Regularity results for minimizers of non-autonomous integral functionals

We establish the higher fractional differentiability for the minimizers of non-autonomous integral functionals of the form \begin{equation} \mathcal{F}(u,\Omega):=\int_\Omega \left[ f(x,Du)- g \cdot u \right] dx , \notag \end{equation} under \((p,q)\)-growth conditions. Besides a suitable differentiability assumption on the partial map \(x \mapsto D_\xi f(x,\xi)\), we do not need to assume any differentiability assumption on the function \(g\). Moreover, we show that the higher differentiability result holds true also assuming strict convexity and growth conditions on \(f\) only at infinity.