Partial Lipschitz regularity of minimizers of asymptotically convex functionals with general growth conditions

We consider the partial Lipschitz continuity of minimizers of functionals of the formv↦∫Ωf(x,v,Dv)dx, where Ω⊆Rn is open and bounded. The map (x,u,ξ)↦f(x,u,ξ) appearing in the above functional is assumed to be Hölder continuous with respect to its first two arguments, of class C2 in its third argument, and asymptotically related to the more regular map (x,u,ξ)↦a(x,u)G(ξ) as |ξ|→+∞. The main novelty is the allowance that, even as |ξ|→+∞, the map f may retain full dependence on the minimizer u, not only its gradient Du and the spatial coordinate x.