On the Relaxation and the Lavrentieff Phenomenon of Variational Integrals with Pointwise Measurable Gradient Constraints

Relaxation problems for a functional of the type $G(u) = \int_\Omega g(x,\nabla u)dx$ are analyzed, where $\Omega$ is a bounded smooth open subset of $R^N$ and $g$ is a Carath´eodory function. The admissible functions u are forced to satisfy a pointwise gradient constraint of the type $\nabla u(x) \in C(x)$ for a.e. $ x \in \Omega, C(x)$ being, for every $x \in \Omega$, a bounded convex subset of $R^N$, in general varying with $x$ not necessarily in a smooth way. The relaxed functionals $G_{PC^1}(\Omega)$ and $G_{W^{1,\infty}(\Omega)$ of $G$ obtained letting $u$ vary respectively in $PC^1(\Omega)$, the set of the piecewise $C^1$-functions in $\Omega$, and in $W^{1,\infty}(\Omega)$ in the definition of $G$ are considered. For both of them integral representation results are proved, with an explicit representation formula for the density of $G_{PC^1}(\Omega)$. Examples are proposed showing that in general the two densities are different, and that the one of $G_{W^{1,\infty}}(\Omega)$ is not obtained from $g$ simply by convexification arguments. Eventually, the results are discussed in the framework of Lavrentieff phenomenon, showing by means of an example that deep differences occur between $G_{PC^1}(\Omega)$ and $G_{W^{1,\infty}(\Omega)}$. Results in more general settings are also obtained.