Estimates on the boundary of a domain for the first derivatives of functions satisfying an elliptic or parabolic inequality
type: Item
instance of: scholarly article
title: Estimates on the boundary of a domain for the first derivatives of functions satisfying an elliptic or parabolic inequality
zbMATH Open document ID: 0707.35023
author: Olga Aleksandrovna Ladyženskaja
author: Nina Uraltseva
published in: Proceedings of the Steklov Institute of Mathematics
publication date: 1989-01-01T00:00:00Z
review text: The authors investigate boundary regularity of solutions of the Dirichlet problem for a quasilinear elliptic equation in nondivergence form: \[ a^{ij}(x,u,Du)D_{ij}u+a(x,u,Du)=0\text{ in } \Omega,\quad u=0\text{ on } \partial \Omega, \] where \(\Omega\) is a domain in \({\mathbb{R}}^ n\), under fairly weak hypotheses. Specifically if there are positive constants \(\nu\leq \mu\) such that \[ \nu | \xi |^ 2\leq a^{ij}(x,u,Du)\xi_ i\xi_ j\leq \mu | \xi |^ 2\text{ for all } x\in \Omega,\quad \xi \in {\mathbb{R}}^ n, \] and a nonnegative constant \(\mu_ 1\) and functions b, \(\Phi\) in \(L^ q(\Omega)\), for some \(q>n\), such that \[ | a(x,u,Du)| \leq \mu,\quad | Du|^ 2+b(x)| Du| +\Phi (x), \] and if \(\partial \Omega \in W^{2,q}\), then the normal derivative exists on \(\partial \Omega\) and is bounded and Hölder continuous there. Moreover the norms can be estimated in terms of the data. Analogous results for parabolic problems are also proved.
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35K30
zbMATH DE Number: 4161189
zbMATH Keywords: boundary regularity
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: quasilinear elliptic equation
zbMATH Keywords: nondivergence form
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4161189