Regularity results for very degenerate elliptic equations

We consider a family of elliptic equations introduced in the context of traffic congestion. They have the form del . (del F(del u)) = f where F is a convex function which vanishes inside some convex set and is elliptic outside. Under some natural assumptions on F and f, we prove that the function del F(del u) is continuous in any dimension, extending a previous result valid only in dimension 2 (Santambrogio and Vespri, 2010 [14]).