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On the distance between homotopy classes in W 1/p,p (π•Š 1 ;π•Š 1 )
Itai Shafrir1
1 Department of Mathematics, Technion - I.I.T., 32 000 Haifa, Israel
Confluentes Mathematici, Volume 10 (2018) no. 1, pp. 125-136.

For every p∈(1,∞) there is a natural notion of topological degree for maps in W 1/p,p (π•Š 1 ;π•Š 1 ) which allows us to write that space as a disjoint union of classes,

W1/p,p(π•Š1;π•Š1)=⋃dβˆˆβ„€β„°d.

For every pair d 1 ,d 2 βˆˆβ„€, we show that the distance

DistW1/p,p(β„°d1,β„°d2):=supfβˆˆβ„°d1infgβˆˆβ„°d2dW1/p,p(f,g)

equals the minimal W 1/p,p -energy in β„° d 1 -d 2 . In the special case p=2 we deduce from the latter formula an explicit value: Dist W 1/2,2 (β„° d 1 ,β„° d 2 )=2Ο€|d 2 -d 1 | 1/2 .

Received:
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
License: CC-BY-NC-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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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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