On the distance between homotopy classes in W 1/p,p (π•Š 1 ;π•Š 1 )
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 .

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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/

[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, BirkhΓ€user 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 (π•Š N ;π•Š 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 (Ξ©;π•Š 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

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