Lipschitz regularity for minima without strict convexity of the Lagrangian

Abstract

We give, in a non-smooth setting, some conditions under which (some of) the minimizers of ∫ Ω f ( ∇ u ( x ) ) d x + g ( x , u ( x ) ) d x 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 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 ∫ Ω f ( ∇ u ( x ) ) d x on u 0 + W 0 1 , 1 ( Ω ) , whenever they exist, are Lipschitz. A relaxation result follows.