Sull’equazione det Du = f senza ipotesi di segno

We consider the nonlinear problemdet?u (x) = f (x) x ? u (x) = x x ? ?where k ? 1 is an integer, is a bounded smooth domain in Rn and f ? Ck ???? satisfiesZf (x) dx = meas.The positivity of f is a standard assumption in the literature. In a recent joint paper withB.Dacorogna and O.Kneuss (EPFL) we prove the existence of a solution u ? Ck ????;Rn with no assumptions on the sign of f. Here we state this theorem together with somerelated results and we outline the main features of the problem