Fractional Sobolev Regularity for the Brouwer Degree

Abstract

We prove that if $\Omega \subset {\mathbb{R}}^n$ is a bounded open set and $n\alpha >{\dim }_b(\partial \Omega )=d$, then the Brouwer degree deg$(v,\Omega ,\cdot )$ of any Hölder 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 $0\le \beta <\frac{n}{p}-\frac{d}{\alpha }$. 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 $\beta \ge 0$ and $p\ge 1$ 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 $\mbox{deg}\, (v, Ω, \cdot)\notin W^{β, p}$, where $B_1\subset {\mathbb{R}}^n$ is the unit ball.

arXiv comment12 pages, 1 figure