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 $\lambda$, and the aspect ratio $\rho$, which is the ratio of the inner and outer square edge lengths. The WORS exists for all geometrical parameters and for all temperatures; we prove that the WORS is globally stable for either $\lambda$ small enough or for $\rho$ sufficiently close to unity. We study three different types of critical points in this model 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. In the two-dimensional case, we use $\Gamma$-convergence techniques to identify the energy-minimizing competitors. We decompose the second variation of the Landau-de Gennes energy into three separate components to study the effects of different types of perturbations on the WORS solution and find that it is most susceptible to in-plane perturbations. In the three-dimensional setting, we numerically find up to $28$ critical points for moderately large values of $\rho$ and we find two critical points with the full five degrees of freedom for very small values of $\rho$, with an escaped profile around the isotropic square inclusion.