Variations, approximation, and low regularity in one dimension

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We investigate the properties of minimizers of one-dimensional
variational problems when the Lagrangian has no higher smoothness than
continuity. An elementary approximation result is proved, but it is shown that
this cannot be in general of the form of a standard Lipschitz "variation". Part
of this investigation, but of interest in its own right, is an example of a nowhere
locally Lipschitz minimizer which serves as a counter-example to any putative
Tonelli partial regularity statement. Under these low assumptions we find it
nonetheless remains possible to derive necessary conditions for minimizers, in
terms of approximate continuity and equality of the one-sided derivatives.
Original languageEnglish
Article number57
Number of pages60
JournalCalculus of Variations and Partial Differential Equations
Issue number3
Early online date13 Apr 2017
Publication statusPublished - Jun 2017


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