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 $\varOmega\subset\mathbb{ R}^{ n}$.
Categories: math.AP