Lower semicontinuity of L-infinity functionals

We study the lower semicontinuity properties and existence of a minimizer o f the functional F(u) = ess sup(x is an element of Omega) f(x, u(x), Du(x)) on W-1,W-infinity(Omega; R-m). We introduce the notions of Morrey quasiconv exity, polyquasiconvexity, and rank-one quasiconvexity, all stemming from t he notion of quasiconvexity (= convex level sets) of f in the last variable . We also formally derive the Aronsson-Euler equation for such problems. (C ) 2001 Editions scientifiques et medicales Elsevier SAS.