On the Relaxation of Variational Integrals with Pointwise Continuous-Type Gradient Constraints

Relaxation problems for a functional of the type $G(u) =int_Omega g(x,∇u) dx$ are analyzed, where $Omega$ is a bounded smooth subset of $R^N$ and $g$ is a Carath´eodory function, when the admissible $u$ are forced to satisfy a pointwise gradient constraint of the type $ abla u(x) in C(x)$ for a.e. $x in Omega$, $C(x)$ being, for every $x in Omega$, a bounded convex subset of $R^N$ . The relaxed functionals G_{PC^1(Omega)$, and $G_W^{1,infty(Omega)$ of $G$ obtained letting $u$ vary in $PC^1(Omega)$, the set of the piecewise $C^1$-functions in $Omega$, and in $W^{1,infty}(Omega)$ respectively in the definition of $G$ are considered. Identity and integral representation results are proved under continuity-type assumptions on $C$, together with the description of the common density by means of convexification arguments. Classical relaxation results are extended to the case of the continuous variable dependence of $C$, and the non-identity features described in the measurable dependence case by De Arcangelis, Monsurr`o and Zappale (2004) are shown to be non-occurring. Proofs are based on the properties of certain limits of multifunctions, and on an approximation result for functions $u$ in $W^1,infty(Omega)$, with $ abla u(x) in C(x)$ for a.e. $x in Omega$ by $PC^1(Omega)$ ones satisfying the same condition. Results in more general settings are also obtained.