Nematic liquid crystal director structures in rectangular regions

We consider a shallow rectangular well of nematic liquid crystal subject to weak anchoring on the sides of the well. By considering weak anchoring instead of infinitely strong anchoring, we are able to analyse nematic equilibria in the well without the need to exclude point defects at the corners, as done in previous work in the area. For relatively weak anchoring, we are able to derive analytic expressions for the director alignment angle in terms of an infinite series of modes, involving roots of a transcendental equation. The analytic forms of the director configuration are then used to calculate critical anchoring strengths at which uniform and distorted director structures exchange stability. We also consider the asymptotic behaviour of the director structure and energy for very strong anchoring. We show that in both cases - for the transitions from uniform to distorted states and the limit of infinitely strong anchoring - the approximate analytic expansions agree very well with corresponding numerical calculations of the full model.