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

Abstract

The original proof of Dacorogna-Moser theorem on the prescribed Jacobian PDE, $\text{det}\nabla \phi =f$, 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 $\phi$ satisfying \[ \text{supp}(f-1)\subset\varOmega\Longrightarrow\text{supp}(φ-\text{id})\subset\varOmega \] can be found and $\phi$ 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 $\Omega \subset {\mathbb{R}}^n$.

there is an Addendum - arXiv:1705.01416, generalizing the result to the case of the pullback between any two (nondegenerate) prescribed volume forms that coincide near the boundary and have the same total volume over a domain

Cited by in corpus (1)

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