Supplementary material from "Elastic anisotropy in the reduced Landau–de Gennes model"

We study the effects of elastic anisotropy on Landau–de Gennes critical points, for nematic liquid crystals, on a square domain. The elastic anisotropy is captured by a parameter, L₂, and the critical points are described by 3 d.f. We analytically construct a symmetric critical point for all admissible values of L₂, which is necessarily globally stable for small domains, i.e. when the square edge length, lambda, is small enough. We perform asymptotic analyses and numerical studies to discover at least five classes of these symmetric critical points—the WORS, Ring pm, Constant and pWORS solutions, of which the WORS, Ring⁺ and Constant solutions can be stable. Furthermore, we demonstrate that the novel Constant solution is energetically preferable for large lambda and large L₂, and prove associated stability results that corroborate the stabilizing effects of L₂ for reduced Landau–de Gennes critical points. We complement our analysis with numerically computed bifurcation diagrams for different values of L₂, which illustrate the interplay of elastic anisotropy and geometry for nematic solution landscapes, at low temperatures.

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