Regularity and selecting principles for implicit ordinary differential equations

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January 2009, 11(1): 87-101. doi: 10.3934/dcdsb.2009.11.87

Regularity and selecting principles for implicit ordinary differential equations

Bernard Dacorogna ^1, and Alessandro Ferriero ^2,

1.	Department of Mathématics, EPFL, 1015 Lausanne
2.	Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco,28049 Madrid, Spain

Received October 2007 Revised April 2008 Published November 2008

Abstract
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Implicit Ordinary or Partial Differential Equations have been widely studied in recent times, essentially from the existence of solutions point of view. One of the main issues is to select a meaningful solution among the infinitely many ones. The most celebrated principle is the viscosity method. This selection principle is well adapted to convex Hamiltonians, but it is not always applicable to the non-convex setting.
In this work we present an alternative selecting principle that singles out the most regular solutions (which do not always coincide with the viscosity ones). Our method is based on a general regularity theorem for Implicit ODEs. We also provide several examples.

Keywords: Regularity, Implicit ODEs, Differential Inclusions, Viscosity Solutions.

Mathematics Subject Classification: Primary: 34A60, 34H05, 35F30, 49J45, 49L25.

Citation: Bernard Dacorogna, Alessandro Ferriero. Regularity and selecting principles for implicit ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 87-101. doi: 10.3934/dcdsb.2009.11.87

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