A note on the strong maximum principle for elliptic differential inequalities
type: Item
instance of: scholarly article
title: A note on the strong maximum principle for elliptic differential inequalities
zbMATH Open document ID: 0952.35163
DOI: 10.1016/S0021-7824(99)00146-4
author: Patrizia Pucci
author: James Serrin
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 2000-06-22T00:00:00Z
review text: The paper gives necessary and sufficient conditions for the validity of the strong maximum principle and the compact support principle for quasilinear elliptic differential inequalities in divergence form. More precisely, consider the differential inequalities \[ \text{ div} \big\{ A(|Du|)Du\big\}-f(u)\leq 0,\qquad u\geq 0,\tag{1} \] \[ \text{ div} \big\{ A(|Du|)Du\big\}-f(u)\geq 0,\qquad u\geq 0,\tag{2} \] in a (possibly unbounded) domain \(D\subset {\mathbb{R}}^n,\) \(n\geq 2,\) where \(A=A(t)\) and \(f(t)\) satisfy the following conditions: \(A\in C(0,\infty),\) \(\Omega(t)=tA(t)\) is strictly increasing in \((0,\infty)\) and \(\lim_{t\to 0}\Omega(t)=0,\) \(f\in C[0,\infty),\) \(f(0)=0\) and \(f\) is non-decreasing on some interval \([0,\delta),\) \(\delta>0.\) Set \(F(t)=\int_0^t f(s)ds,\) \(s>0\) and \[ H(t)=t\Omega(t)-\int_0^t \Omega(s)ds,\quad t\geq 0. \] By a solution of \((1)\) or \((2)\) one means a non-negative function \(u\in C^1(D)\) which satisfies \((1)\) or \((2)\) in a distribution sense. By definition, the strong maximum principle holds for \((1)\), if \(u(x_0)=0\) for some solution \(u\) of \((1)\) and some \(x_0\in D\) implies \(u\equiv 0\) in \(D\), while \((2)\) obeys the compact support principle, if \(\lim_{|x|\to\infty} u(x)=0\) for some solution \(u\) to \((2)\) in an exterior domain \(D\) yields that \(u\) has compact support in \(D.\) The main results of the paper are as follows: 1) The strong maximum principle holds for \((1)\) if and only if either \(f(s)\equiv 0\) for \(s\in [0,\mu),\) \(\mu>0,\) or \(f(s)>0\) for \(s\in(0,\delta)\) and \(\int_0^\delta {{ds}\over {H^{-1}(F(s))}}=\infty.\) 2) The compact support principle holds for \((2)\) if and only if \(f(s)>0\) for \(s\in(0,\delta)\) and \(\int_0^\delta {{ds}\over {H^{-1}(F(s))}}<\infty\).
Mathematics Subject Classification ID: 35R45
Mathematics Subject Classification ID: 35B50
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 1463866
zbMATH Keywords: elliptic differential inequalities in divergence form
zbMATH Keywords: strong maximum principle
zbMATH Keywords: compact support principle
reviewed by: Dian K. Palagachev
MaRDI profile type: Publication
Wikidata QID: Q126789345
description: scientific article; zbMATH DE number 1463866