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

Pedro Teixeira

doi:10.3934/dcds.2017173

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2017, Volume 37, Issue 7: 4071-4089. Doi: 10.3934/dcds.2017173

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

Pedro Teixeira

Centro de Matemática da Universidade do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal

Received: July 31, 2016

Revised: February 28, 2017

Published: August 2017

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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 $C^{r+1, α}$ if $f$ is $C^{r, α}$, the domain of $f$ being a bounded connected open $C^{r+2, α}$$ set $\varOmega\subset\mathbb{R}^{n}$ .

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Mathematics Subject Classification: Primary: 35F30.

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Figure 6.1. Finding hb t satisfying$\int_\mathit{\Omega } {\left( {f/\widetilde f} \right)} {h_{\widehat t}} = {\rm{meas}}{\mkern 1mu} \;\mathit{\Omega }$. The functions ht are seen in the background (bell shaped).

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Figure 8.1. Extending $\mathit{g} \in {\mathit{C}^1}\left( U \right)\mathit{ }$ to the whole $\mathit{\Omega }$

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