Lipschitz regularity for minima without strict convexity of the Lagrangian

We give, in a non-smooth setting, some conditions under which (some of) the minimizers of among the functions in W^1,1(Ω) that lie between two Lipschitz functions are Lipschitz. We weaken the usual strict convexity assumption in showing that, if just the faces of the epigraph of a convex function are bounded and the boundary datum u₀ satisfies a generalization of the Bounded Slope Condition introduced by A. Cellina then the minima of on , whenever they exist, are Lipschitz. A relaxation result follows.