The distributional hyper-Jacobian determinants in fractional Sobolev spaces

2018/08/22 by Tu, Qiang, Wu, Chuanxi, Qiu, Xueting
#42B35 #46E35 #46F10 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1808.07190

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 mth-Jacobian minors of degree r are weak continuous in fractional Sobolev spaces W^m-(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 mth-Jacobian minors of degree r are well defined in W^s,p if and only if W^s,p ⊆ W^m-(m)/(r),m in case m=1,2.

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