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 form upsilon bar right arrow integral(Omega) f(x, upsilon, D upsilon)dx, where Q subset of R-n is open and bounded. The map (x, u, xi) bar right arrow f(x, u, xi) appearing in the above functional is assumed to be Holder continuous with respect to its first two arguments, of class l(2) in its third argument, and asymptotically related to the more regular map (x, u, xi) bar right arrow a(x, u)G(xi) as vertical bar xi vertical bar -> +infinity. The main novelty is the allowance that, even as vertical bar xi vertical bar -> +infinity, the map f may retain full dependence on the minimizer u, not only its gradient Du and the spatial coordinate x. (C) 2017 Elsevier Inc. All rights reserved.