Semicontinuity and relaxation of L∞-functionals

Fixed a bounded open set Ω of R , we completely characterize the weak* lower semicontinuity of functionals of the form F (u,A) = ess sup x∈A f(x, u(x), Du(x)) 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 F 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 F coincides with the level convex envelope of F . Mathematics Subject Classification (2000): 47J20, 58B20, 49J45.