Pathological solutions to the Euler–Lagrange equation and existence/regularity of minimizers in one-dimensional variational problems

Richard Gratwick; Aidys Sedipkov; Mikhail Sychev; Aris Tersenov

doi:10.1016/j.crma.2017.01.020

Calculus of variations

Pathological solutions to the Euler–Lagrange equation and existence/regularity of minimizers in one-dimensional variational problems

Presented by: John Ball

Richard Gratwick ¹; Aidys Sedipkov ^2,³; Mikhail Sychev ^2,³; Aris Tersenov ^2,³

¹ Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK
² Laboratory of Differential Equations and Related Problems of Analysis, Sobolev Institute of Mathematics, Koptuyg Avenue, 4, Novosibirsk 630090, Russia
³ Novosibirsk State University, Russia

Comptes Rendus. Mathématique, Volume 355 (2017) no. 3, pp. 359-362.

Abstract
Résumé

In this paper, we prove that if $L (x, u, v) \in C^{3} (R^{3} \to R)$ , $L_{v v} > 0$ and $L \geq α | v | + β$ , $α > 0$ , then all problems (1), (2) admit solutions in the class $W^{1, 1} [a, b]$ , which are in fact $C^{3}$ -regular provided there are no pathological solutions to the Euler equation (5). Here $u \in C^{3} [c, d [$ is called a pathological solution to equation (5) if the equation holds in $[c, d [$ , $| \dot{u} (x) | \to \infty$ as $x \to d$ , and ${‖ u ‖}_{C [c, d]} < \infty$ . We also prove that the lack of pathological solutions to the Euler equation results in the lack of the Lavrentiev phenomenon, see Theorem 9; no growth assumptions from below are required in this result.

Dans cette Note, nous démontrons que si $L (x, u, v) \in C^{3} (R^{3} \to R)$ , $L_{v v} > 0$ et $L \geq α | v | + β$ , $α > 0$ , alors tous les problèmes (1)–(2) admettent des solutions dans la classe $W^{1, 1} [a, b]$ , qui sont en fait $C^{3}$ -régulières pourvu que l'équation d'Euler (5) n'ait pas de solution pathologique. Ici, une solution $u \in C^{3} [c, d [$ de (5) est dite pathologique si l'équation est satisfaite dans $[c, d [$ , $| \dot{u} (x) | \to \infty$ lorsque $x \to d$ et ${‖ u ‖}_{C [c, d]} < \infty$ . Nous montrons également (voir Théorème 9), que l'absence de solution pathologique à l'équation d'Euler entraîne l'absence de phénomène de Lavrentiev ; aucune hypothèse de croissance minimale n'est requise pour ce résultat.

Received: 2016-10-27
Accepted: 2017-01-31
Published online: 2017-02-09

DOI: 10.1016/j.crma.2017.01.020

Author's affiliations:

Richard Gratwick ¹; Aidys Sedipkov ^{2, 3}; Mikhail Sychev ^{2, 3}; Aris Tersenov ^{2, 3}

¹ Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK
² Laboratory of Differential Equations and Related Problems of Analysis, Sobolev Institute of Mathematics, Koptuyg Avenue, 4, Novosibirsk 630090, Russia
³ Novosibirsk State University, Russia

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     author = {Richard Gratwick and Aidys Sedipkov and Mikhail Sychev and Aris Tersenov},
     title = {Pathological solutions to the {Euler{\textendash}Lagrange} equation and existence/regularity of minimizers in one-dimensional variational problems},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {359--362},
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Richard Gratwick; Aidys Sedipkov; Mikhail Sychev; Aris Tersenov. Pathological solutions to the Euler–Lagrange equation and existence/regularity of minimizers in one-dimensional variational problems. Comptes Rendus. Mathématique, Volume 355 (2017) no. 3, pp. 359-362. doi : 10.1016/j.crma.2017.01.020. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.01.020/

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Cited by Sources:

^☆ This research was partially supported by the European Research Council/ERC Grant Agreement No. 291497 and by the grants RFBR N 15-01-08275 and 0314-2015-0012 from the Presidium of RAS.

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