Sull’equazione det Du = f senza ipotesi di segno

Abstract

We consider the nonlinear problem

det?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 ????
satisfies
Z

f (x) dx = meas
.
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 ? Ck ????
;Rn 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.