Dacorogna-Moser theorem on the Jacobian determinant equation with control of support

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 $\varOmega\subset\mathbb{R}^{n}$.