LOCAL LIPSCHITZ REGULARITY OF MINIMA FOR A SCALAR PROBLEM OF THE CALCULUS OF VARIATIONS

Abstract

We consider a functional I(u) = ∫Ωf(∇ u(x)) dx on u0 + W1,1(Ω). Under the assumption that f is just convex, we prove a new Comparison Principle, we improve and give a short proof of Cellina's Comparison result for a new class of minimizers. We then extend a local Lipschitz regularity result obtained recently by Clarke for a wider class of functions f and boundary data u0 satisfying a new one-sided Bounded Slope Condition. A relaxation result follows.