Sull'equazione det Du = f senza ipotesi di segno

We consider the nonlinear problem det Du=f in a bounded smooth domain in R^n, with boundary condition u(x) =x where f is in C^{k}(O), kge 1, satisfying a suitable compatibility condition. The positivity of f is a standard assumption in the literature. In a recent joint paper with B.Dacorogna and O.Kneuss (EPFL) we prove the existence of a solution u in C^k(overline{O};R^n) with no assumptions on the sign of f. Here we state this theorem together with some related results and we outline the main features of the problem