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

Abstract

We establish the higher fractional differentiability of bounded minimizers to a class of obstacle problems with non-standard growth conditions of the form \begin{gather*} \min \biggl\{ \displaystyle\int_Ω F(x,Dw)dx \ : \ w \in \mathcal{K}_ψ(Ω) \biggr\}, \end{gather*} where $Ω$ is a bounded open set of $\mathbb{R}^n$, $n \geq 2$, the function $ψ\in W^{1,p}(Ω)$ is a fixed function called \textit{obstacle} and $\mathcal{K}_ψ(Ω) := \{ w \in W^{1,p}(Ω) : w \geq ψ\ \text{a.e. in} \ Ω\}$ 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 $x \mapsto D_ξF(x,ξ)$ belong to some suitable Besov space. The main novelty is that such assumptions are not related to the dimension $n$.