classification ID: 35A05
Mathematics Subject Classification ID: 35B35
zbMATH DE Number: 3944419
zbMATH Keywords: eigenfunctions of elliptic operators
zbMATH Keywords: elliptic problem
author: Yu V. Egorov
author: V. A. Kondrat'ev
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3944419