[formula omitted]-convergence for power-law functionals with variable exponents.

We study the Γ -convergence of the functionals F n (u) : = | | f (⋅ , u (⋅) , D u (⋅)) | | p n (⋅) and F n (u) : = ∫ Ω 1 p n (x) f p n (x) (x , u (x) , D u (x)) d x defined on X ∈ { L 1 (Ω , R d) , L ∞ (Ω , R d) , C (Ω , R d) } (endowed with their usual norms) with effective domain the Sobolev space W 1 , p n (⋅) (Ω , R d). Here Ω ⊆ R N is a bounded open set, N , d ≥ 1 and the measurable functions p n : Ω → [ 1 , + ∞) satisfy the conditions ess sup Ω p n ≤ β ess inf Ω p n < + ∞ for a fixed constant β > 1 and ess inf Ω p n → + ∞ as n → + ∞. We show that when f (x , u , ⋅) is level convex and lower semicontinuous and it satisfies a uniform growth condition from below, then, as n → ∞ , the sequence (F n) n Γ -converges in X to the functional F represented as F (u) = | | f (⋅ , u (⋅) , D u (⋅)) | | ∞ on the effective domain W 1 , ∞ (Ω , R d). Moreover we show that the Γ - lim n F n is given by the functional F (u) : = 0 i f | | f (⋅ , u (⋅) , D u (⋅)) | | ∞ ≤ 1 , + ∞ o t h e r w i s e i n X. [ABSTRACT FROM AUTHOR]