Boundary behavior of Dirichlet eigenfunction of second order elliptic operators
type: Item
instance of: scholarly article
title: Boundary behavior of Dirichlet eigenfunction of second order elliptic operators
zbMATH Open document ID: 0676.47030
DOI: 10.1007/BF02570861
author: D. J. Harris
author: Robert M. Kauffman
author: W. D. Evans
published in: Mathematische Zeitschrift
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/183758
review text: We investigate the rate of decay of solutions of a second-order elliptic equation \(\tau f=\lambda f\) in an arbitrary domain \(\Omega \subset {\mathbb{R}}^ n\) which satisfy zero boundary conditions, and then apply the results to a study of the behavior of the associated heat semigroup in various weighted \(L^ 2\) spaces. Our results concern the behavior at infinity and at points on the boundary \(\partial \Omega\) of \(\Omega\), especially irregular ones like corners and cusps. The decay rates we obtain are expressed in terms of weighted \(L^ 2\) estimates which are governed by the sharpness of the cusps and the properties of the coefficients of \(\tau\). At a cusp we are able to estimate correctly the rate of decay in problems for which the answer is already known: as far as we know, rather general results fail to do this.
Mathematics Subject Classification ID: 47F05
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 35P20
Mathematics Subject Classification ID: 35K05
zbMATH DE Number: 4107513
zbMATH Keywords: rate of decay of solutions of a second-order elliptic equation
zbMATH Keywords: heat semigroup in various weighted \(L^ 2\) spaces
zbMATH Keywords: corners
zbMATH Keywords: cusps
zbMATH Keywords: boundary behavior of Dirichlet eigenfunctions
reviewed by: W. D. Evans
MaRDI profile type: Publication
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OpenAlex ID: W2073390541
description: scientific article; zbMATH DE number 4107513