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.
Similar content being viewed by others
References
Ambrosio, L., Ascenzi, O., Buttazzo, G.: Lipschitz regularity for minimizers of integral functionals with highly discontinuous integrands. J. Math. Anal. Appl. 142, 301–316 (1989)
Ball, J., Mizel, V.: Singular minimizers for regular one-dimensional problems in the calculus of variations. Bull. Am. Math. Soc. 11, 143–146 (1984)
Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976)
Clarke, F., Vinter, R.: Regularity properties of solutions to the basic problem in the calculus of variations. Trans. Am. Math. Soc. 289, 73–98 (1985)
Clarke, F., Vinter, R.: Existence and regularity in the small in the calculus of variations. J. Differ. Equ. 59, 336–354 (1985)
Gratwick, R., Preiss, D.: A one-dimensional variational problem with continuous Lagrangian and singular minimizer. Arch. Ration. Mech. Anal. 202, 177–211 (2011)
Loewen, P.: On the Lavrentiev phenomenon. Can. Math. Bull. 30, 102–108 (1987)
Willem, M.: Analyse Convexe et Optimisation Editions CIACO. ISBN: 2-87085-202-9 (1985)
Zaslavski, A.: Nonoccurence of the Lavrentiev phenomenon for non-convex variational problems. Ann. Inst. Henri Poincare (C) Non Linear Anal. 22, 579–596 (2005)
Acknowledgements
I thank an anonymous referee who has carefully read the paper and substantially improved the exposition.
Author information
Authors and Affiliations
Corresponding author
Additional information
This paper is dedicated to Michel Théra in honour of his 70th birthday.
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10013-018-0285-z