Regularity Results for Bounded Solutions to Obstacle Problems with Non-standard Growth Conditions

In this paper, we consider a class of obstacle problems of the type min ∫ Ω f ( x , D v ) d x : v ∈ K ψ ( Ω ) where ψ is the obstacle, K ψ ( Ω ) = { v ∈ u 0 + W 0 1 , p ( Ω , R ) : v ≥ ψ a.e. in Ω } , with u 0 ∈ W 1 , p ( Ω ) a fixed boundary datum, the class of the admissible functions and the integrand f ( x , Dv ) satisfies non standard ( p , q )-growth conditions. We prove higher differentiability results for bounded solutions of the obstacle problem under dimension-free conditions on the gap between the growth and the ellipticity exponents. Moreover, also the Sobolev assumption on the partial map x ↦ A ( x , ξ ) is independent of the dimension n and this, in some cases, allows us to manage coefficients in a Sobolev class below the critical one W 1 , n .