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

Abstract

AbstractIn this paper, we consider a class of obstacle problems of the type $$\begin{aligned} \min \left\{ \int _{\Omega }f(x, Dv)\, {\mathrm d}x\,:\, v\in {\mathcal {K}}_\psi (\Omega )\right\} \end{aligned}$$ min ∫ Ω f ( x , D v ) d x : v ∈ K ψ ( Ω ) where $$\psi $$ ψ is the obstacle, $${\mathcal {K}}_\psi (\Omega )=\{v\in u_0+W^{1, p}_{0}(\Omega , {\mathbb {R}}): v\ge \psi \text { a.e. in }\Omega \}$$ K ψ ( Ω ) = { v ∈ u 0 + W 0 1 , p ( Ω , R ) : v ≥ ψ a.e. in Ω } , with $$u_0 \in W^{1,p}(\Omega )$$ 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 $$x\mapsto A(x, \xi )$$ 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 $$W^{1,n}$$ W 1 , n .