Semicontinuity and supremal representation in the calculus of variations

摘要

We study the weak* lower semicontinuity properties of functionals of the form [GRAPHICS] where Omega is a bounded open set of R-N and u epsilon W-1,W-infinity(Omega). Without a continuity assumption on f (., xi) 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.