# CONFLUENTES MATHEMATICI

On the distance between homotopy classes in ${W}^{1/p,p}\left({𝕊}^{1};{𝕊}^{1}\right)$
Confluentes Mathematici, Volume 10 (2018) no. 1, pp. 125-136.

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

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

For every pair ${d}_{1},{d}_{2}\in ℤ$, we show that the distance

 ${Dist}_{{W}^{1/p,p}}\left({ℰ}_{{d}_{1}},{ℰ}_{{d}_{2}}\right):=\underset{f\in {ℰ}_{{d}_{1}}}{sup}\phantom{\rule{4pt}{0ex}}\underset{g\in {ℰ}_{{d}_{2}}}{inf}\phantom{\rule{4pt}{0ex}}{d}_{{W}^{1/p,p}}\left(f,g\right)$

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

Revised:
Accepted:
Published online:
DOI: 10.5802/cml.48
Classification: 46E35
Keywords: ${\protect \mathbb{S}}^1$-valued maps, Fractional Sobolev spaces
Itai Shafrir 1

1 Department of Mathematics, Technion - I.I.T., 32 000 Haifa, Israel
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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, Volume 10 (2018) no. 1, pp. 125-136. doi : 10.5802/cml.48. https://cml.centre-mersenne.org/articles/10.5802/cml.48/

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