Higher differentiability results in the scale of Besov spaces to a class of double-phase obstacle problems

Abstract

We study the higher fractional differentiability properties of the gradient of the solutions to variational obstacle problems of the form \begin {gather*} \min \biggl\{ \int_{\Omega} F(x,w,Dw) d x \ : \ w \in \mathcal{K}_{\psi}(\Omega) \biggr\}, \end {gather*} with $F$ double phase functional of the form \begin {equation*} F(x,w,z)=b(x,w)(|z|^p+a(x)|z|^q), \end {equation*} where $\Omega$ is a bounded open subset of $\mathbb{R}^n$ , $\psi \in W^{1,p}(\Omega)$ 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 . Assuming that the gradient of the obstacle belongs to a suitable Besov space, we are able to prove that the gradient of the solution preserves some fractional differentiability property .