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
[Solutions pathologiques à l'équation d'Euler–Lagrange et existence/régularité des minimiseurs des problèmes variationnels en dimension un]

Présenté par : 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.

Résumé
Abstract

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.

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.

Reçu le : 2016-10-27
Accepté le : 2017-01-31
Publié le : 2017-02-09

DOI : 10.1016/j.crma.2017.01.020

Affiliations des auteurs :

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

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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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     number = {3},
     year = {2017},
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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/

Bibliographie
Cité par

[1] J.M. Ball; V.J. Mizel One-dimensional variational problems whose minimizers do not satisfy the Euler–Lagrange equation, Arch. Ration. Mech. Anal., Volume 90 (1985) no. 4, pp. 325-388

[2] G. Buttazzo; M. Giaquinta; S. Hildebrandt One-Dimensional Variational Problems. An Introduction, Oxford University Press, UK, 1998

[3] F.H. Clarke; R.B. Vinter Regularity properties of solutions to the basic problem in the calculus of variations, Trans. Amer. Math. Soc., Volume 289 (1984) no. 1, pp. 73-98

[4] F.H. Clarke; R.B. Vinter On the conditions under which the Euler equation or the maximum principle hold, Appl. Math. Optim., Volume 12 (1984), pp. 73-79

[5] A.M. Davie Singular minimizers in the calculus of variations, Arch. Ration. Mech. Anal., Volume 101 (1988), pp. 161-177

[6] R. Gratwick Singular sets and the Lavrentiev phenomenon, Proc. R. Soc. Edinb., Sect. A, Volume 145 (2015) no. 3, pp. 513-533

[7] R. Gratwick; M.A. Sychev; A.S. Tersenov Regularity and singularity phenomena for one-dimensional variational problems with singular ellipticity, Pure Appl. Funct. Anal., Volume 1 (2016) no. 3, pp. 397-416

[8] V.J. Mizel; M.A. Sychev A condition on the value function both necessary and sufficient for full regularity of minimizers of one-dimensional variational problems, Trans. Amer. Math. Soc., Volume 350 (1998), pp. 119-133

[9] M.A. Sychev On a classical problem of the calculus of variations, Sov. Math. Dokl., Volume 44 (1992), pp. 116-120

[10] M.A. Sychev On the question of regularity of the solutions of variational problems, Russian Acad. Sci. Sb. Math., Volume 75 (1993), pp. 535-556

[11] M.A. Sychev Examples of classically unsolvable regular scalar variational problems satisfying standard growth conditions, Sib. Math. J., Volume 37 (1996), pp. 1212-1227

[12] M.A. Sychev Another theorem of classical solvability ‘in small’ for one-dimensional variational problems, Arch. Ration. Mech. Anal., Volume 202 (2011), pp. 269-294

[13] M.A. Sychev; N.N. Sycheva On Legendre and Weierstrass conditions in one-dimensional variational problems, J. Convex Anal. (2017) (in press)

[14] L. Tonelli Sur une méthode direte du calcul des variations, Rend. Circ. Mat. Palermo, Volume 39 (1915), pp. 233-264

[15] L. Tonelli Fondamenti di calcolo delle Variazioni, vol. II, Zanichelli, Bologna, Italy, 1921

Cité par 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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