# Fractional Sobolev Regularity for the Brouwer Degree

## TL;DR

This paper establishes fractional Sobolev regularity results for the Brouwer degree of Hölder continuous functions on bounded domains, extending previous work and proving the optimality of the exponent ranges.

## Contribution

It extends Olbermann's summability results for the Brouwer degree to fractional Sobolev spaces and provides a more elementary proof, also demonstrating the optimality of the exponent ranges.

## Key findings

- Brouwer degree belongs to W^{eta,p} under certain conditions
- Extension of Olbermann's summability result
- Optimality of the exponent range is proven

## Abstract

We prove that if $\Omega\subset \mathbb R^n$ is a bounded open set and $n\alpha> {\rm dim}_b (\partial \Omega) = d$, then the Brouwer degree deg$(v,\Omega,\cdot)$ of any H\"older function $v\in C^{0,\alpha}\left (\Omega, \mathbb R^{n}\right)$ belongs to the Sobolev space $W^{\beta, p} (\mathbb R^n)$ for every $0\leq \beta < \frac{n}{p} - \frac{d}{\alpha}$. This extends a summability result of Olbermann and in fact we get, as a byproduct, a more elementary proof of it. Moreover we show the optimality of the range of exponents in the following sense: for every $\beta\geq 0$ and $p\geq 1$ with $\beta > \frac{n}{p} - \frac{n-1}{\alpha}$ there is a vector field $v\in C^{0, \alpha} (B_1, \mathbb R^n)$ with $\mbox{deg}\, (v, \Omega, \cdot)\notin W^{\beta, p}$, where $B_1 \subset \mathbb R^n$ is the unit ball.