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_{\Omega} F(x,Dw)dx \ : \ w \in \mathcal{K}_{\psi}(\Omega) \biggr\}, \end{gather*} where $\Omega$ is a bounded open set of $\mathbb{R}^n$, $n \geq 2$, the function $\psi \in W^{1,p}(\Omega)$ is a fixed function called \textit{obstacle} and $\mathcal{K}_{\psi}(\Omega) := \{ w \in W^{1,p}(\Omega) : w \geq \psi \ \text{a.e. in} \ \Omega \}$ is the class of admissible functions. If the obstacle $\psi$ 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 $x \mapsto D_\xi F(x,\xi)$ belong to some suitable Besov space. The main novelty is that such assumptions are not related to the dimension $n$.