A relaxation result in the vectorial setting and $L^p$-approximation for $L^{\mathrm{\infty }}$-functionals

We provide relaxation for not lower semicontinuous supremal functionals of the type $W^{1,\infty}(\Omega;\mathbb R^d) \ni u \mapsto\supess_{ x \in \Omega}f(\nabla u(x))$inthevectorialcase,where$\Omega\subset \mathbb R^N$isaLipschitz,boundedopenset,and$f$ is level convex. The connection with indicator functionals is also enlightened, thus extending previous lower semicontinuity results in that framework. Finally we discuss the $L^p$-approximation of supremal functionals, with non-negative, coercive densities $f=f(x,\xi )$, which are only $\L^N \otimes \B_{d \times N}$-measurable.