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# 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 $C^{r+1, α}$ if $f$ is $C^{r, α}$, the domain of $f$ being a bounded connected open $C^{r+2, α}$$set$\varOmega\subset\mathbb{R}^{n}$. Mathematics Subject Classification: Primary: 35F30.  Citation: • 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). Figure 8.1. Extending$\mathit{g} \in {\mathit{C}^1}\left( U \right)\mathit{ }$to the whole$\mathit{\Omega }$•  R. Abraham, J. Marsden and T. Ratiu, Manifolds, Tensor Analysis, and Applications, Global Analysis Pure and Applied: Series B, 2. Addison-Wesley Publishing Co. , Reading Mass. , 1983. A. Avila , On the regularization of conservative maps, Acta Math., 205 (2010) , 5-18. doi: 10.1007/s11511-010-0050-y. E. Bierstone , Differentiable functions, Bol.Soc.Brasil, 11 (1980) , 139-189. doi: 10.1007/BF02584636. G. Csató, B. Dacorogna and O. Kneuss, The Pullback Equation for Differential Forms, Progress in Nonlinear Differential Equations and their Applications, 83. Birkhäuser/Springer, 2012. doi: 10.1007/978-0-8176-8313-9. B. Dacorogna, Direct Methods in the Calculus of Variations, Second edition. Applied Mathematical Sciences, 78. Springer, New York, 2008. B. Dacorogna and J. Moser , On a partial differential equation involving the Jacobian determinant, Ann. Inst. H. Poincaré Anal. Non Linéaire, 7 (1990) , 1-26. doi: 10.1016/S0294-1449(16)30307-9. D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001. M. Hirsch, Differential Topology, Corrected reprint of the 1976 original edition. Graduate Texts in Mathematics 33. Springer-Verlag, New York, 1994. C. Matheus, A remark on the Jacobian determinant PDE, https://matheuscmss.wordpress.com/2013/07/06/a-remark-on-the-jacobian-determinant-pde/ R. Seeley , Extension of$C^{∞}\$ functions defined in a half space, Proc. Amer. Math. Soc., 15 (1964) , 625-626.  doi: 10.2307/2034761. F. Takens , Homoclinic points in conservative systems, Invent. math., 18 (1972) , 267-292.  doi: 10.1007/BF01389816.

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