A new condition ensuring gradient continuity for minimizers of non-autonomous functionals with mild phase transition

We show that local minimizers of the non-autonomous functional P-log(u,Omega) = integral(Omega)|Du|(p)(1+a(x)loge+|Du|)dx, p > 1, have continuous gradient provided that the function a()is (almost everywhere) non-negative and weakly differentiable, and moreover its gradient locally belongs to the Lorentz-ZygmundspaceL(n,1)logL. This gives a precise insight of the fact that for this type of two-phase functionals the lack of uniform ellipticity can be overcome by additional regularity of the switching coefficient a(); the novelty is that the condition is not pointwise, but has integral character, and actually improves the known results ensuring regularity for minimizers of such functionals.