A relaxation result in the vectorial setting and $L^p$-approximation for $L^\infty$-functionals

Abstract

We provide relaxation for not lower semicontinuous supremal functionals of the type W 1,∞ (Ω; R d ) ∋ u → ess sup x∈Ω f (∇u(x)) in the vectorial case, where Ω ⊂ R N is a Lipschitz, bounded open set, and f is level convex. The connection with indicator functionals is also enlightened, thus extending previous lower semicontinuity results in that framework. Finally we discuss the L p -approximation of supremal functionals, with non-negative, coercive densities f = f (x, ξ), which are only L N ⊗ B d×N -measurable.