The equivalence of the uniqueness of the Dirichlet problem and the maximum principle for elliptic systems
type: Item
instance of: scholarly article
title: The equivalence of the uniqueness of the Dirichlet problem and the maximum principle for elliptic systems
zbMATH Open document ID: 0818.35012
author: Chiping Zhou
published in: Journal of Mathematical Analysis and Applications
publication date: 1994-07-04T00:00:00Z
review text: The main result of this paper is stated in the following theorem: Let \(D\) be a bounded domain in \(\mathbb{R}^ n\). Let \(U\) be a complex-valued vector function satisfying \((*)\) \(\Delta U + C(x)U = 0\) in \(D\), where \(C(x)\) is a complex-valued bounded Hölder continuous matrix. Then a necessary and sufficient condition for existence of a positive constant \(K\) such that \(\| U \|_ D \leq K \| U \|_{\partial D}\) for all \(C^ 2(D) \cap C(\overline D)\) solutions \(U\) of \((*)\) is that the Dirichlet problem \(\Delta U + C(x)U = 0\) in \(D\), \(U = 0\) on \(\partial D\), has only the trivial solution \(U \equiv 0\).
Mathematics Subject Classification ID: 35B50
Mathematics Subject Classification ID: 35J55
zbMATH DE Number: 599945
zbMATH Keywords: maximum principle for elliptic systems
reviewed by: G. A. Philippin
MaRDI profile type: Publication
full work available at URL: jmaa.1994.1141
OpenAlex ID: W1990300133
DOI: 10.1006/JMAA.1994.1141
description: scientific article; zbMATH DE number 599945