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Variational methods for non-variational problems

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.

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Correspondence to Pablo Pedregal.

Additional information

Research supported in part by MTM2013-47053-P of the MCyT (Spain) and PEII-2014-010-P of the Conserjería de Cultura (JCCM) of the JCCM (Castilla-La Mancha).

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Pedregal, P. Variational methods for non-variational problems. SeMA 74, 299–317 (2017). https://doi.org/10.1007/s40324-017-0119-z

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Keywords

  • Non-local functionals
  • Non-linear problems
  • Optimality conditions

Mathematics Subject Classification

  • 49J05
  • 49K05
  • 49M05