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July  2017, 37(7): 4071-4089. doi: 10.3934/dcds.2017173

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

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

Received  August 2016 Revised  March 2017 Published  April 2017

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

Citation: Pedro Teixeira. Dacorogna-Moser theorem on the Jacobian determinant equation with control of support. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 4071-4089. doi: 10.3934/dcds.2017173
References:
[1]

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.

[2]

A. Avila, On the regularization of conservative maps, Acta Math., 205 (2010), 5-18. doi: 10.1007/s11511-010-0050-y.

[3]

E. Bierstone, Differentiable functions, Bol.Soc.Brasil, 11 (1980), 139-189. doi: 10.1007/BF02584636.

[4]

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.

[5]

B. Dacorogna, Direct Methods in the Calculus of Variations, Second edition. Applied Mathematical Sciences, 78. Springer, New York, 2008.

[6]

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.

[7]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001.

[8]

M. Hirsch, Differential Topology, Corrected reprint of the 1976 original edition. Graduate Texts in Mathematics 33. Springer-Verlag, New York, 1994.

[9]

C. Matheus, A remark on the Jacobian determinant PDE, https://matheuscmss.wordpress.com/2013/07/06/a-remark-on-the-jacobian-determinant-pde/

[10]

R. Seeley, Extension of $C^{∞}$ functions defined in a half space, Proc. Amer. Math. Soc., 15 (1964), 625-626. doi: 10.2307/2034761.

[11]

F. Takens, Homoclinic points in conservative systems, Invent. math., 18 (1972), 267-292. doi: 10.1007/BF01389816.

show all references

References:
[1]

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.

[2]

A. Avila, On the regularization of conservative maps, Acta Math., 205 (2010), 5-18. doi: 10.1007/s11511-010-0050-y.

[3]

E. Bierstone, Differentiable functions, Bol.Soc.Brasil, 11 (1980), 139-189. doi: 10.1007/BF02584636.

[4]

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.

[5]

B. Dacorogna, Direct Methods in the Calculus of Variations, Second edition. Applied Mathematical Sciences, 78. Springer, New York, 2008.

[6]

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.

[7]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001.

[8]

M. Hirsch, Differential Topology, Corrected reprint of the 1976 original edition. Graduate Texts in Mathematics 33. Springer-Verlag, New York, 1994.

[9]

C. Matheus, A remark on the Jacobian determinant PDE, https://matheuscmss.wordpress.com/2013/07/06/a-remark-on-the-jacobian-determinant-pde/

[10]

R. Seeley, Extension of $C^{∞}$ functions defined in a half space, Proc. Amer. Math. Soc., 15 (1964), 625-626. doi: 10.2307/2034761.

[11]

F. Takens, Homoclinic points in conservative systems, Invent. math., 18 (1972), 267-292. doi: 10.1007/BF01389816.

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 $\mathit{\Omega }$
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