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Itai Shafrir
On the distance between homotopy classes in $W^{1/p,p}({\protect \mathbb{S}}^1;{\protect \mathbb{S}}^1)$
Confluentes Mathematici, 10 no. 1 (2018), p. 125-136
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Class. Math.: 46E35
Keywords: ${\protect \mathbb{S}}^1$-valued maps, Fractional Sobolev spaces

Résumé - Abstract

For every $p\in (1,\infty )$ there is a natural notion of topological degree for maps in $W^{1/p,p}({\mathbb{S}}^1;{\mathbb{S}}^1)$ which allows us to write that space as a disjoint union of classes,

$$W^{1/p,p}({\mathbb{S}}^1;{\mathbb{S}}^1)=\bigcup _{d\in {\mathbb{Z}}}\mathcal{E}_d.$$

For every pair $d_1,d_2\in {\mathbb{Z}}$, we show that the distance

$$\operatorname{Dist}_{W^{1/p,p}}({\mathcal{E}}_{d_1}, {\mathcal{E}}_{d_2}):=\sup _{f\in {\mathcal{E}}_{d_1}}\ \inf _{g\in {\mathcal{E}}_{d_2}}\ d_{W^{1/p,p}}(f, g)$$

equals the minimal $W^{1/p,p}$-energy in $\mathcal{E}_{d_1-d_2}$. In the special case $p=2$ we deduce from the latter formula an explicit value: $\operatorname{Dist}_{W^{1/2,2}}({\mathcal{E}}_{d_1}, {\mathcal{E}}_{d_2})=2\pi |d_2-d_1|^{1/2}$.


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