We establish local C 1;˛regularity for some ˛2.0; 1/ and C ˛-regularity for any ˛2.0; 1/ of local minimizers of the functionalwhere ' 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 ' in terms of a single condition for the map .x; t/ 7 ! '.x; t/, rather than separately in the xand 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 '.x; t/ with particular structure, including p-, Orlicz-, p.x/and double phasegrowth, our single condition implies known, essentially optimal, regularity conditions. Hence, we handle regularity theory for the above functional in a universal way.

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