Feller semigroups on \(\mathbb{R}^N\)
type: Item
instance of: scholarly article
title: Feller semigroups on \(\mathbb{R}^N\)
zbMATH Open document ID: 1014.35050
published in: Semigroup Forum
publication date: 2002-12-01T00:00:00Z
review text: The parabolic problem \[ \begin{cases} D_tu(t,x)= Au(t,x),\quad & t>0,\;x\in \mathbb{R}^N,\\ u(0,x)= f(x),\quad & x\in \mathbb{R}^N,\end{cases} \] where \(A\) is the second-order elliptic partial differential operator \[ Au(x):= \sum^N_{i,j=1} a_{ij} (x)D_{ij} u(x)+ \sum^N_{i=1} b_i(x)D_i u(x) \] with continuous, unbounded coefficients in \(\mathbb{R}^N\) is considered. Particularly the existence and the qualitative properties of the semigroup generated by \(A\) in the space of bounded continuous functions and the associated resolvent equation \[ \lambda u-Au=f,\;\lambda > 0,\tag{1} \] are studied. The techniques used here are based on comparison with subsolutions or supersolutions of the problem (1) obtained by the maximum principle. These functions (named Lyapunov functions) act as barriers for this problem and allow to deduce sufficient and necessary conditions for the well-posedness of problem (1). Lyapunov functions are constructed using radial operators i.e. operators \(B\) such that \(Bu\) depends only upon \(|x|\) if \(u=u (|x|)\). So radial operators can be regarded as ordinary differential operators on \((0,\infty)\) and allow to exploit the one-dimensional theory, especially W. Feller's results.
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 60J35
Mathematics Subject Classification ID: 47D07
zbMATH DE Number: 1837428
zbMATH Keywords: subsolutions or supersolutions
zbMATH Keywords: Lyapunov functions
zbMATH Keywords: radial operators
author: Giorgio Metafune
author: Diego Pallara
author: Markus Wacker
reviewed by: A. Cichocka
MaRDI profile type: Publication
full work available at URL: s002330010129
OpenAlex ID: W1988425663
DOI: 10.1007/S002330010129
description: scientific article; zbMATH DE number 1837428