On a new necessary condition in the Calculus of Variations for Lagrangians that are highly discontinuous in the state and velocity

Abstract

We consider a local minimizer, in the sense of the $W^{1,1}$ norm, of the classical problem of the calculus of variations

where $\Lambda :[a,b]\times {\mathbb{R}}^n\times {\mathbb{R}}^n\to \mathbb{R}\cup \{+\infty \}$ is just Borel measurable, $C$ is a cone, $\Sigma$ is a nonempty subset of ${\mathbb{R}}^n$ and $\Psi$ is an arbitrary extended valued function: this allows to cover any kind of endpoint constraints. We do not assume further assumptions than Borel measurability and a local Lipschitz condition on $\Lambda$ with respect to $t$, allowing $\Lambda (t,x,\xi )$ to be possibly discontinuous, nonconvex in $x$ or $\xi$. This article reconsiders the results obtained in two recent papers by the authors: we refer to [5, 4] for further details and proofs. Consider a local minimizer $x_{\ast }$, in the sense of the norm of the absolutely continuous functions. We illustrate a new necessary condition: there exists an absolutely continuous function $p$ such that, for almost every $t$ in $[a,b]$,

and moreover, $p^{\prime }$ belongs to a suitable generalized subdifferential of $s\mapsto \Lambda (s,x_{\ast }(t),x_{\ast }^{\prime }(t))$ at $s=t$. The proof of (W) takes full advantage of a classical reparametrization technique, and of recent versions of the maximum principle. The variational inequality turns out to be equivalent to a generalized Erdmann–Du Bois-Reymond (EDBR) type necessary condition, that we are able to express in terms of the classical tools of convex analysis (e.g. convex subdifferentials): in the autonomous, real valued case it holds true for every Borel Lagrangian. More regularity is required to reformulate the (EDBR) condition in terms of the limiting subdifferential.

From (W) we deduce the Lipschitz regularity of the local minimizers for (P) if the Lagrangian satisfies a growth condition, less restrictive than superlinearity, inspired by those introduced in [8, 17]. In the autonomous case the result implies the most general Lipschitz regularity theorem present in the literature, for Lagrangians that are just Borel, and is new in the case of an extended valued Lagrangian.