Calderón–Zygmund type estimates for a class of obstacle problems with growth

For local minimizers u ∈ W loc 1 , p ( ⋅ ) ( Ω ) of quasiconvex integral functionals of the type F [ u ] : = ∫ Ω f ( x , D u ( x ) ) d x with p ( x ) growth in the class K : = { u ∈ W loc 1 , p ( ⋅ ) ( Ω ) : u ⩾ ψ } , where ψ ∈ W loc 1 , p ( ⋅ ) ( Ω ) is a given obstacle function, we show estimates of Calderón–Zygmund type, i.e. | D ψ | p ( ⋅ ) ∈ L loc q ⇒ | D u | p ( ⋅ ) ∈ L loc q , for any q > 1 , provided that the modulus of continuity ω of the exponent function p satisfies the condition ω ( ρ ) log 1 ρ → 0 as ρ → 0 .