Prescribing the Jacobian in critical spaces

Pierre Bousquet (1) , Petru Mironescu (2)

1 LATP - Laboratoire d'Analyse, Topologie, Probabilités
2 EDPA - Équations aux dérivées partielles, analyse

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.

Mots clés: Jacobian; sphere-valued maps

Domaines: Analyse classique [math.CA]

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