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Analysis and Stochastics of Growth Processes and Interface Models$
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Peter Mörters, Roger Moser, Mathew Penrose, Hartmut Schwetlick, and Johannes Zimmer

Print publication date: 2008

Print ISBN-13: 9780199239252

Published to Oxford Scholarship Online: September 2008

DOI: 10.1093/acprof:oso/9780199239252.001.0001

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Liquid Crystals and Harmonic Maps in Polyhedral Domains

Liquid Crystals and Harmonic Maps in Polyhedral Domains

Chapter:
(p.306) 14 Liquid Crystals and Harmonic Maps in Polyhedral Domains
Source:
Analysis and Stochastics of Growth Processes and Interface Models
Author(s):

Apala Majumdar

Jonathan Robbins

Maxim Zyskin

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199239252.003.0014

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.

Keywords:   harmonic unit-vector field, homotopy class, Dirichlet energy, liquid crystal

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