# Coarea formulae and chain rules for the Jacobian determinant in fractional Sobolev spaces

## TL;DR

This paper establishes weak and strong coarea and chain rule formulas for the Jacobian determinant in fractional Sobolev spaces, extending classical results to fractional and H"older functions and connecting to open problems in geometric analysis.

## Contribution

It introduces new coarea and chain rule formulas for distributional Jacobians in fractional Sobolev spaces, broadening the scope of geometric measure theory.

## Key findings

- Weak coarea and chain rule formulas for $Ju$ when $sp>n-1$ and $s> (n-1)/n
- Strong formulas for $Ju$ when $sp extgreater n$ and $s extgreater n/(n+1)
- Chain rule for H"older functions related to open problems in geometric analysis

## Abstract

We prove weak and strong versions of the coarea formula and the chain rule for distributional Jacobian determinants $Ju$ for functions $u$ in fractional Sobolev spaces $W^{s,p}(\Omega)$, where $\Omega$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary. The weak forms of the formulae are proved for the range $sp>n-1$, $s> \frac{n-1}{n}$, while the strong versions are proved for the range $sp\geq n$, $s\geq \frac{n}{n+1}$. We also provide a chain rule for distributional Jacobian determinants of H\"older functions and point out its relation to two open problems in geometric analysis.