The \(L ^{p}\) resolvents of elliptic operators with uniformly continuous coefficients.
type: Item
instance of: scholarly article
title: The \(L ^{p}\) resolvents of elliptic operators with uniformly continuous coefficients.
zbMATH Open document ID: 1057.47052
DOI: 10.1016/S0022-0396(02)00109-2
author: Yoichi Miyazaki
published in: Journal of Differential Equations
publication date: 2003-04-28T00:00:00Z
review text: The author considers a strongly elliptic operator \(A\) of order \(2m\) in divergence form with bounded measurable coefficients and uniformly continuous top order coefficients. For \(1< p<\infty\), \(A\) is a bounded linear operator from the \(L^p\) Sobolev space \(H^{m,p}\) into \(H^{-m,p}\). The Mihlin multiplier theorem is used to prove that for appropriate complex \(\lambda\), the resolvent \((A-\lambda)^{-1}: H^{-m,p}\to H^{m,p}\) exists and satisfies \[ \|(A- \lambda)^{-1}\|_{H^{-i,p}\to H^{j,p}}\leq c|\lambda|^{-1+(i+ j)/2m} \] for \(0\leq i\), \(j\leq m\). A key component of the proof is the construction of a parametrix for \(A-\lambda\).
Mathematics Subject Classification ID: 47F05
Mathematics Subject Classification ID: 35J30
zbMATH DE Number: 1902860
zbMATH Keywords: \(L^p\) theory
zbMATH Keywords: elliptic operator
zbMATH Keywords: non-smooth coefficient
zbMATH Keywords: divergence form
zbMATH Keywords: resolvent
zbMATH Keywords: uniformly continuous
reviewed by: Alan J. Pryde
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1902860