*Published Paper*

**Inserted:** 11 sep 2019

**Last Updated:** 11 sep 2019

**Journal:** Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl.

**Volume:** 30

**Number:** 3

**Pages:** 649-663

**Year:** 2019

**Doi:** 10.4171/RLM/865

**Links:**
Link to the article

**Abstract:**

We consider a local minimizer, in the sense of the $W^{1,1}$ norm, of the classical problem of the calculus of variations \[\begin{cases} \text{Minimize}\quad &\displaystyle I(x):=\int_a^b\Lambda(t,x(t), x'(t))\,dt+\Psi(x(a), x(b))\\ \text{subject to:} &x\in W^{1,1}([a,b];\mathbb R),\\ &x'(t)\in C\,\text{ a.e., } \,x(t)\in \Sigma\quad\forall t\in [a,b].\\ \end{cases}\tag{P} \] 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. In the case where $\Lambda$ is real valued 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 $\xi$. In the case of an extended valued Lagrangian, we further impose the l.s.c. and a condition on the effective domain of $\Lambda(t,x,\cdot)$. This article is a survey of two recent papers of the authors (accepted in J. Differential Equations and in J. Optim. Theory Appl.). Consider a local minimizer $x_*$, 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 $[a,b]$, \[\Lambda\Big(t, x_*(t),\dfrac{x_*'(t)}{v}\Big)v-\Lambda(t, x_*(t), x_*'(t))\ge p(t)(v-1)\quad\forall v>0,\tag{W}\] and moreover, $p'$ belongs to a suitable generalized subdifferential of $t\mapsto\Lambda (t, x_*(t), x_*'(t))$. The proof of (W) takes full advantage of a classical reparametrization technique, and of the most recent versions of the Maximum Principle. The variational inequality turns out to be equivalent to a generalized Erdmann -- DuBois-Reymond (EDBR) necessary type condition, that we are able to express just in terms of the classical tools of convex analysis (e.g. convex subdifferentials): in the autonomous, real valued case it hold true for \emph{every} Borel Lagrangian. Instead, one needs more regularity to formulate a version of the (EDBR) in terms of the limiting subdifferential.

We deduce the Lipschitz regularity of the local minimizers of (P) if the Lagrangian satisfies a growth condition, less restrictive than superlinearity, inspired by those introduced by Cellina & al. 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.

**Keywords:**
Lipschitz regularity, Maximum Principle, nonautonomous, Du Bois-Reymond, Weierstrass