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On the Euler–Lagrange Equation in Calculus of Variations

Abstract

In 1985, Clarke and Vinter proved that, in the classical Bolza problem of the calculus of variations, if the Lagrangian is coercive and autonomous, all minimizers are Lipschitz and satisfy the Euler–Lagrange equation. I give a short and direct proof of this result.

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Acknowledgements

I thank an anonymous referee who has carefully read the paper and substantially improved the exposition.

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Correspondence to Ivar Ekeland.

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This paper is dedicated to Michel Théra in honour of his 70th birthday.

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Ekeland, I. On the Euler–Lagrange Equation in Calculus of Variations. Vietnam J. Math. 46, 359–363 (2018). https://doi.org/10.1007/s10013-018-0285-z

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  • DOI: https://doi.org/10.1007/s10013-018-0285-z

Keywords

  • Euler–Lagrange equation
  • Lavrentiev phenomenon

Mathematics Subject Classification (2010)

  • 49J05
  • 49J30