Gradient regularity for a class of elliptic obstacle problems

Abstract

We prove some regularity results for a priori bounded local minimizers of non-autonomous integral functionals of the form $$\mathcal{F}(v,\hat{I}\text{c◯})={\int }_{\hat{I}}\text{c◯}F(x,Dv)dx,$$ under the constraint $v\ge \ddot{I}ˆ$ a.e. in $\hat{I}\text{c◯}$, where $\ddot{I}ˆ$ is a fixed obstacle function. Assuming that the coefficients of the partial map $x\mapsto D_{\hat{I}}¾F(x,\hat{I}¾)$ satisfy a suitable Sobolev regularity, we are able to obtain higher differentiability and Lipschitz continuity results for the local minimizers.