Elastic anisotropy of nematic liquid crystals in the two-dimensional Landau-de Gennes model

We study the effects of elastic anisotropy on the Landau-de Gennes critical points for nematic liquid crystals, in a square domain. The elastic anisotropy is captured by a parameter, L2, and the critical points are described by three degrees of freedom. 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, λ, is small enough. We perform asymptotic analyses and numerical studies to discover at least 5 classes of these symmetric critical points the WORS, Ring±, 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 λ and large L2, and prove associated stability results that corroborate the stabilising 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.