### Abstract

A unit-vector field n:P → S2 on a convex polyhedron P ⊂ R3 satisfies tangent boundary conditions if, on each face of P, n takes values tangent to that face. Tangent unit-vector fields are necessarily discontinuous at the vertices of P. We consider fields which are continuous elsewhere. We derive a lower bound E− P (h) for the infimum Dirichlet energy Einf P (h) for such tangent unitvector fields of arbitrary homotopy type h. E− P (h) is expressed as a weighted sum of minimal connections, one for each sector of a natural partition of S2 induced by P. For P a rectangular prism, we derive an upper bound for Einf P (h) whose ratio to the lower bound may be bounded independently of h. The problem is motivated by models of nematic liquid crystals in polyhedral geometries. Our results improve and extend several previous results.

Original language | Undefined/Unknown |
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Pages (from-to) | 77-103 |

Number of pages | 27 |

Journal | Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire |

Volume | 25 |

Issue number | 1 |

Early online date | 28 Dec 2006 |

DOIs | |

Publication status | Published - Jan 2008 |

### Keywords

- harmonic maps
- tangent boundary conditions
- minimal connection
- liquid crystals
- bistability

## Cite this

Majumdar, A., Robbins, J. M., & Zyskin, M. (2008). Energies of S2-valued harmonic maps on polyhedra with tangent boundary conditions.

*Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire*,*25*(1), 77-103. https://doi.org/10.1016/j.anihpc.2006.11.003