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 f(Omega) f(del u(x))dx + g(x,u(x)) dx among the functions in W-1,W-1(Omega) 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 f : R-n --> R are bounded and the boundary datum u(0) satisfies a generalization of the Bounded Slope Condition introduced by A. Cellina then the minima of f Omega f (del u (x)) dx on 1, 1 (Q) whenever they exist, are Lipschitz. A relaxation result follows. u(0) + W-0(1,1) (C) 2007 Elsevier Inc. All rights reserved.