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 $\Omega$ is a bounded open set of ${\mathbb{R}}^n$, $n\ge 2$, the function $\psi \in W^{1,p}(\Omega )$ is a fixed function called \textit{obstacle} and ${\mathcal{K}}_{\psi }(\Omega ):=\{w\in W^{1,p}(\Omega ):w\ge \psi \text{a.e. in}\Omega \}$ 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 $n$.

arXiv commentarXiv admin note: text overlap with arXiv:2109.01584