Partial Differential Equations/Topology
Tangent unit-vector fields: Nonabelian homotopy invariants and the Dirichlet energy
[Champs de vecteurs unités tangents : Invariants homotopiques non abelien et énergie de Dirichlet]
Comptes Rendus. Mathématique, Tome 347 (2009) no. 19-20, pp. 1159-1164.

Soit O un polygone géodésique fermé de S2. On dit qu'une application de O dans S2 vérifie des conditions aux limites tangentes si elle associe à chaque côté de O la géodésique qui le contient. Dans le cas où O est un octant de S2, on calcule l'infimum d'énergie de Dirichlet, E(H), pour des applications tangentes continues d'un type d'homotopie quelconque H. L'expression de E(H) utilise un invariant topologique, la longueur nominale, lié au groupe fondamentel (non abélien) de la sphère S2 à n trous ponctuels, π1(S2{s1,,sn},). Les réeultats obtenus ont des applications pratiques, notamment dans la modélisation des systèmes contenant de cristaux liquides nématiques.

Let O be a closed geodesic polygon in S2. Maps from O into S2 are said to satisfy tangent boundary conditions if the edges of O are mapped into the geodesics which contain them. Taking O to be an octant of S2, we evaluate the infimum Dirichlet energy, E(H), for continuous tangent maps of arbitrary homotopy type H. The expression for E(H) involves a topological invariant – the spelling length – associated with the (nonabelian) fundamental group of the n-times punctured two-sphere, π1(S2{s1,,sn},). These results have applications for the theoretical modelling of nematic liquid crystal devices.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2009.09.002
Majumdar, Apala 1 ; Robbins, J.M. 2 ; Zyskin, Maxim 3

1 Mathematical Institute, University of Oxford, 24 - 29 St. Giles, Oxford OX1 3LB, UK
2 School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK
3 Department of Mathematics, SETB 2.454 - 80 Fort Brown, Brownsville, TX 78520, USA
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Majumdar, Apala; Robbins, J.M.; Zyskin, Maxim. Tangent unit-vector fields: Nonabelian homotopy invariants and the Dirichlet energy. Comptes Rendus. Mathématique, Tome 347 (2009) no. 19-20, pp. 1159-1164. doi : 10.1016/j.crma.2009.09.002. http://www.numdam.org/articles/10.1016/j.crma.2009.09.002/

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