  Using Landau-de Gennes theory to describe nematic order, we study a
frustrated cell consisting of nematic liquid crystal confined between two
parallel plates. We prove the uniqueness of equilibrium states for a small cell
width. Letting the cell width grow, we study the behaviour of this unique
solution. Restricting ourselves to a certain interval of temperature, we prove
that this solution becomes unstable at a critical value of the cell width.
Moreover, we show that this loss of stability comes with the appearance of two
new solutions: there is a symmetric pitchfork bifurcation. This picture agrees
with numerical simulations performed by P. Palffy-Muhorray, E.C. Gartland and
J.R. Kelly. Some of the methods that we use in the present paper apply to other
situations, and we present the proofs in a general setting. More precisely, the
paper contains the proof of a general uniqueness result for a class of
perturbed quasilinear elliptic systems, and general considerations about
symmetric solutions and their stability, in the spirit of Palais' Principle of
Symmetric Criticality.
