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

Abstract

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 ${\mathcal{K}}_{\psi }(\Omega )$ 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.

arXiv admin note: text overlap with arXiv:2007.04064, arXiv:1910.04506