Chapter

Liquid Crystals and Harmonic Maps in Polyhedral Domains

Apala Majumdar, Jonathan Robbins and Maxim Zyskin

in Analysis and Stochastics of Growth Processes and Interface Models

Published in print July 2008 | ISBN: 9780199239252
Published online September 2008 | e-ISBN: 9780191716911 | DOI: http://dx.doi.org/10.1093/acprof:oso/9780199239252.003.0014
 Liquid Crystals and Harmonic Maps in Polyhedral Domains

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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

Chapter.  9820 words.  Illustrated.

Subjects: Probability and Statistics

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