2021/05/21 by Yucen Han, Han, Yucen, Joseph Harris +5
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.AP #math.MP
paper · pdf · doi:10.48550/arxiv.2105.10253
13 figures
arxiv created 2021/05/21 · arxiv updated 2021/05/24
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.