# The Well Order Reconstruction Solution for Three-Dimensional Wells, in the Landau-de Gennes theory

## TL;DR

This paper investigates the existence, stability, and numerical solutions of Well Order Reconstruction Solutions (WORS) in three-dimensional nematic wells, extending previous 2D results to 3D with various boundary conditions.

## Contribution

It proves the existence and stability of WORS in 3D wells for arbitrary heights and explores their behavior under different boundary conditions and well sizes.

## Key findings

- WORS exist in 3D wells for all heights with natural and realistic boundary conditions.
- WORS are stable for small well sizes and unstable as the size increases.
- Numerical solutions reveal complex 3D defect structures and effects of surface anchoring.

## Abstract

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 $\lambda$, and the well height denoted by $\epsilon$. The WORS are distinctive equilibria reported in [10] 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 centre. 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 $\lambda$ small enough in both cases and unstable as $\lambda$ increases. We numerically compute novel mixed 3D solutions for large $\lambda$ and $\epsilon$ 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.