*Submitted Paper*

**Inserted:** 27 sep 2019

**Last Updated:** 18 may 2020

**Year:** 2019

**Abstract:**

We provide relaxation for not lower semicontinuous supremal functionals of the type $W^{1,\infty}(\Omega;\mathbb R^d) \ni u \mapsto{\rm esssup}_{ x \in \Omega}f(\nabla u(x))$ in the vectorial case, where $\Omega\subset \mathbb R^N$ is a Lipschitz, bounded open set, 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 $\mathcal L^N \otimes \mathcal B_{d \times N}$-measurable.

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