Order Reconstruction for Nematics on Squares with Isotropic Inclusions: A Landau--De Gennes Study

We prove the existence of a well order reconstruction solution (WORS)-type Landau- de Gennes critical point on a square domain with an isotropic concentric square inclusion, with tangent boundary conditions on the outer square edges. There are two geometrical parameters-the outer square edge length λ , and the aspect ratio ρ , which is the ratio of the inner and outer square edge lengths. The WORS exists for all geometrical parameters and all temperatures, and is globally stable for either λ small enough or for ρ sufficiently close to unity. We study three different types of Landau-de Gennes critical points in this setting: Critical points with the minimal two degrees of freedom consistent with the imposed boundary conditions, critical points with three degrees of freedom, and critical points with five degrees of freedom. We identify the competitors for the WORS in the two- and three-dimensional settings. In the three-dimensional setting, we numerically find up to 28 critical points for moderately large values of ρ , of which diagonal solutions are global energy minimizers when they exist. We find two nonenergy minimizing critical points with five degrees of freedom for very small values of ρ , with an escaped profile around the isotropic square inclusion.