  The original proof of Dacorogna-Moser theorem on the prescribed Jacobian PDE,
$\text{det}\,\nabla\varphi=f$, can be modified in order to obtain control of
support of the solutions from that of the initial data, while keeping optimal
regularity. Briefly, under the usual conditions, a solution diffeomorphism
$\varphi$ satisfying \[
\text{supp}(f-1)\subset\varOmega\Longrightarrow\text{supp}(\varphi-\text{id})\subset\varOmega
\] can be found and $\varphi$ is still of class $C^{r+1,\alpha}$ if $f$ is
$C^{r,\alpha}$, the domain of $f$ being a bounded connected open
$C^{r+2,\alpha}$ set $\varOmega\subset\mathbb{R}^{n}$.
