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∈Kψ(Ω)}where ψ is the obstacle, Kψ(Ω)={v∈u0+W01,p(Ω,R):v≥ψa.e. inΩ}, with u∈ W1,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 W1,n.