Semicontinuity and relaxation of $L^{infty}$-functionals

Fixed a bounded open set $\Og$ of $\R^N$, we completely characterize the weak* lower semicontinuity of functionals of the form $$ F(u,A)=\supess_{x \in A} f(x,u(x),Du (x)) $$ defined for every $u \in W^{1,\infty}(\Omega)$ and for every open subset $A\subset \Om$. Without a continuity assumption on $f( \cdot,u,\xi)$ we show that the {\sl supremal} functional $F$ is weakly* lower semicontinuous if and only if it can be represented through a {\sl level convex} function. Then we study the properties of the lower semicontinuous envelope $\overline F$ of $F$. A complete relaxation theorem is shown in the case where $f$ is a continuous function. In the case $f=f(x,\xi)$ is only a Carath\'eodory function, we show that $\overline F$ coincides with the level convex envelope of $F$.