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{∫ _Ωf(x, Dv) dx : v∈𝒦_ψ (Ω )} where ψ is the obstacle, 𝒦_ψ (Ω )={v∈ u_0+W^1, p_0(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 .