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_Ωf\left(x, Dv(x)\right)dx\,:\, v\in \mathcal{K}_ψ(Ω)\right\}, \end{equation*} where the function $f$ satisfies $p-$growth conditions with respect to the gradient variable, for $1<p<2$, and $\mathcal{K}_ψ(Ω)$ is the class of admissible functions. Here we show that, if the obstacle $ψ$ is bounded, then a Sobolev regularity assumption on the gradient of the obstacle $ψ$ transfers to the gradient of the solution, provided the partial map $x\mapsto D_ξf(x,ξ)$ 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. $f(x, ξ)\approx a(x)|ξ|^p$ with $1<p<2,$ and where the map $a$ belongs to a Sobolev space.

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