Semicontinuity and relaxation of L ∞-functionals

Abstract

Fixed a bounded open set Ω of ℝ N, we completely characterize the weak * lower semicontinuity of functionals of the form F(u,A) = ess sup f(x, u(x), Du(x)) χ∈A defined for every u ∈ W 1,∞(Ω) and for every open subset A ⊂ Ω. Without a continuity assumption on f(·,u,ξ) we show that the supremal functional F is weakly * lower semicontinuous if and only if it can be represented through a level convex function. Then we study the properties of the lower semicontinuous envelope F̄ of F. A complete relaxation theorem is shown in the case where f is a continuous function. In the case f = f(x,ξ) is only a Carathéodory function, we show that F̄ coincides with the level convex envelope of F. © de Gruyter 2009.