Regularity results for solutions to a class of non-autonomous obstacle problems with sub-quadratic growth conditions

We establish some higher differentiability results for solution to non-autonomous obstacle problems of the form \begin{equation*} \min \left\{\int_{\Omega}f\left(x, Dv(x)\right)dx\,:\, v\in \mathcal{K}_\psi(\Omega)\right\}, \end{equation*} where the function $f$ satisfies $p-$growth conditions with respect to the gradient variable, for $1<p<2$, and $\mathcal{K}_\psi(\Omega)$ is the class of admissible functions. Here we show that, if the obstacle $\psi$ is bounded, then a Sobolev regularity assumption on the gradient of the obstacle $\psi$ transfers to the gradient of the solution, provided the partial map $x\mapsto D_\xi f(x,\xi)$ belongs to a Sobolev space, $W^{1, p+2}$. The novelty here is that we deal with subquadratic growth conditions with respect to the gradient variable, i.e. $f(x, \xi)\approx a(x)|\xi|^p$ with $1<p<2,$ and where the map $a$ belongs to a Sobolev space.