Distances between classes of sphere-valued Sobolev maps

Haïm Brézis; Petru Mironescu; Itai Shafrir

doi:10.1016/j.crma.2016.05.001

Mathematical analysis

Distances between classes of sphere-valued Sobolev maps
[Distances entre classes d'applications de Sobolev à valeurs dans une sphère]

Présenté par : Haïm Brézis

Haïm Brézis ^1,² ; Petru Mironescu ³ ; Itai Shafrir ²

¹ Department of Mathematics, Rutgers University, USA
² Department of Mathematics, Technion – I.I.T., 32 000 Haifa, Israel
³ Université de Lyon, Université Lyon-1, CNRS UMR 5208, Institut Camille-Jordan, 43, bd. du-11-Novembre-1918, 69622 Villeurbanne cedex, France

Comptes Rendus. Mathématique, Volume 354 (2016) no. 7, pp. 677-684.

Résumé
Abstract

On introduit une relation d'équivalence sur $W^{s, p} (S^{N}; S^{N})$ liée au degré topologique, et on présente des estimées pour les distances (au sens usuel et au sens de Hausdorff) entre les classes d'équivalence. Dans certains cas particuliers, il s'agit même de formules exactes. On considère ensuite des questions semblables pour $W^{1, p} (Ω; S^{1})$ .

We introduce an equivalence relation on $W^{s, p} (S^{N}; S^{N})$ involving the topological degree, and we evaluate the distances (in the usual sense and in the Hausdorff sense) between the equivalence classes. In some special cases, we even obtain exact formulas. Next we discuss related issues for $W^{1, p} (Ω; S^{1})$ .

Reçu le : 2016-04-23
Accepté le : 2016-04-24
Publié le : 2016-05-18

DOI : 10.1016/j.crma.2016.05.001

Affiliations des auteurs :

Haïm Brézis ^{1, 2} ; Petru Mironescu ³ ; Itai Shafrir ²

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     author = {Ha{\"\i}m Br\'ezis and Petru Mironescu and Itai Shafrir},
     title = {Distances between classes of sphere-valued {Sobolev} maps},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {677--684},
     publisher = {Elsevier},
     volume = {354},
     number = {7},
     year = {2016},
     doi = {10.1016/j.crma.2016.05.001},
     language = {en},
}

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Haïm Brézis; Petru Mironescu; Itai Shafrir. Distances between classes of sphere-valued Sobolev maps. Comptes Rendus. Mathématique, Volume 354 (2016) no. 7, pp. 677-684. doi : 10.1016/j.crma.2016.05.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.05.001/

Bibliographie
Cité par

[1] F. Bethuel; X.M. Zheng Density of smooth functions between two manifolds in Sobolev spaces, J. Funct. Anal., Volume 80 (1988), pp. 60-75

[2] H. Brézis, P. Mironescu, Sobolev maps with values into the circle, Birkhäuser, in preparation.

[3] H. Brézis; P. Mironescu; A. Ponce $W^{1, 1}$ -maps with values into $S^{1}$ , Geometric Analysis of PDE and Several Complex Variables, Contemp. Math., vol. 368, American Mathematical Society, Providence, RI, USA, 2005, pp. 69-100

[4] H. Brézis, P. Mironescu, I. Shafrir, Distances between classes in $W^{1, 1} (Ω; S^{1})$ , in preparation.

[5] H. Brézis; P. Mironescu; I. Shafrir Distances between homotopy classes of $W^{s, p} (S^{N}; S^{N})$ , ESAIM Control Optim. Calc. Var. (2016) (in press, hal-01257581)

[6] H. Brézis; L. Nirenberg Degree theory and BMO. I. Compact manifolds without boundaries, Sel. Math. New Ser., Volume 1 (1995), pp. 197-263

[7] S. Levi; I. Shafrir On the distance between homotopy classes of maps between spheres, J. Fixed Point Theory Appl., Volume 15 (2014), pp. 501-518

[8] J. Rubinstein; I. Shafrir The distance between homotopy classes of $S^{1}$ -valued maps in multiply connected domains, Isr. J. Math., Volume 160 (2007), pp. 41-59

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