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, Volume 10 (2018) no. 1, pp. 125-136.

Abstract

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

Received: 2017-09-06
Revised: 2017-12-24
Accepted: 2017-12-27
Published online: 2018-09-09

MR

DOI: 10.5802/cml.48

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

Author's affiliations:

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

@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},
     pages = {125--136},
     publisher = {Institut Camille Jordan},
     volume = {10},
     number = {1},
     year = {2018},
     doi = {10.5802/cml.48},
     mrnumber = {3869013},
     language = {en},
     url = {https://cml.centre-mersenne.org/articles/10.5802/cml.48/}
}

TY  - JOUR
AU  - Itai Shafrir
TI  - On the distance between homotopy classes in $W^{1/p,p}({\protect \mathbb{S}}^1;{\protect \mathbb{S}}^1)$
JO  - Confluentes Mathematici
PY  - 2018
SP  - 125
EP  - 136
VL  - 10
IS  - 1
PB  - Institut Camille Jordan
UR  - https://cml.centre-mersenne.org/articles/10.5802/cml.48/
DO  - 10.5802/cml.48
LA  - en
ID  - CML_2018__10_1_125_0
ER  -

%0 Journal Article
%A Itai Shafrir
%T On the distance between homotopy classes in $W^{1/p,p}({\protect \mathbb{S}}^1;{\protect \mathbb{S}}^1)$
%J Confluentes Mathematici
%D 2018
%P 125-136
%V 10
%N 1
%I Institut Camille Jordan
%U https://cml.centre-mersenne.org/articles/10.5802/cml.48/
%R 10.5802/cml.48
%G en
%F CML_2018__10_1_125_0

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/

References
Cited by

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

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

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

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

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

[6] Haim Brezis; Petru Mironescu; Itai Shafrir Distances between classes in $W^{1, 1} (Ω; 𝕊^{1})$ , Calc. Var. Partial Differential Equations, Volume 57 (2018) no. 1, Art. 14, 32 p pages | MR | Zbl

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

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

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

Cited by Sources: