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\nobreak (0,1) of local minimizers of the functional v\ \mapsto\ \int_\Omega \phi(x,|Dv|)\,dx, where \phi 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 \phi in terms of a single condition for the map (x,t)\mapsto \phi(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 \phi(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.

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