A comparison principle 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, the Lipschitz continuity of minimizers. To prove the main theorem we give a result on the existence of a representative of a given Sobolev function that is absolutely continuous along the trajectories of a suitable autonomous system. (C) 2000 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.