  In this work we recover the Oseen-Frank theory of nematic liquid crystals as
a $\Gamma$-limit of a particular mean-field free energy as the sample size
becomes infinitely large. The Frank constants are not necessarily all equal.
Our analysis takes place in a broader framework however, also providing results
for more general systems such as biaxial or polar molecules. We start from a
mesoscopic model describing a competition between entropy and a non-local
pairwise interactions between molecules. We assume the interaction potential is
separable so that the energy can be reduced to a model involving a macroscopic
order parameter. We assume only integrability and decay properties of the
macroscopic interaction, but no regularity assumptions are required. In
particular, singular interactions are permitted. The analysis becomes
significantly simpler on a translationally invariant domain, so we first
consider periodic domains with increasing domain of periodicity. Then we tackle
the problem on a Lipschitz domain with non-local boundary conditions by
considering an asymptotically equivalent problem on periodic domains. We
conclude by applying the results to a case which reduces to the Oseen-Frank
model of elasticity, and give expressions for the Frank constants in terms of
integrals of the interaction kernel.
