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

We consider a local minimizer, in the sense of the $W^{1,m}$ norm ($mge 1$), of the classical problem (P) of the calculus of variations Minimize $ I(x):=int_a^bLambda(t,x(t), x'(t)),dt+Psi(x(a), x(b))$ subject to: $xin W^{1,m}([a,b];R^n)$, $x'(t)in C, ext{ a.e., } ,x(t)in Sigmaquadorall tin [a,b]$ where $L:[a,b] imesR^n imesR^n oRcup{+infty}$ is just Borel measurable, $C$ is a cone, $Sigma$ is a nonempty subset of $R^n$ and $Psi$ is an arbitrary possibly extended valued function. When $L$ is real valued, we merely assume a local Lipschitz condition on $L$ with respect to $t$, allowing $L(t,x,\xi)$ to be discontinuous, and nonconvex in $x$ or $\xi$. In the case of an extended valued Lagrangian, we impose the lower semicontinuity of $L(cdot,x,cdot)$, and a condition on the effective domain of $L(t,x,cdot)$. We use a recent variational Weierstrass type inequality to show that the minimizers satisfy a relaxation result and a Erdmann Du Bois-Reymond convex inclusion which, remarkably, holds whenever $L(x,\xi)$ is autonomous and just Borel. Under a growth condition, weaker than superlinearity, we infer their Lipschitz continuity.