On the Lower Semicontinuity of Supremal Functionals

Abstract

In this paper we study the lower semicontinuity problem for a supremal functional of the form F(u,Ω) = ess sup f(x, u(x),Du(x)) with respect to the strong convergence in L ∞ (Ω), furnishing a comparison with the analogoustheory developed by Serrin for integrals. A sort of Mazur's lemma for gradients of uniformly converging sequences is proved.