Some \(L_{p}\)-estimates for elliptic and parabolic operators with measurable coefficients
type: Item
instance of: scholarly article
title: Some \(L_{p}\)-estimates for elliptic and parabolic operators with measurable coefficients
zbMATH Open document ID: 1258.35069
published in: Discrete and Continuous Dynamical Systems. Series B
publication date: 2012-09-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/1201.4573
reviewed by: Dian K. Palagachev
Mathematics Subject Classification ID: 35J15
Mathematics Subject Classification ID: 35K10
Mathematics Subject Classification ID: 60H10
zbMATH DE Number: 6081735
zbMATH Keywords: elliptic and parabolic equations
zbMATH Keywords: measurable coefficients
zbMATH Keywords: occupation measures for diffusion processes
author: N. V. Krylov
MaRDI profile type: Publication
OpenAlex ID: W2962966817
arXiv ID: 1201.4573
DOI: 10.3934/DCDSB.2012.17.2073
review text: The very interesting paper under review deals with linear elliptic and parabolic operators with measurable coefficients of the form NEWLINE\[NEWLINE {\mathcal L}=\partial_t+a^{ij}(t,x)D_{ij}+b^i(t,x)d_i - c(t,x),\quad x\in\mathbb R^d NEWLINE\]NEWLINE with a uniformly nondegenerate and bounded matrix \(\{a^{ij}\}.\) The author proves two types of \(L_{p}\)-results for the strong solutions to the equation \({\mathcal L}u,\) used recently in the strong solvability theory of fully nonlinear elliptic and parabolic second order equations developed in [\textit{H. Dong, N. V. Krylov} and \textit{X. Li}, ``On fully nonlinear elliptic and parabolic equations with VMO coefficients in domains'', Algebra Anal. 24, No. 1, 53--94 (2012; \url{http://mi.mathnet.ru/aa1269})]. The first one is an estimate for the \(L_{\gamma}\)-norm of the second order derivatives in terms of \(|{\mathcal L}u|_{L_{d+1}}\) and \(\sup|u|,\) where \(\gamma\in(0,1).\) The second result deals with estimates of the resolvent operators in \(L_{p}\) when the first-order coefficients are summable to an appropriate power.
description: scientific article; zbMATH DE number 6081735