2009/05/11 by A Majumdar, Apala Majumdar, JM Robbins +5
Materials Science · Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #FOS: Physical sciences #Liquid Crystal Research Advancements #Mathematical Physics (math-ph) #Mathematics and Applications #math-ph #math.MP
paper · pdf · doi:10.48550/arxiv.0905.1585
21 pages, 8 figures; in "Analysis and Stochastics of Growth Processes and Interface Models", P Morters et al. eds., Oxford University Press 2008, http://www.oup.com/uk/catalogue/?ci=9780199239252
arxiv created 2009/05/11 · openalex publication_date 2009/05/11 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Unit-vector fields \nvec on a convex polyhedron P subject to tangent boundary conditions provide a simple model of nematic liquid crystals in prototype bistable displays. The equilibrium and metastable configurations correspond to minimisers and local minimisers of the Dirichlet energy, and may be regarded as S2-valued harmonic maps on P. We consider unit-vector fields which are continuous away from the vertices of P. 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 of P. In certain cases, this lower bound can be improved by incorporating certain nonabelian homotopy invariants. For a rectangular prism, upper bounds for the infimum Dirichlet energy 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. However, since the homotopy classes are not weakly closed, the infimum may not be realised; the existence and regularity properties of continuous local minimisers of given homotopy type are open questions. Numerical results suggest that some homotopy classes always contain smooth minimisers, while others may or may not depending on the geometry of P. Numerical results modelling a bistable device suggest that the observed nematic configurations may be distinguished topologically.