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