Semicontinuity and supremal representation in the Calculus of Variations
نویسنده
چکیده
We study the weak* lower semicontinuity properties of functionals of the form F (u) = ess sup x∈Ω f(x,Du(x)) where Ω is a bounded open set of R and u ∈ W 1,∞(Ω). Without a continuity assumption on f(·, ξ) we show that the supremal functional F is weakly* lower semicontinuous if and only if it is a level convex functional (i.e. it has convex sub levels). In particular if F is weakly* lower semicontinuous, than it can be represented through a level convex function. Finally a counterexample shows that it is not possible to represent F through the level convex envelope of f . Mathematics Subject Classification (2000): 47J20, 58B20, 49J45.
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