With cedram.org Confluentes Mathematici Search for an article Search within the site English français Table of contents for this issue | Previous article Itai ShafrirOn the distance between homotopy classes in $W^{1/p,p}({\protect \mathbb{S}}^1;{\protect \mathbb{S}}^1)$Confluentes Mathematici, 10 no. 1 (2018), p. 125-136, doi: 10.5802/cml.48 Article PDF Class. Math.: 46E35Keywords: ${\protect \mathbb{S}}^1$-valued maps, Fractional Sobolev spaces Résumé - AbstractFor every $p\in (1,\infty )$ there is a natural notion of topological degree for maps in $W^{1/p,p}({\mathbb{S}}^1;{\mathbb{S}}^1)$ which allows us to write that space as a disjoint union of classes, $$W^{1/p,p}({\mathbb{S}}^1;{\mathbb{S}}^1)=\bigcup _{d\in {\mathbb{Z}}}\mathcal{E}_d.$$ For every pair $d_1,d_2\in {\mathbb{Z}}$, we show that the distance $$\operatorname{Dist}_{W^{1/p,p}}({\mathcal{E}}_{d_1}, {\mathcal{E}}_{d_2}):=\sup _{f\in {\mathcal{E}}_{d_1}}\ \inf _{g\in {\mathcal{E}}_{d_2}}\ d_{W^{1/p,p}}(f, g)$$ equals the minimal $W^{1/p,p}$-energy in $\mathcal{E}_{d_1-d_2}$. In the special case $p=2$ we deduce from the latter formula an explicit value: $\operatorname{Dist}_{W^{1/2,2}}({\mathcal{E}}_{d_1}, {\mathcal{E}}_{d_2})=2\pi |d_2-d_1|^{1/2}$. Bibliography[1] Jean Bourgain, Haim Brezis & Petru Mironescu, “Lifting, degree, and distributional Jacobian revisited”, Comm. Pure Appl. Math. 58 (2005) no. 4, p. 529-551 [2] Jean Bourgain, Haim Brezis & Hoai-Minh Nguyen, “A new estimate for the topological degree”, C. R. Math. Acad. Sci. Paris 340 (2005) no. 11, p. 787-791 [3] Anne Boutet de Monvel-Berthier, Vladimir Georgescu & Radu Purice, “A boundary value problem related to the Ginzburg-Landau model”, Comm. Math. Phys. 142 (1991) no. 1, p. 1-23 [4] Haim Brezis, New questions related to the topological degree, The unity of mathematics, Progr. Math. 244, Birkhäuser Boston, Boston, MA, 2006, p. 137–154 [5] Haim Brezis, Petru Mironescu & Itai Shafrir, “Distances between homotopy classes of $W^{s,p}(\mathbb{S}^N;\mathbb{S}^N)$”, ESAIM Control Optim. Calc. Var. 22 (2016) no. 4, p. 1204-1235 [6] Haim Brezis, Petru Mironescu & Itai Shafrir, “Distances between classes in $W^{1,1}(\Omega ;{\mathbb{S}}^1)$”, Calc. Var. Partial Differential Equations 57 (2018) no. 1 [7] Haim Brezis & Louis Nirenberg, “Degree theory and BMO. I. Compact manifolds without boundaries”, Selecta Math. (N.S.) 1 (1995) no. 2, p. 197-263 [8] Petru Mironescu, “Profile decomposition and phase control for circle-valued maps in one dimension”, C. R. Math. Acad. Sci. Paris 353 (2015) no. 12, p. 1087-1092 [9] Hoai-Minh Nguyen, “Optimal constant in a new estimate for the degree”, J. Anal. Math. 101 (2007), p. 367-395 © 2012Confluentes Mathematici Published by the Institut Camille Jordan CNRS - UMR 5208 and the Unité de Mathématiques Pures et Appliquées CNRS - UMR 5669 of Université de Lyon eISSN : 1793-7434