A Du Bois-Reymond Convex Inclusion for Nonautonomous Problems of the Calculus of Variations and Regularity of Minimizers

Abstract

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

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathrm{Minimize}}\quad &{}\displaystyle I(x):=\int _a^b\varLambda (t,x(t), x'(t))\,dt+\varPsi (x(a), x(b))\\ \text {subject to:} &{}x\in W^{1,m}([a,b];\mathbb {R}^n),\\ &{}x'(t)\in C\,\text { a.e., } \,x(t)\in \varSigma \quad \forall t\in [a,b].\\ \end{array}\right. } \end{aligned}$$

(P)

where $\varLambda :[a,b]\times \mathbb {R}^n\times \mathbb {R}^n\rightarrow \mathbb {R}\cup \{+\infty \}$ is just Borel measurable, C is a cone, $\varSigma$ is a nonempty subset of $\mathbb {R}^n$ and $\varPsi$ is an arbitrary possibly extended valued function. When $\varLambda$ is real valued, we merely assume a local Lipschitz condition on $\varLambda$ with respect to t, allowing $\varLambda (t,x,\xi )$ to be discontinuous and nonconvex in x or $\xi$. In the case of an extended valued Lagrangian, we impose the lower semicontinuity of $\varLambda (\cdot ,x,\cdot )$, and a condition on the effective domain of $\varLambda (t,x,\cdot )$. We use a recent variational Weierstrass type inequality to show that the minimizers satisfy a relaxation result and an Erdmann – Du Bois-Reymond convex inclusion which, remarkably, holds whenever $\varLambda (x,\xi )$ is autonomous and just Borel. Under a growth condition, weaker than superlinearity, we infer the Lipschitz continuity of minimizers.