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 ∫ Ω f (∇u(x)) dx + g(x,u(x)) dx among the functions in W1,1(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 f :Rn → R are bounded and the boundary datum u0 satisfies a generalization of the Bounded Slope Condition introduced by A. Cellina then the minima of ∫ Ω f (∇u(x)) dx on u0 +W 0 (Ω), whenever they exist, are Lipschitz. A relaxation result follows. © 2007 Elsevier Inc. All rights reserved. MSC: 49J30; 49J10