September  2014, 13(5): 1779-1787. doi: 10.3934/cpaa.2014.13.1779

Multiple Jacobian equations

1. 

Section de Mathématiques, Station 8, EPFL, 1015 Lausanne

2. 

Department of Mathematics, UC Berkeley, Berkeley, CA, 94720, United States

Received  July 2013 Revised  November 2013 Published  June 2014

The existence, regularity and uniqueness of a local diffeomorphism $\varphi$ satisfying \begin{eqnarray} g_{i}(\varphi) \det\nabla\varphi=f_{i}\quad for\ every\ 1\leq i\leq n \end{eqnarray} is discussed.
Citation: Bernard Dacorogna, Olivier Kneuss. Multiple Jacobian equations. Communications on Pure and Applied Analysis, 2014, 13 (5) : 1779-1787. doi: 10.3934/cpaa.2014.13.1779
References:
[1]

G. Csató, B. Dacorogna and O. Kneuss, The Pullback Equation for Differential Forms, Birkhäuser/Springer, New York, 2012. doi: 10.1007/978-0-8176-8313-9.

[2]

B. Dacorogna and N. Fusco, Semi-continuité des fonctionnelles avec contraintes du type $\det\nabla u>0$, Boll. Un. Mat. Ital., 4-B (1985), 179-189.

[3]

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.

[4]

J. Moser J, On the volume elements on a manifold, Trans. Amer. Math. Soc., 120 (1965), 286-294.

show all references

References:
[1]

G. Csató, B. Dacorogna and O. Kneuss, The Pullback Equation for Differential Forms, Birkhäuser/Springer, New York, 2012. doi: 10.1007/978-0-8176-8313-9.

[2]

B. Dacorogna and N. Fusco, Semi-continuité des fonctionnelles avec contraintes du type $\det\nabla u>0$, Boll. Un. Mat. Ital., 4-B (1985), 179-189.

[3]

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.

[4]

J. Moser J, On the volume elements on a manifold, Trans. Amer. Math. Soc., 120 (1965), 286-294.

[1]

Neil S. Trudinger. On the local theory of prescribed Jacobian equations. Discrete and Continuous Dynamical Systems, 2014, 34 (4) : 1663-1681. doi: 10.3934/dcds.2014.34.1663

[2]

Angelo Favini, Rabah Labbas, Keddour Lemrabet, Stéphane Maingot, Hassan D. Sidibé. Resolution and optimal regularity for a biharmonic equation with impedance boundary conditions and some generalizations. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 4991-5014. doi: 10.3934/dcds.2013.33.4991

[3]

Niklas Hartung. Efficient resolution of metastatic tumor growth models by reformulation into integral equations. Discrete and Continuous Dynamical Systems - B, 2015, 20 (2) : 445-467. doi: 10.3934/dcdsb.2015.20.445

[4]

Louis Tebou. Simultaneous controllability of some uncoupled semilinear wave equations. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3721-3743. doi: 10.3934/dcds.2015.35.3721

[5]

Cătălin-George Lefter, Elena-Alexandra Melnig. Feedback stabilization with one simultaneous control for systems of parabolic equations. Mathematical Control and Related Fields, 2018, 8 (3&4) : 777-787. doi: 10.3934/mcrf.2018034

[6]

Daniel Guo, John Drake. A global semi-Lagrangian spectral model of shallow water equations with time-dependent variable resolution. Conference Publications, 2005, 2005 (Special) : 355-364. doi: 10.3934/proc.2005.2005.355

[7]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems and Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056

[8]

Ai-Li Yang, Yu-Jiang Wu. Newton-MHSS methods for solving systems of nonlinear equations with complex symmetric Jacobian matrices. Numerical Algebra, Control and Optimization, 2012, 2 (4) : 839-853. doi: 10.3934/naco.2012.2.839

[9]

Guji Tian, Xu-Jia Wang. Partial regularity for elliptic equations. Discrete and Continuous Dynamical Systems, 2010, 28 (3) : 899-913. doi: 10.3934/dcds.2010.28.899

[10]

Isabelle Déchène. On the security of generalized Jacobian cryptosystems. Advances in Mathematics of Communications, 2007, 1 (4) : 413-426. doi: 10.3934/amc.2007.1.413

[11]

Lan Qiao, Sining Zheng. Non-simultaneous blow-up for heat equations with positive-negative sources and coupled boundary flux. Communications on Pure and Applied Analysis, 2007, 6 (4) : 1113-1129. doi: 10.3934/cpaa.2007.6.1113

[12]

Huaiyu Jian, Xiaolin Liu, Hongjie Ju. The regularity for a class of singular differential equations. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1307-1319. doi: 10.3934/cpaa.2013.12.1307

[13]

Wenxiong Chen, Congming Li. Regularity of solutions for a system of integral equations. Communications on Pure and Applied Analysis, 2005, 4 (1) : 1-8. doi: 10.3934/cpaa.2005.4.1

[14]

Martí Prats. Beltrami equations in the plane and Sobolev regularity. Communications on Pure and Applied Analysis, 2018, 17 (2) : 319-332. doi: 10.3934/cpaa.2018018

[15]

Zsolt Páles, Vera Zeidan. $V$-Jacobian and $V$-co-Jacobian for Lipschitzian maps. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 623-646. doi: 10.3934/dcds.2011.29.623

[16]

Andrew D. Lewis, David R. Tyner. Geometric Jacobian linearization and LQR theory. Journal of Geometric Mechanics, 2010, 2 (4) : 397-440. doi: 10.3934/jgm.2010.2.397

[17]

Ronen Peretz, Nguyen Van Chau, L. Andrew Campbell, Carlos Gutierrez. Iterated images and the plane Jacobian conjecture. Discrete and Continuous Dynamical Systems, 2006, 16 (2) : 455-461. doi: 10.3934/dcds.2006.16.455

[18]

Jian-Feng Cai, Raymond H. Chan, Zuowei Shen. Simultaneous cartoon and texture inpainting. Inverse Problems and Imaging, 2010, 4 (3) : 379-395. doi: 10.3934/ipi.2010.4.379

[19]

Huilian Jia, Lihe Wang, Fengping Yao, Shulin Zhou. Regularity theory in Orlicz spaces for the poisson and heat equations. Communications on Pure and Applied Analysis, 2008, 7 (2) : 407-416. doi: 10.3934/cpaa.2008.7.407

[20]

Luis Barreira, Claudia Valls. Stable manifolds with optimal regularity for difference equations. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1537-1555. doi: 10.3934/dcds.2012.32.1537

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (94)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]