An Analysis of Equilibria in Dense Nematic Liquid Crystals

Abstract

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 $L^\infty$ 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 $L^p$ 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.