Gradient regularity for a class of elliptic obstacle problems

We prove some regularity results for a priori bounded local minimizers of non-autonomous integral functionals of the form $$\mathcal{F}(v,\mathrm{\Omega })={\int }_{\mathrm{\Omega }}F(x,Dv)dx,$$ under the constraint $v\ge \psi$ a.e. in $\mathrm{\Omega }$, where $\psi$ is a fixed obstacle function. Assuming that the coefficients of the partial map $x\mapsto D_{\xi }F(x,\xi )$ satisfy a suitable Sobolev regularity, we are able to obtain higher differentiability and Lipschitz continuity results for the local minimizers.