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

J. Bernis ; P. Bettiol ; C. Mariconda

Journal of convex analysis, Tome 27 (2020) no. 1, pp. 179-204

Résumé

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.

Détail

Classification : 49N60, 49K15
Mots-clés : Calculus of variations, minimizer regularity, higher order problems, necessary conditions, Weierstrass inequality, Erdmann-Du Bois-Reymond condition