An analysis of equilibria in dense nematic liquid crystals

This paper is concerned with the rigorous analysis of a recently proposed model of Zheng et. al. for describing nematic liquid crystals within the dense regime, with the orientation distribution function as the variable. A key feature of the model is that in high density regimes all non-trivial minimisers are zero on a set of positive measure so that L^infinity variations cannot generally be taken about minimisers. In particular, it is unclear if the Euler-Lagrange equation is well defined, and if local minimisers satisfy it. It will be shown that there exists an analogue of the Euler-Lagrange equation that is satisfied by L^p local minimisers by reducing the minimisation problem to an equivalent finite-dimensional saddle-point problem, obtained by observing that on certain subsets of the domain the free-energy functional is convex so that duality methods can be applied. This analogue of the Euler-Lagrange equation is then shown to be equivalent to a vanishing variation criteria on a certain family of non-linear curves on which the free- energy functional is sufficiently smooth. All critical points of the finite-dimensional saddle-point problem also correspond to all probability distributions where these non-linear variations vanish. Furthermore, the analysis provides results on some qualitative phase behaviour of the model.