The Well Order Reconstruction Solution for Three Dimensional Wells in the Landau-de Gennes Theory

We study nematic equilibria on three-dimensional square wells, with emphasis on Well Order Reconstruction Solutions (WORS) as a function of the well size, characterized by λ, and the well height denoted by ϵ. The WORS are distinctive equilibria reported in Kralj and Majumdar (2014) for square domains, without taking the third dimension into account, which have two mutually perpendicular defect lines running along the square diagonals, intersecting at the square center. We prove the existence of WORS on three-dimensional wells for arbitrary well heights, with (i) natural boundary conditions and (ii) realistic surface energies on the top and bottom well surfaces, along with Dirichlet conditions on the lateral surfaces. Moreover, the WORS is globally stable for λ small enough in both cases and unstable as λ increases. We numerically compute novel mixed 3D solutions for large λ and ϵ followed by a numerical investigation of the effects of surface anchoring on the WORS, exemplifying the relevance of the WORS solution in a 3D context.