Boundary behavior of solutions of elliptic equations in nondivergence form
type: Item
instance of: scholarly article
title: Boundary behavior of solutions of elliptic equations in nondivergence form
zbMATH Open document ID: 1296.35051
author: Yongpan Huang
author: Dongsheng Li
author: Lihe Wang
published in: Manuscripta Mathematica
publication date: 2014-06-27T00:00:00Z
review text: The interesting paper under review deals with the boundary behaviour of the viscosity solutions to the Dirichlet problem \[ \begin{cases} -a^{ij}(x)\dfrac{\partial^2u(x)}{\partial x_i\partial x_j}=f(x) & \text{in}\;\Omega,\\ u(x)=g(x) & \text{on}\;\partial\Omega, \end{cases} \] where \(\Omega\subset \mathbb{R}^n\) is a bounded Lipschitz domain, the symmetric matrix \(\{a^{ij}(x)\}\) is uniformly elliptic, \(a^{ij},f\in C(\overline{\Omega})\) and \(g\in C(\partial\Omega)\). Introducing in a suitable way the notion of \(C^{1,\text{Dini}}\) regularity of \(\partial\Omega\) and \(g\) at a point \(x\in\partial\Omega,\) the authors prove that the solution of the above problem is Lipschitz continuous at \(x\in \partial\Omega\) if \(\partial\Omega\) and \(g\) are \(C^{1,\text{Dini}}\) in \(x.\) If, in addition, \(\partial\Omega\) is punctually \(C^1\) in \(x\) then the solution results differentiable at \(x\). The main tools used in the proofs rely on the Aleksandrov-Bakelman-Pucci maximum principle, the Harnack inequality and barrier technique.
reviewed by: Dian K. Palagachev
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35B60
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35D40
zbMATH DE Number: 6309556
zbMATH Keywords: second order elliptic equations
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: viscosity solution
zbMATH Keywords: regularity
zbMATH Keywords: Aleksandrov-Bakelman-Pucci maximum principle
zbMATH Keywords: Harnack inequality
zbMATH Keywords: barrier
MaRDI profile type: Publication
full work available at URL: s00229-013-0643-9
OpenAlex ID: W2096080319
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DOI: 10.1007/S00229-013-0643-9
description: scientific article