Multistability for a Reduced Nematic Liquid Crystal Model in the Exterior of 2D Polygons

We study nematic equilibria in an unbounded domain, with a two-dimensional regular polygonal hole with $K$ edges, in a reduced Landau-de Gennes framework. This complements our previous work on the "interior problem" for nematic equilibria confined inside regular polygons (SIAM Journal on Applied Mathematics, 80(4):1678-1703, 2020). The two essential dimensionless model parameters are $\lambda$--the ratio of the edge length of polygon hole to the nematic correlation length, and an additional degree of freedom, $\gamma^*$--the nematic director at infinity. In the $\lambda\to 0$ limit, the limiting profile has two interior point defects outside a generic polygon hole, except for a triangle and a square. For a square hole, the limiting profile has either no interior defects or two line defects depending on $\gamma^*$, and for a triangular hole, there is a unique interior point defect outside the hole. In the $\lambda\to\infty$ limit, there are at least $\binom{K}{2}$ stable states %differentiated by the location of two bend vertices and the multistability is enhanced by $\gamma^*$, compared to the interior problem. Our work offers new insights into how to tune the existence, location, and dimensionality of defects.