*Published Paper*

**Inserted:** 6 oct 2008

**Last Updated:** 10 nov 2014

**Journal:** Adv. Calc. Var.

**Year:** 2008

**Abstract:**

Fixed a bounded open set $\Omega$ of $\bf R^N$, we completely characterize the weak$^*$ lower semicontinuity of functionals of the form \[ F(u,A)=\hbox{ess-sup}_{x \in A} f(x,u(x),Du (x)) \] defined for every $u \in W^{1,\infty}(\Omega)$ and for every open subset $A\subset \Omega$. Without a continuity assumption on $f( \cdot,u,\xi)$ we show that the supremal functional $F$ is weakly$^*$ lower semicontinuous if and only if it can be represented through a 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 Carathéodory function, we show that $\overline F$ coincides with the level convex envelope of $F$.

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