A Comparison Principle and the Lipschitz Continuity for Minimizers

We give some conditions that ensure the validity of a Comparison Principle for the minimizers of integral functionals, without assuming the validity of the Euler-Lagrange equation. We deduce a weak Maximum Principle for (possibly) degenerate elliptic equations and, together with a generalization of the Bounded Slope Condition, a result on the Lipschitz continuity of minimizers.