*Accepted Paper*

**Inserted:** 30 jun 2015

**Journal:** NODEA

**Year:** 2015

**Abstract:**

In this paper we show that if the supremal functional $F(V,B)=\esssup_{x \in B} f(x,DV (x))$ is sequentially weak** lower semicontinuous on $ W^{1,\infty}(B, \R^d)$ for every open set $B\subseteq \Omega$ (where $\Omega$ is a fixed open set of $\R^N$), then $f(x,\cdot)$ is rank-one level convex for a.e $x\in \Omega$. Next, we provide an example of a weak Morrey quasiconvex function which is not strong Morrey quasiconvex.
Finally we discuss the $L^p$-approximation of a supremal functional $F$ via $\Gamma$-convergence when $f$ is a non-negative and coercive Carath\'eodory function.**