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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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PRINTED FROM OXFORD SCHOLARSHIP ONLINE (oxford.universitypressscholarship.com). (c) Copyright Oxford University Press, 2021. All Rights Reserved. An individual user may print out a PDF of a single chapter of a monograph in OSO for personal use. date: 18 June 2021

Liquid Crystals and Harmonic Maps in Polyhedral Domains

Liquid Crystals and Harmonic Maps in Polyhedral Domains

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

Apala Majumdar

Jonathan Robbins

Maxim Zyskin

Oxford University Press

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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