# CONFLUENTES MATHEMATICI

On the distance between homotopy classes in ${W}^{1/p,p}\left({𝕊}^{1};{𝕊}^{1}\right)$
Confluentes Mathematici, Tome 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: ${Dist}_{{W}^{1/2,2}}\left({ℰ}_{{d}_{1}},{ℰ}_{{d}_{2}}\right)=2\pi {|{d}_{2}-{d}_{1}|}^{1/2}$.

Reçu le : 2017-09-07
Révisé le : 2017-12-25
Accepté le : 2017-12-28
Publié le : 2018-09-10
DOI : https://doi.org/10.5802/cml.48
Classification : 46E35
Mots clés: ${𝕊}^{1}$-valued maps, Fractional Sobolev spaces
@article{CML_2018__10_1_125_0,
author = {Itai Shafrir},
title = {On the distance between homotopy classes in $W^{1/p,p}({\protect \mathbb{S}}^1;{\protect \mathbb{S}}^1)$},
journal = {Confluentes Mathematici},
publisher = {Institut Camille Jordan},
volume = {10},
number = {1},
year = {2018},
pages = {125-136},
doi = {10.5802/cml.48},
language = {en},
url = {cml.centre-mersenne.org/item/CML_2018__10_1_125_0/}
}
Shafrir, Itai. On the distance between homotopy classes in $W^{1/p,p}({\protect \mathbb{S}}^1;{\protect \mathbb{S}}^1)$. Confluentes Mathematici, Tome 10 (2018) no. 1, pp. 125-136. doi : 10.5802/cml.48. https://cml.centre-mersenne.org/item/CML_2018__10_1_125_0/

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