Γ-convergence for power-law functionals with variable exponents. (April 2021) - Eleuteri, Michela; Prinari, Francesca
Abstract: 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 .