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

TL;DR

This paper investigates the fractional differentiability of solutions to double-phase obstacle problems, showing that under certain conditions, the gradient of solutions retains higher regularity in Besov spaces.

Contribution

It establishes higher fractional differentiability results for the gradient of solutions to double-phase obstacle problems within Besov space frameworks.

Findings

Gradient of solutions retains fractional differentiability

Results depend on obstacle's Besov space regularity

Advances regularity theory for double-phase variational 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.