# CONFLUENTES MATHEMATICI

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

For every $p\beta \left(1,\mathrm{\beta }\right)$ there is a natural notion of topological degree for maps in ${W}^{1/p,p}\left({\mathrm{\pi }}^{1};{\mathrm{\pi }}^{1}\right)$ which allows us to write that space as a disjoint union of classes,

 ${W}^{1/p,p}\left({\mathrm{\pi }}^{1};{\mathrm{\pi }}^{1}\right)=\underset{d\beta \mathrm{\beta €}}{\beta }{\mathrm{\beta °}}_{d}.$

For every pair ${d}_{1},{d}_{2}\beta \mathrm{\beta €}$, we show that the distance

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

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

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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