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Journal Articles Journal d'analyse mathématique Year : 2014

Prescribing the Jacobian in critical spaces

Abstract

We consider the Sobolev space $X=W^{s,p}({\mathbb S}^m ; {\mathbb S}^{k-1})$. We prove the existence of a robust distributional Jacobian $Ju$ for $u\in X$ provided $sp\ge k-1$. This generalizes a result of Bourgain, Brezis and the second author (Comm. Pure Appl. Math. 2005), where the case $m=k$ is considered. In the critical case where $sp=k-1$, we identify the image of the map $X\ni u\mapsto Ju$. This extends a result of Alberti, Baldo and Orlandi (J. Eur. Math. Soc. 2003) for $s=1$ and $p=k-1$. We also present a new, analytical, dipole construction method.
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Dates and versions

hal-00747444 , version 1 (31-10-2012)

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  • HAL Id : hal-00747444 , version 1

Cite

Pierre Bousquet, Petru Mironescu. Prescribing the Jacobian in critical spaces. Journal d'analyse mathématique, 2014, 122, pp.317--373. ⟨hal-00747444⟩
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