Regularity for non-autonomous functionals with almost linear growth

We consider non-autonomous functionals F (u; Omega) = integral(Omega) f(x, Du) dx, where the density f : Omega x R(nN) -> R has almost linear growth, i.e., f(x, xi) approximate to vertical bar xi vertical bar log (1 + vertical bar xi vertical bar). We prove partial C (1,gamma) -regularity for minimizers u: R(n) superset of Omega -> R(N) under the assumption that D (xi) f (x, xi) is Holder continuous with respect to the x-variable. If the x-dependence is C (1) we can improve this to full regularity provided additional structure conditions are satisfied.