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:
 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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