AN ANALYSIS OF EQUILIBRIA IN DENSE NEMATIC LIQUID CRYSTALS

This paper is concerned with the rigorous analysis of a recently proposed model of Nascimento 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 nontrivial minimizers are zero on a set of positive measure so that L variations cannot generally be taken about minimizers. In particular, it is unclear if the Euler Lagrange equation is well defined, and if local minimizers satisfy it. It will be shown that there exists an analogue of the Euler Lagrange equation that is satisfied by LP local minimizers by reducing the minimization 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 criterion on a certain family of nonlinear 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 nonlinear variations vanish. Furthermore, the analysis provides results on some qualitative phase behavior of the model.