Regularity results for minimizers of non-autonomous integral functionals

Abstract

We establish the higher fractional differentiability for the minimizers of non-autonomous integral functionals of the form \begin{equation} \mathcal{F}(u,Ω):=\int_Ω\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.