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

This paper concerns an

-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}} ∫ a b Λ ( t , x ( t ) , … , x ( N ) ( t ) ) d t

, in which the Lagrangian

\Lambda Λ

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 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 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.