Calculus of Variations and Geometric Measure Theory
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C. De Lellis - D. Inauen

Fractional Sobolev regularity for the Brouwer degree

created by delellis on 04 Sep 2017


Accepted Paper

Inserted: 4 sep 2017
Last Updated: 4 sep 2017

Journal: Comm. Partial Differential Equations
Year: 2017


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.


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