The regularity of mappings with a convex potential

In this work, we apply the techniques developed in [Cl] to the problem of mappings with a convex potential between domains. That is, given two bounded domains Q, Q2 of Rn and two nonnegative real functions fi defined in Qi that are bounded away from zero and infinity, we want to study the map vf = V W for a Lipschitz convex ,v, such that V ,/ maps Ql onto Q?2 in the a.e. sense and in some (weak) sense. (1) f2(VyV) det Dij V = f1 (X) . In recent work Y. Brenier showed existence and uniqueness of such a map (provided that JaQil = 0) under the obvious compatibility condition