# 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/

[1] Jean Bourgain; Haim Brezis; Petru Mironescu Lifting, degree, and distributional Jacobian revisited, Comm. Pure Appl. Math., Tome 58 (2005) no. 4, pp. 529-551 | Article | MR 2119868

[2] Jean Bourgain; Haim Brezis; Hoai-Minh Nguyen A new estimate for the topological degree, C. R. Math. Acad. Sci. Paris, Tome 340 (2005) no. 11, pp. 787-791 | Article | MR 2139888

[3] Anne Boutet de Monvel-Berthier; Vladimir Georgescu; Radu Purice A boundary value problem related to the Ginzburg-Landau model, Comm. Math. Phys., Tome 142 (1991) no. 1, pp. 1-23 http://projecteuclid.org/euclid.cmp/1104248488 | MR 1137773

[4] Haim Brezis New questions related to the topological degree, The unity of mathematics (Progr. Math.) Tome 244, Birkhäuser Boston, Boston, MA, 2006, pp. 137-154 | Article | MR 2181804

[5] Haim Brezis; Petru Mironescu; Itai Shafrir Distances between homotopy classes of ${W}^{s,p}\left({𝕊}^{N};{𝕊}^{N}\right)$, ESAIM Control Optim. Calc. Var., Tome 22 (2016) no. 4, pp. 1204-1235 | Article | MR 3570500

[6] Haim Brezis; Petru Mironescu; Itai Shafrir Distances between classes in ${W}^{1,1}\left(\Omega ;{𝕊}^{1}\right)$, Calc. Var. Partial Differential Equations, Tome 57 (2018) no. 1, Art. 14, 32 p pages

[7] Haim Brezis; Louis Nirenberg Degree theory and BMO. I. Compact manifolds without boundaries, Selecta Math. (N.S.), Tome 1 (1995) no. 2, pp. 197-263 | Article | MR 1354598

[8] Petru Mironescu Profile decomposition and phase control for circle-valued maps in one dimension, C. R. Math. Acad. Sci. Paris, Tome 353 (2015) no. 12, pp. 1087-1092 | Article | MR 3427913

[9] Hoai-Minh Nguyen Optimal constant in a new estimate for the degree, J. Anal. Math., Tome 101 (2007), pp. 367-395 | Article | MR 2346551