Maximal regularity for local minimizers of non-autonomous functionals

Abstract

We establish local $C^{1,\alpha }$-regularity for some $\alpha \in (0,1)$ and $C^{\alpha }$-regularity for any $\alpha \in (0,1)$ of local minimizers of the functional

where $\varphi$ satisfies a $(p,q)$-growth condition. Establishing such a regularity theory with sharp, general conditions has been an open problem since the 1980s. In contrast to previous results, we formulate the continuity requirement on $\varphi$ in terms of a single condition for the map $(x,t)\mapsto \varphi (x,t)$, rather than separately in the $x$- and $t$-directions. Thus we can obtain regularity results for functionals without assuming that the gap $q/p$ between the upper and lower growth bounds is close to $1$. Moreover, for $\varphi (x,t)$ with particular structure, including $p$-, Orlicz-, $p(x)$- and double phase-growth, our single condition implies known, essentially optimal, regularity conditions. Hence, we handle regularity theory for the above functional in a universal way.