Journal Home Page

Cumulative Index

List of all Volumes

Complete Contents
of this Volume

Previous Article

Next Article

Journal of Convex Analysis 27 (2020), No. 1, 179--204
Copyright Heldermann Verlag 2020

Higher Order Problems in the Calculus of Variations:
Du Bois-Reymond Condition and Regularity of Minimizers

Julien Bernis
Laboratoire de Mathématiques, Université de Brest, 29200 Brest, France

Piernicola Bettiol
Laboratoire de Mathématiques, Université de Brest, 29200 Brest, France

Carlo Mariconda
Dip. di Matematica, Università di Padova, 35121 Padova, Italy


This paper concerns an $N$-order problem in the calculus of variations of minimizing the functional $\smash{\int_{a}^{b}{\Lambda(t,x(t),\ldots,x^{(N)}(t))\mathrm{d}t}}$, in which the Lagrangian $\Lambda$ is a Borel measurable, non autonomous, and possibly extended valued function. Imposing some additional assumptions on the Lagrangian, such as an integrable boundedness of the partial proximal subgradients (up to the ($N\!-\!2$)-order variable), a growth condition (more general than superlinearity w.r.t. the last variable) and, when the Lagrangian is extended valued, the lower semicontinuity, we prove that the $N$-th derivative of a reference minimizer is essentially bounded. We also provide necessary optimality conditions in the Euler-Lagrange form and, for the first time for higher order problems, in the Erdmann-Du Bois-Reymond form. The latter can be also expressed in terms of a (generalized) convex subdifferential, and is valid even without requiring neither a particular growth condition nor convexity in any variable.

Keywords: Calculus of variations, minimizer regularity, higher order problems, necessary conditions, Weierstrass inequality, Erdmann-Du Bois-Reymond condition.

MSC: 49N60, 49K15.

[ Fulltext-pdf  (688  KB)] for subscribers only.