On the distance between homotopy classes in $W^{1/p,p}({\protect \mathbb{S}}^1;{\protect \mathbb{S}}^1)$

Itai Shafrir

doi:10.5802/cml.48

On the distance between homotopy classes in

W^{1 / p, p} (𝕊^{1}; 𝕊^{1})

Itai Shafrir ¹

¹ Department of Mathematics, Technion - I.I.T., 32 000 Haifa, Israel

Confluentes Mathematici, Tome 10 (2018) no. 1, pp. 125-136

Résumé

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

W^{1 / p, p} (𝕊^{1}; 𝕊^{1}) = ⋃_{d \in ℤ} ℰ_{d} .

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

{Dist}_{W^{1 / p, p}} (ℰ_{d_{1}}, ℰ_{d_{2}}) : = sup_{f \in ℰ_{d_{1}}} inf_{g \in ℰ_{d_{2}}} d_{W^{1 / p, p}} (f, g)

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: ${Dist}_{W^{1 / 2, 2}} (ℰ_{d_{1}}, ℰ_{d_{2}}) = 2 π {| d_{2} - d_{1} |}^{1 / 2}$ .

Reçu le : 2017-09-06
Révisé le : 2017-12-24
Accepté le : 2017-12-27
Publié le : 2018-09-09

MR

DOI : 10.5802/cml.48

Classification : 46E35
Keywords: ${\protect \mathbb{S}}^1$-valued maps, Fractional Sobolev spaces

Affiliations des auteurs :

Itai Shafrir ¹

¹ Department of Mathematics, Technion - I.I.T., 32 000 Haifa, Israel

Licence :

CC-BY-NC-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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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, Tome 10 (2018) no. 1, pp. 125-136. doi: 10.5802/cml.48

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