### Abstract

This chapter is concerned with harmonic maps from a polyhedron to the unit two-sphere, which provide a model of nematic liquid crystals in bistable displays. This chapter looks at the Dirichlet energy of homo-topy classes of such harmonic maps, subject to tangent boundary conditions, and investigate lower and upper bounds for this Dirichlet energy on each homotopy class; local minimisers of this energy correspond to equilibrium and metastable configurations. A lower bound for the infimum Dirichlet energy for a given homotopy class is obtained as a sum of minimal connections between fractional defects at the vertices. In certain cases, this lower bound can be improved. For a rectangular prism, upper bounds are obtained from locally conformal solutions of the Euler-Lagrange equations, with the ratio of the upper and lower bounds bounded independently of homotopy type.

Original language | English |
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Title of host publication | Analysis and Stochastics of Growth Processes and Interface Models |

Publisher | Oxford University Press |

Chapter | 14 |

Number of pages | 23 |

ISBN (Print) | 9780199239252 |

DOIs | |

Publication status | Published - 2007 |

### Keywords

- harmonic unit-vector field
- homotopy class
- Dirichlet energy
- liquid crystal

## Cite this

Majumdar, A., Robbins, J., & Zyskin, M. (2007). Liquid Crystals and Harmonic Maps on Polyhedral Domains. In

*Analysis and Stochastics of Growth Processes and Interface Models*Oxford University Press. https://doi.org/10.1093/acprof:oso/9780199239252.003.0014