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In this work we recover the Oseen-Frank theory of nematic liquid crystals asa $\Gamma$-limit of a particular mean-field free energy as the sample sizebecomes infinitely large. The Frank constants are not necessarily all equal.Our analysis takes place in a broader framework however, also providing resultsfor more general systems such as biaxial or polar molecules. We start from amesoscopic model describing a competition between entropy and a non-localpairwise interactions between molecules. We assume the interaction potential isseparable so that the energy can be reduced to a model involving a macroscopicorder parameter. We assume only integrability and decay properties of themacroscopic interaction, but no regularity assumptions are required. Inparticular, singular interactions are permitted. The analysis becomessignificantly simpler on a translationally invariant domain, so we firstconsider periodic domains with increasing domain of periodicity. Then we tacklethe problem on a Lipschitz domain with non-local boundary conditions byconsidering an asymptotically equivalent problem on periodic domains. Weconclude by applying the results to a case which reduces to the Oseen-Frankmodel of elasticity, and give expressions for the Frank constants in terms ofintegrals of the interaction kernel.

Oseen-Frank-type theories of ordered media as the $\Gamma$-limit of a non-local mean-field free energy