Semicontinuity and Supremal Representation in the Calculus of Variations

We study the weak* lower semicontinuity properties of functionals of the form where Ω is a bounded open set of R ^N and u∈W ^1,∞(Ω). Without a continuity assumption on f(⋅,ξ) we show that the 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.