Regularity and selecting principles for implicit ordinary differential equations

Bernard Dacorogna; Alessandro Ferriero

doi:10.3934/dcdsb.2009.11.87

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2009, Volume 11, Issue 1: 87-101. Doi: 10.3934/dcdsb.2009.11.87

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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

Received: September 30, 2007

Revised: March 31, 2008

Published: June 2009

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Abstract

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.

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

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