Higher differentiability for bounded solutions to a class of obstacle problems with (p,q)-growth

Abstract

Abstract We establish the higher fractional differentiability of bounded minimizers to a class of obstacle problems with non-standard growth conditions of the form min ⁡ { ∫ Ω F ⁢ ( x , D ⁢ w ) ⁢ 𝑑 x : w ∈ 𝒦 ψ ⁢ ( Ω ) } , \min\bigg{\{}\int_{\Omega}F(x,Dw)dx:w\in\mathcal{K}_{\psi}(\Omega)\biggr{\}}, where Ω is a bounded open set of ℝ n {\mathbb{R}^{n}} , n ≥ 2 {n\geq 2} , the function ψ ∈ W 1 , p ⁢ ( Ω ) {\psi\in W^{1,p}(\Omega)} is a fixed function called obstacle, and 𝒦 ψ ⁢ ( Ω ) := { w ∈ W 1 , p ⁢ ( Ω ) : w ≥ ψ ⁢  a.e. in  ⁢ Ω } \mathcal{K}_{\psi}(\Omega):=\bigl{\{}w\in W^{1,p}(\Omega):w\geq\psi\text{ a.e.% in }\Omega\bigr{\}} is the class of admissible functions. If the obstacle ψ is locally bounded, we prove that the gradient of solution inherits some fractional differentiability property, assuming that both the gradient of the obstacle and the mapping x ↦ D ξ ⁢ F ⁢ ( x , ξ ) {x\mapsto D_{\xi}F(x,\xi)} belong to some suitable Besov space. The main novelty is that the Besov type regularity on the partial map x ↦ D ξ ⁢ F ⁢ ( x , ξ ) {x\mapsto D_{\xi}F(x,\xi)} is not related to the dimension n.