Partial Hölder regularity for asymptotically convex functionals with borderline double-phase growth

We study partial Holder regularity of the local minimizers is an element of(1,1)(loc)(Omega; R)with >= 1 to the integral functional integral(Omega) (,,)in a bounded domain Omega subset of R for >= 2. Under the assumption of asymptotically convex to the borderline double-phase functional integral(Omega) (,)(|| + ()||log( + ||)). where (,) satisfies VMO in and is continuous in, respectively, and()isa strongly log-Holder continuous function, we prove that the local minimizer of such a functional is locally Holder continuous with an explicit Holder exponent in an open set Omega(0) subset of Omega with Hn--(Omega\Omega(0)) = 0 for some small > 0, where H denotes -dimensional Hausdorff measure