Higher differentiability for bounded solutions to a class of obstacle problems with $(p,q)$-growth

TL;DR

This paper proves that bounded solutions to certain obstacle problems with non-standard growth conditions exhibit higher fractional differentiability, even when the obstacle is only locally bounded, under assumptions on Besov space regularity.

Contribution

It establishes fractional differentiability of solutions' gradients for obstacle problems with $(p,q)$-growth, independent of the spatial dimension, under Besov space regularity assumptions.

Findings

Solutions' gradients have fractional differentiability properties.

Regularity results hold without dimension-dependent assumptions.

Applicable to obstacle problems with non-standard growth conditions.

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 $n$.