Harnack's inequality for second order linear ordinary differential inequalities
type: Item
instance of: scholarly article
title: Harnack's inequality for second order linear ordinary differential inequalities
zbMATH Open document ID: 1341.34015
author: Ahmed Mohammed
author: Hannah Turner
published in: Involve
publication date: 2016-03-10T00:00:00Z
reviewed by: Gnana Bhaskar Tenali
Mathematics Subject Classification ID: 34A40
Mathematics Subject Classification ID: 34C11
Mathematics Subject Classification ID: 34A30
Mathematics Subject Classification ID: 26D20
zbMATH DE Number: 6553287
zbMATH Keywords: Harnack's inequality, maximum principles
zbMATH Keywords: ordinary differential inequalities
MaRDI profile type: Publication
OpenAlex ID: W2285475576
DOI: 10.2140/INVOLVE.2016.9.281
review text: Harnack type inequalities for nonnegative solutions of second order linear ordinary differential inequalities of the form \( L u \leq f(x), x \in I :=(A, B)\) where, for \(u \in C^2(I)\), NEWLINENEWLINE\[NEWLINEL u := u'' + p(x) u'(x) + q(x) u(x).NEWLINE\]NEWLINE NEWLINENEWLINEThe coefficients \(p, q\) and \(f\) are assumed to be continuous real valued functions on \(I\). To make the paper self contained, simpler proofs of known results from the literature are also given. The following main result is established for the nonnegative solutions of \(L u \leq 0\) on \(I\).NEWLINENEWLINETheorem. Given \([a,b] \subseteq I,\) for any nonnegative solution of \(L u \leq 0\) on \(I\), there is a positive constant \(C\) that does not depend on \(u\), such that NEWLINENEWLINE\[NEWLINE\max_{a \leq x \leq b}\; u(x) \leq C \;\min_{a \leq x \leq b} \; u(x) .NEWLINE\]NEWLINE NEWLINEThe same cannot be established if \(L u \geq 0\). Another inequality of Harnack type is established for nonnegative solutions of nonhomogeneous equations \(L_0 u = f\) on \(I\), where NEWLINENEWLINE\[NEWLINEL_0 u = u'' + p u ' - q^- u \text{ and } q^-(x) = \max\{-q(x), 0\}, f \geq 0.NEWLINE\]NEWLINE NEWLINEFinally, a Harnack-type inequality for nonnegative solutions of \( L u \leq f\) with no sign restrictions on \(f\) is derived.
description: scientific article; zbMATH DE number 6553287