\(L ^{p }\)-theory for second-order elliptic operators with unbounded coefficients
type: Item
instance of: scholarly article
title: \(L ^{p }\)-theory for second-order elliptic operators with unbounded coefficients
zbMATH Open document ID: 1277.47060
author: Motohiro Sobajima
published in: Journal of Evolution Equations
publication date: 2013-02-18T00:00:00Z
review text: The paper deals with second-order elliptic operators with unbounded coefficients of the form \[ Au := -{\text{div}}(a\nabla u) + F \cdot \nabla u + Vu\quad \text{in}\;L^{p}(\mathbb{R}^{N}),\;1 < p < \infty. \] A new criterion for \(m\)-accretivity and \(m\)-sectoriality of \(A\) in \({L^{p}(\mathbb{R}^{N})}\) is given in terms of certain identities which behave like a sesquilinear form over \(L ^{p } \times L ^{p'}\). The result naturally extends the \textit{T. Kato} criterion from [Spectral theory of differential operators, Proc. Conf., Birmingham/ USA 1981, North-Holland Math. Stud. 55, 253--266 (1981; Zbl 0484.35027)] for the nonnegative selfadjointness to the case of \(p \neq 2.\) The simplicity of the approach adopted is illustrated by the typical example \({Au = -u'' + x^{3}u' + c |x|^{\gamma}u}\) in \({L^p(\mathbb{R})}.\)
reviewed by: Dian K. Palagachev
Mathematics Subject Classification ID: 47F05
Mathematics Subject Classification ID: 47B44
Mathematics Subject Classification ID: 34G10
zbMATH DE Number: 6136964
zbMATH Keywords: second-order elliptic operators
zbMATH Keywords: unbounded coefficients
zbMATH Keywords: \(m\)-accretive operatorss
zbMATH Keywords: \(m\)-sectorial operators
zbMATH Keywords: analytic contraction semigroups
MaRDI profile type: Publication
full work available at URL: s00028-012-0163-1
OpenAlex ID: W2012779994
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DOI: 10.1007/S00028-012-0163-1
description: scientific article; zbMATH DE number 6136964