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SeMA Journal

, Volume 74, Issue 3, pp 299–317 | Cite as

Variational methods for non-variational problems

  • Pablo PedregalEmail author
Article

Abstract

We propose and describe an alternative perspective for the study of systems of boundary value problems governed by ODEs. It is based on a variational approach that seeks to minimize a certain quadratic error understood as a deviation of paths from being a solution of the corresponding system. We distinguish two situations depending on whether the problem has or has not divergence structure. In the first case, the functional is not a typical integral functional as the ones examined in the Calculus of Variations, and we have to resort to the Palais–Smale condition to show existence of minimizers. In the case of a fully non-linear problem, however, the functional is an integral, local functional of second order. We illustrate the method with some numerical simulations.

Keywords

Non-local functionals Non-linear problems Optimality conditions 

Mathematics Subject Classification

49J05 49K05 49M05 

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Copyright information

© Sociedad Española de Matemática Aplicada 2017

Authors and Affiliations

  1. 1.INEI, Universidad de Castilla La ManchaCiudad RealSpain

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