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, withtangent boundary conditions on the outer square edges. There are two geometrical parameters---theouter square edge length\lambda , and the aspect ratio\rho , which is the ratio of the inner and outer squareedge lengths. The WORS exists for all geometrical parameters and all temperatures, and is globallystable for either\lambda small enough or for\rho sufficiently close to unity. We study three different typesof Landau--de Gennes critical points in this setting: critical points with the minimal two degreesof freedom consistent with the imposed boundary conditions, critical points with three degrees offreedom, and critical points with five degrees of freedom. We identify the competitors for the WORSin the two- and three-dimensional settings. In the three-dimensional setting, we numerically find upto 28 critical points for moderately large values of\rho , of which diagonal solutions are global energyminimizers when they exist. We find two nonenergy minimizing critical points with five degrees offreedom for very small values of\rho , with an escaped profile around the isotropic square inclusion.
Wang, Y., Canevari, G., & Majumdar, A. (2019). Order Reconstruction for Nematics on Squares with Isotropic Inclusions: A Landau--De Gennes Study. SIAM Journal on Applied Mathematics, 79(4), 1314-1340. https://doi.org/10.1137/17M1179820