Comptes Rendus
no. 19-20
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]

Présenté par : Haïm Brezis

Apala Majumdar1 ; J.M. Robbins2 ; Maxim Zyskin3
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
Comptes Rendus. Mathématique, Volume 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
Apala Majumdar 1 ; J.M. Robbins 2 ; Maxim Zyskin 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 
@article{CRMATH_2009__347_19-20_1159_0,
     author = {Apala Majumdar and J.M. Robbins and Maxim Zyskin},
     title = {Tangent unit-vector fields: {Nonabelian} homotopy invariants and the {Dirichlet} energy},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1159--1164},
     publisher = {Elsevier},
     volume = {347},
     number = {19-20},
     year = {2009},
     doi = {10.1016/j.crma.2009.09.002},
     language = {en},
}
TY  - JOUR
AU  - Apala Majumdar
AU  - J.M. Robbins
AU  - Maxim Zyskin
TI  - Tangent unit-vector fields: Nonabelian homotopy invariants and the Dirichlet energy
JO  - Comptes Rendus. Mathématique
PY  - 2009
SP  - 1159
EP  - 1164
VL  - 347
IS  - 19-20
PB  - Elsevier
DO  - 10.1016/j.crma.2009.09.002
LA  - en
ID  - CRMATH_2009__347_19-20_1159_0
ER  - 
%0 Journal Article
%A Apala Majumdar
%A J.M. Robbins
%A Maxim Zyskin
%T Tangent unit-vector fields: Nonabelian homotopy invariants and the Dirichlet energy
%J Comptes Rendus. Mathématique
%D 2009
%P 1159-1164
%V 347
%N 19-20
%I Elsevier
%R 10.1016/j.crma.2009.09.002
%G en
%F CRMATH_2009__347_19-20_1159_0
Apala Majumdar; J.M. Robbins; Maxim Zyskin. Tangent unit-vector fields: Nonabelian homotopy invariants and the Dirichlet energy. Comptes Rendus. Mathématique, Volume 347 (2009) no. 19-20, pp. 1159-1164. doi : 10.1016/j.crma.2009.09.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.09.002/

[1] H. Brezis; J.M. Coron; E.H. Lieb Harmonic maps with defects, Comm. Math. Phys., Volume 107 (1986), pp. 649-705

[2] P.G. De Gennes The Physics of Liquid Crystals, Clarendon Press, Oxford, 1974

[3] J. Eells; J.H. Sampson Harmonic mappings of Riemannian manifolds, Amer. J. Math., Volume 86 (1964), pp. 109-160

[4] S. Kitson; A. Geisow Controllable alignment of nematic liquid crystals around microscopic posts: Stabilization of multiple states, Appl. Phys. Lett., Volume 80 (2002), pp. 3635-3637

[5] W. Magnus; A. Karras; D. Solitar Combinatorial Group Theory, Dover, 1976

[6] A. Majumdar; J.M. Robbins; M. Zyskin Lower bounds for energies of harmonic tangent unit-vector fields on convex polyhedra, Lett. Math. Phys., Volume 70 (2004), pp. 169-183

[7] A. Majumdar; J.M. Robbins; M. Zyskin Elastic energy of liquid crystals in convex polyhedra, J. Phys. A: Math. Gen., Volume 37 (2004), p. L573-L580

[8] A. Majumdar; J.M. Robbins; M. Zyskin Elastic energy for reflection-symmetric topologies, J. Phys. A: Math. Gen., Volume 39 (2006), pp. 2673-2687

[9] A. Majumdar; J.M. Robbins; M. Zyskin Tangent unit-vector fields: Nonabelian homotopy invariants and the Dirichlet energy (2009, in preparation; preprint) | arXiv

[10] J.M. Robbins; M. Zyskin Classification of unit-vector fields in convex polyhedra with tangent boundary conditions, J. Phys. A: Math. Gen., Volume 37 (2004), pp. 10609-10623

[11] M. Spivak A Comprehensive Introduction to Differential Geometry, vol. 2, Publish or Perish Press, Berkeley, CA, 1990

Cité par Sources :

Commentaires - Politique