Cartesian currents in fractional Sobolev spaces and applications to functions of bounded higher variation

TL;DR

This paper extends the theory of distribution Jacobians and functions of bounded higher variation to fractional Sobolev spaces, establishing weak continuity, structure properties, and coarea formulas for these generalized functions.

Contribution

It introduces weak continuity results for distribution Jacobian minors in fractional Sobolev spaces and extends the concept of functions of bounded higher variation to these spaces.

Findings

Weak continuity of distribution Jacobian minors in fractional Sobolev spaces

Introduction of currents associated with graphs of fractional Sobolev maps

Extension of functions of bounded higher variation with new coarea and chain rule results

Abstract

In this paper we establish weak continuity results for the distribution Jacobian minors in fractional sobolev spaces, which can be seen as a extension of recent work of Brezis and Nguyen on the distributional Jacobian determinant. Then we apply results to introduce the currents associated with graphs of maps in fractional Sobolev spaces and study some relevant properties such as structure properties, weak continuity and so on. As another application, we extend the definition of functions of bounded higher variation, which defined by Jerrard and Soner in $W^{1,N-1}\cap L^{\infty}(\Omega,\mathbb{R}^N)$ ( $\mbox{dim}\Omega\geq N$), to $W^{1-\frac{1}{N},N}(\Omega,\mathbb{R}^N)$ and give some results including weak coarea formula, strong coarea formula and chain rule.