Higher fractional differentiability for solutions to a class of obstacle problems with non-standard growth conditions

TL;DR

This paper proves that solutions to certain obstacle problems with non-standard growth conditions exhibit higher fractional differentiability, assuming the obstacle's gradient has specific Besov space regularity.

Contribution

It establishes higher fractional differentiability for solutions to obstacle problems with non-standard growth, under new assumptions on the obstacle's gradient.

Findings

Fractional differentiability of solutions is enhanced under Besov space conditions.

Results extend regularity theory to non-standard growth obstacle problems.

Provides new regularity transfer results for variational inequalities.

Abstract

We here establish the higher fractional differentiability for solutions to a class of obstacle problems with non-standard growth conditions. We deal with the case in which the solutions to the obstacle problems satisfy a variational inequality of the form \begin{equation*} \displaystyle\int_{\Omega} \langle \mathcal{A}(x,Du) ,D(\varphi-u) \rangle dx \geq 0 \qquad \forall \varphi \in \mathcal{K}_\psi(\Omega), \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 some suitable Besov space, we are able to prove that some fractional differentiability property transfers to the gradient of the solution.