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

TL;DR

This paper modifies the proof of the Dacorogna-Moser theorem to ensure the support of solutions to the Jacobian determinant PDE is controlled by the support of the initial data, while maintaining optimal regularity.

Contribution

It provides a modified proof of the Dacorogna-Moser theorem that guarantees support control of solutions based on initial data support, preserving regularity.

Findings

Support of solutions can be controlled by initial data support.

Solutions retain optimal regularity under the modified proof.

The method applies to bounded domains with smooth boundaries.

Abstract

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