"Supremal Representation of L^{infty} Functionals"

We study the weak* lower semicontinuity properties of functionals of the form $$ F(u)=\supess_{x \in \Og} f(x,Du (x)) $$ where $\Og$ is a bounded open set of $\R^N$ and $u \in W^{1,\infty}(\Omega).$ Without a continuity assumption on $f( \cdot,\xi)$ we show that the {\sl supremal} functional $F$ is weakly* lower semicontinuous if and only if it is a level convex functional (i.e. it has convex sub-levels). In particular if $F$ is weakly* lower semicontinuous, then it can be represented through a level convex function. Finally a counterexample shows that in general it is not possible to represent $F$ through the level convex envelope of $f$