Energies of S 2 -valued harmonic maps on polyhedra with tangent boundary conditions
Annales de l'I.H.P. Analyse non linéaire, Volume 25 (2008) no. 1, p. 77-103
@article{AIHPC_2008__25_1_77_0,
     author = {Majumdar, A. and Robbins, J. M. and Zyskin, M.},
     title = {Energies of ${S}^{2}$-valued harmonic maps on polyhedra with tangent boundary conditions},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {25},
     number = {1},
     year = {2008},
     pages = {77-103},
     doi = {10.1016/j.anihpc.2006.11.003},
     zbl = {1141.35005},
     mrnumber = {2383079},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2008__25_1_77_0}
}
Majumdar, A.; Robbins, J. M.; Zyskin, M. Energies of ${S}^{2}$-valued harmonic maps on polyhedra with tangent boundary conditions. Annales de l'I.H.P. Analyse non linéaire, Volume 25 (2008) no. 1, pp. 77-103. doi : 10.1016/j.anihpc.2006.11.003. http://www.numdam.org/item/AIHPC_2008__25_1_77_0/

[1] Abramowitz M., Stegun I.A., Handbook of Mathematical Functions, Dover Publications, 1965.

[2] Birkhoff G., Tres observaciones sobre el algebra lineal, Univ. Nac. Tacumán Rev. Ser. A 5 (1946) 147-151. | MR 20547 | Zbl 0060.07906

[3] Brezis H., The interplay between analysis and topology in some nonlinear PDE problems, Bull. Amer. Math. Soc. 40 (2006) 179-201. | MR 1962295 | Zbl 1161.35354

[4] Brezis H., Coron J.-M., Lieb E.H., Harmonic maps with defects, Comm. Math. Phys. 107 (1986) 649-705. | MR 868739 | Zbl 0608.58016

[5] Davidson A.J., Mottram N.J., Flexoelectric switching in a bistable nematic device, Phys. Rev. E 65 (2002) 051710.

[6] De Gennes P.-G., Prost J., The Physics of Liquid Crystals, second ed., Oxford University Press, 1995.

[7] Denniston C., Yeomans J.M., Flexoelectric surface switching of bistable nematic devices, Phys. Rev. Lett. 87 (2001) 275505.

[8] Eells J., Fuglede B., Harmonic Maps Between Riemannian Polyhedra, Cambridge Tracts in Mathematics, vol. 142, Cambridge University Press, 2001. | MR 1848068 | Zbl 0979.31001

[9] Gromov M., Schoen R., Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one, Publ. IHES 76 (1992) 165-246. | Numdam | MR 1215595 | Zbl 0896.58024

[10] Hardt R.M., Singularities of harmonic maps, Bull. Amer. Math. Soc. 34 (1997) 15-34. | MR 1397098 | Zbl 0871.58026

[11] J.C. Jones, J.R. Hughes, A. Graham, P. Brett, G.P. Bryan-Brown, E.L. Wood, Zenithal bistable devices: Towards the electronic book with a simple LCD, In: Proc IDW, 2000, pp. 301-304.

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

[13] Kléman M., Points, Lines and Walls, John Wiley and Sons, Chichester, 1983. | MR 734901

[14] Kleman M., Lavrentovich O.D., Soft Condensed Matter, Springer, 2002.

[15] Lavrentovich O.D., Topological defects in dispersed liquid crystals, or words and worlds around liquid crystal drops, Liquid Crystals 24 (1998) 117-125.

[16] Lin F.H., Poon C.C., On nematic liquid crystal droplets, in: Elliptic and Parabolic Methods in Geometry, A.K. Peters, 1996, pp. 91-121. | MR 1417951 | Zbl 0876.49038

[17] A. Majumdar, Liquid crystals and tangent unit-vector fields in polyhedral geometries, PhD thesis, University of Bristol, 2006.

[18] Majumdar A., Robbins J.M., Zyskin M., Elastic energy of liquid crystals in convex polyhedra, J. Phys. A 37 (2004) L573-L580, J. Phys. A 38 (2005) 7595. | MR 2169580 | Zbl 1064.58019

[19] Majumdar A., Robbins J.M., Zyskin M., Lower bound for energies of harmonic tangent unit-vector fields on convex polyhedra, Lett. Math. Phys. 70 (2004) 169-183. | MR 2109431 | Zbl 1059.58009

[20] Majumdar A., Robbins J.M., Zyskin M., Elastic energy for reflection-symmetric topologies, J. Phys. A 39 (2006) 2673-2687. | MR 2213361 | Zbl 1099.82019

[21] Majumdar A., Robbins J.M., Zyskin M., Topology and bistability in liquid crystal devices, math-ph/0611016.

[22] Mermin N.D., The topological theory of defects in ordered media, Rev. Mod. Phys. 51 (C) (1979) 591-651. | MR 541885 | Zbl 0711.55009

[23] C.J.P. Newton, T.P. Spiller, Bistable nematic liquid crystal device modelling, In: Proc. 17th IDRC (SID), 1997, p. 13.

[24] Robbins J.M., Zyskin M., Classification of unit-vector fields in convex polyhedra with tangent boundary conditions, J. Phys. A 37 (2004) 10609-10623. | MR 2098054 | Zbl 1138.58309

[25] Stewart I.W., The Static and Dynamic Continuum Theory of Liquid Crystals, Taylor and Francis, London, 2004.

[26] Virga E.G., Variational Theories for Liquid Crystals, Chapman and Hall, 1994. | MR 1369095 | Zbl 0814.49002

[27] Volovik G.E., Lavrentovich O.D., Topological dynamics of defects - boojums in nematic drops, Sov. Phys. JETP 58 (1983) 1159.

[28] M. Zyskin, Homotopy classification of director fields on polyhedral domains with tangent and periodic boundary conditions, Preprint, 2005.