We establish local -regularity for some and -regularity for any of local minimizers of the functional
where satisfies a -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 , rather than separately in the - and -directions. Thus we can obtain regularity results for functionals without assuming that the gap between the upper and lower growth bounds is close to . Moreover, for with particular structure, including -, Orlicz-, - 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.
Cite this article
Peter Hästö, Jihoon Ok, Maximal regularity for local minimizers of non-autonomous functionals. J. Eur. Math. Soc. 24 (2022), no. 4, pp. 1285–1334DOI 10.4171/JEMS/1118