  We consider the relationship between three continuum liquid crystal theories:
Oseen-Frank, Ericksen and Landau-de Gennes. It is known that the function space
is an important part of the mathematical model and by considering various
function space choices for the order parameters $s$, ${\bf n}$, and ${\bf Q}$,
we establish connections between the variational formulations of these
theories. We use these results to derive a version of the Oseen-Frank theory
using special functions of bounded variation. This proposed model can describe
both orientable and non-orientable defects. Finally we study a number of
frustrated nematic and cholesteric liquid crystal systems and show that the
model predicts the existence of point and surface discontinuities in the
director.
