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
julien.bernis@univ-brest.fr
Piernicola Bettiol
Laboratoire de Mathématiques, Université de Brest, 29200 Brest, France
piernicola.bettiol@univ-brest.fr
Carlo Mariconda
Dip. di Matematica, Università di Padova, 35121 Padova, Italy
carlo.mariconda@unipd.it
[Abstract-pdf]
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.
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