Mathematical Questions of Liquid Crystal Theory

A liquid crystal is a mesomorphic phase of a material which occurs between its liquid and solid phases. Frequently the material is composed of rod-like molecules which display orientational order, unlike a liquid, but lacking the lattice structure of a solid. It may flow easily and so may also be thought of as an anisotropic fluid. This anisotropy is evident in the way it transmits light; for example, a nematic liquid crystal is optically uniaxial. We take the opportunity of this note to discuss a few of the analytical issues which arise in the attempt to study static equilibrium configurations. One attractive feature of this subject is that it has a well developed continuum description in the Ericksen-Leslie [E1,[L1] theory. Some of the questions have significance in the context of harmonic mappings into spheres and we shall attempt to clarify these connections. We refer to the articles by F. Leslie [L2],[L3] in these proceedings both for other aspects of the static theory and for an introduction to flow problems.

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