On the lower semicontinuity and approximation of $L^infty$ functionals

In this paper we show that if the supremal functional F(V,B) = ess sup x∈B f(x, DV (x)) is sequentially weak* lower semicontinuous on W1,∞(B, Rd) for every open set B ⊆ Ω (where Ω is a fixed open set of RN ), then f(x, ·) is rank-one level convex for a.e x ∈ Ω. Next, we provide an example of a weak Morrey quasiconvex function which is not strong Morrey quasiconvex. Finally we discuss the Lp-approximation of a supremal functional F via Γ-convergence when f is a non-negative and coercive Carath´eodory function.