
Journal of Convex Analysis 27 (2020), No. 1, 179204 Copyright Heldermann Verlag 2020 Higher Order Problems in the Calculus of Variations: Du BoisReymond Condition and Regularity of Minimizers Julien Bernis Laboratoire de Mathématiques, Université de Brest, 29200 Brest, France julien.bernis@univbrest.fr Piernicola Bettiol Laboratoire de Mathématiques, Université de Brest, 29200 Brest, France piernicola.bettiol@univbrest.fr Carlo Mariconda Dip. di Matematica, Università di Padova, 35121 Padova, Italy carlo.mariconda@unipd.it [Abstractpdf] 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 EulerLagrange form and, for the first time for higher order problems, in the ErdmannDu BoisReymond 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, ErdmannDu BoisReymond condition. MSC: 49N60, 49K15. [ Fulltextpdf (688 KB)] for subscribers only. 