# $W^{s,\frac{n}{s}}$-maps with positive distributional Jacobians

## TL;DR

This paper extends the continuity and degree estimate results for maps with positive Jacobians from classical Sobolev spaces to fractional Sobolev spaces, broadening the understanding of their regularity and topological properties.

## Contribution

It proves that maps in fractional Sobolev spaces with positive distributional Jacobians are continuous, generalizing classical results to fractional settings.

## Key findings

- Maps in $W^{s,rac{n}{s}}$ with positive Jacobian are continuous for $s \\geq \frac{n}{n+1}$.
- Degree estimates for sense-preserving maps extend to fractional Sobolev spaces.
- Positive Jacobian implies regularity properties in fractional Sobolev spaces.

## Abstract

We extend the well-known result that any $f \in W^{1,n}(\Omega,\mathbb{R}^n)$, $\Omega \subset \mathbb{R}^n$ with strictly positive Jacobian is actually continuous: it is also true for fractional Sobolev spaces $W^{s,\frac{n}{s}}(\Omega)$ for any $s \geq \frac{n}{n+1}$, where the sign condition on the Jacobian is understood in a distributional sense. Along the way we also obtain extensions to fractional Sobolev spaces $W^{s,\frac{n}{s}}$ of the degree estimates known for $W^{1,n}$-maps with positive or non-negative Jacobian, such as the sense-preserving property.