Geodesic
Phenomena of compensation and estimates for partial differential equations
Documenta mathematica, ICM Berlin 1998, Vol. III (1998), pp. 21-30.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Quantities like the Jacobian determinant of a mapping play an important role in several partial differential equations in Physics and Geometry. The algebraic structure of such nonlinearities allow to improve slightly the integrability or the regularity of these quantities, sometimes in a crucial way. Focused on the instance of $${\partial a\over \partial x}{\partial b\over \partial y}- {\partial a\over \partial y}{\partial b\over \partial x}, \qquad {where $a$ and }b\in H^1(\bbfR^2),$$ we review some results obtained on that quantity for 30 years and applications to partial differential equations arising in Geometry, in particular concerning the conformal parametrisations of constant mean curvature surfaces and the harmonic mappings between Riemannian manifolds.
Classification : 35J60, 58E20, 53A10, 49Q05
Keywords: compensation phenomena, harmonic maps, Jacobian determinant, regularity, conformal parametrisations 
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     author = {H\'elein, Fr\'ed\'eric},
     title = {Phenomena of compensation and estimates for partial differential equations},
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     volume = {ICM Berlin 1998, Vol. III},
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Hélein, Frédéric. Phenomena of compensation and estimates for partial differential equations. Documenta mathematica, ICM Berlin 1998, Vol. III (1998), pp. 21-30. http://geodesic.mathdoc.fr/item/DOCMA_1998__S9__a76/