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 λ-the ratio of the edge length of polygon hole to the nematic correlation length, and an additional degree of freedom, ϒ*-the nematic director at infinity. In the λ → 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 ϒ*, and for a triangular hole, there is a unique interior point defect outside the hole. In the λ → ∞ limit, there are at least (K 2) stable states and the multistability is enhanced by ϒ*, compared to the interior problem. Our work offers new insights into how to tune the existence, location, and dimensionality of defects.