Multiple Jacobian equations

Bernard Dacorogna; Olivier Kneuss

doi:10.3934/cpaa.2014.13.1779

Article Contents

2014, Volume 13, Issue 5: 1779-1787. Doi: 10.3934/cpaa.2014.13.1779

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Multiple Jacobian equations

Bernard Dacorogna^1, and
Olivier Kneuss^2,

1.
Section de Mathématiques, Station 8, EPFL, 1015 Lausanne
2.
Department of Mathematics, UC Berkeley, Berkeley, CA, 94720

Received: June 30, 2013

Revised: October 31, 2013

Published: August 2015

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Abstract

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.

Keywords:

Mathematics Subject Classification: 35F50.

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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.