Regularity and selecting principles for implicit ordinary differential equations
type: Item
instance of: scholarly article
title: Regularity and selecting principles for implicit ordinary differential equations
zbMATH Open document ID: 1170.34007
author: Alessandro Ferriero
author: Bernard Dacorogna
published in: Discrete and Continuous Dynamical Systems. Series B
publication date: 2009-02-19T00:00:00Z
review text: The authors study implicit scalar differential equations of the form \(f(t,u(t),u'(t))=0\) on some interval \((a,b)\). Solutions are sought in the space of absolute continuous functions with bounded derivatives and which are zero at the boundaries. As can be seen from the simple example \(|u'(t)|=1\) on \((0,1)\), there exist in general infinitely many solutions. For selecting a meaningful solution, the authors propose a selecting principle which singles out the most regular solution. The regularity of a solution is measured by the number of discontinuity points of its derivative. The authors give sufficient conditions which ensure the existence of such most regular solutions and they also formulate conditions which ensure that there exists exactly one positive and one negative most regular solution. The selecting principle is compared with two other selecting principle, the so called viscosity method and the maximal solution principle. Furthermore, several examples are given which also show the limits of the selecting principles. The paper is well structured and nicely written.
reviewed by: Stephan Trenn
Mathematics Subject Classification ID: 34A09
Mathematics Subject Classification ID: 34A60
Mathematics Subject Classification ID: 49L25
zbMATH DE Number: 5509382
zbMATH Keywords: implicit ODEs
zbMATH Keywords: differential inclusions
zbMATH Keywords: viscosity solutions
zbMATH Keywords: regularity
MaRDI profile type: Publication
full work available at URL: dcdsb.2009.11.87
OpenAlex ID: W2001916183
DOI: 10.3934/DCDSB.2009.11.87
description: scientific article; zbMATH DE number 5509382