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arXiv · 1808.07190

The distributional hyper-Jacobian determinants in fractional Sobolev spaces

Qiang Tu·Chuanxi Wu·Xueting Qiu
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Abstract

In this paper we give a positive answer to a question raised by Baer-Jerison in connection with hyper-Jacobian determinants and associated minors in fractional Sobolev spaces. Inspired by recent works of Brezis-Nguyen and Baer-Jerison on the Jacobian and Hessian determinants, we show that the distributional $m$th-Jacobian minors of degree $r$ are weak continuous in fractional Sobolev spaces $W^{m-\frac{m}{r},r}$, and the result is optimal, satisfying the necessary conditions, in the frame work of fractional Sobolev spaces. In particular, the conditions can be removed in case $m=1,2$, i.e., the $m$th-Jacobian minors of degree $r$ are well defined in $W^{s,p}$ if and only if $W^{s,p} \subseteq W^{m-\frac{m}{r},m}$ in case $m=1,2$.

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Qiang Tu, Chuanxi Wu, Xueting Qiu. 2018-08-22. The distributional hyper-Jacobian determinants in fractional Sobolev spaces. https://arxiv.org/abs/1808.07190

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