We prove existence of
\({u\in C^{k}(\overline{\Omega};\mathbb{R}^{n})}\) satisfying
$\left\{\begin{array}{ll} det\nabla u(x) =f(x) \, x\in \Omega\\ u(x) =x \quad\quad\quad\quad x\in\partial\Omega\end{array}\right.$
where
k ≥ 1 is an integer,
\({\Omega}\) is a bounded smooth domain and
\({f\in C^{k}(\overline{\Omega}) }\) satisfies
$\int\limits_{\Omega}f(x) dx={\rm meas} \Omega$
with no sign hypothesis on
f.