Estimates of the nontangential maximal function for solutions of a second-order elliptic equation
type: Item
instance of: scholarly article
title: Estimates of the nontangential maximal function for solutions of a second-order elliptic equation
zbMATH Open document ID: 1271.35014
DOI: 10.1134/S1064562412050225
author: A. K. Gushchin
published in: Doklady Mathematics
publication date: 2013-01-14T00:00:00Z
review text: This article considers linear second-order elliptic PDEs without lower order terms with \(L^p\)-Dirichlet boundary values on a \(C^1\) boundary. The authors consider weak solutions in \(W^1_{2,\mathrm{loc}}\) where the \(L^p\)-boundary condition is realized in the sense of parallel surfaces. The normal of the boundary and the coefficients in the PDE are assumed to be Dini continuous. They achieve an \(L^p\)-bound for the tangential maximal function on the boundary which they show to be dominated by the \(L^p\)-norm of the boundary data. This result is then used to show that the unique weak solution of the boundary value problem has a nontangential limit at a.e. boundary point. The limit coincides a.e. with the given boundary data.
reviewed by: Carla Cederbaum
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35J15
Mathematics Subject Classification ID: 35D30
Mathematics Subject Classification ID: 35B45
zbMATH DE Number: 6124984
zbMATH Keywords: second-order elliptic Dirichlet problem
zbMATH Keywords: weak solutions
zbMATH Keywords: a priori estimate
zbMATH Keywords: nontangential maximal function
zbMATH Keywords: harmonic functions
zbMATH Keywords: Lusin area integral
MaRDI profile type: Publication
cites work: Solvability of the Dirichlet problem for a second-order elliptic equation with a boundary function from \(L_{p}\)
cites work: Continuity of Solutions of Parabolic and Elliptic Equations
cites work: Q5187597
cites work: Weighted norm inequalities for the Lusin area integral and the nontangential maximal functions for functions harmonic in a Lipschitz domain
cites work: Q5607484
cites work: An Interpolation Problem for Bounded Analytic Functions
cites work: Interpolations by bounded analytic functions and the corona problem
cites work: Lp Estimates for (Pluri-) Subharmonic Functions.
cites work: Distribution function inequalities for the area integral
description: scientific article; zbMATH DE number 6124984