  We study nematic liquid crystal configurations in confined geometries within
the continuum Landau--De Gennes theory. These nematic configurations are
mathematically described by symmetric, traceless two-tensor fields, known as
$\Qvec$-tensor order parameter fields. We obtain explicit upper bounds for the
order parameters of equilibrium liquid crystal configurations in terms of the
temperature, material constants, boundary conditions and the domain geometry.
These bounds are compared with the bounds predicted by the statistical
mechanics definition of the $\Qvec$-tensor order parameter. They give
quantitative information about the temperature regimes for which the Landau-De
Gennes definition and the statistical mechanics definition of the
$\Qvec$-tensor order parameter agree and the temperature regimes for which the
two definitions fail to agree. For the temperature regimes where the two
definitions do not agree, we discuss possible alternatives.
