The Journal of Geometric Analysis Volume 4, Number 3, 1994 On Ericksen's Model for Liquid Crystals By Fang Hua Lin and Chi Cheung Poon ABSTRACT. We consider a new mathematical model for nematic liquid crystal proposed by J. Ericksen using Landau's order parameter approach. In this model, the static configuration of liquid crystal can be described by a map minimizing certain degenerate variational integral. Here we prove that minimizers exist and are Holder continuous. 1. Introduction For the small molecule nematic liquid crystals commonly used in display devices, the Oseen- Frank theory has been quite successful in describing many static phenomena. For phenomena involv- ing flow, the Ericksen-Leslie equations have been used rather nicely: they reduce to the Oseen-Frank equations for statics. There are, however, several new issues that have arisen in recent studies which motivate one to consider modifying the theory. For one thing, physicists are interested in seeing the development of the theory of defects that might be at rest, or moving. The static theory of point defects, based on Oseen-Frank equations, is now rather well developed, as discussed in surveys by Ericksen [El ] and Kinderlehrer [K]. Though one believes that the Ericksen-Leslie theory is capable of describing moving point defects, no real progress has been made, despite several serious efforts. More serious difficulties are encountered with some observed kinds of disclinations (line defects) in nematics, or wall (surface defects) in polymers. In [E2], Ericksen proposed a new model using Landau's order parameter approach. In this model, the orientation of liquid crystals is described by a unit vector field, n, denoting the optical director, and a scalar function, s, of position denoting the orientational order of the liquid crystals. In the static case, after various physical reasonings, Ericksen has derived a general form of bulk energy density, see (2.6) below. By a particular choice of material constants involved, Ambrosio [A 1], [A2], Hardt and Lin [HL], and Lin [L] have made various careful studies of energy minimizing configurations. Their works show not only the consistency of the new theory with the old ones but Math Subject Classification 35, 49. Key Words and Phrases Existence and regularity results, variational problem. The research of the first author is supported by an NSF-PYI grant. @ 1994 The Joumal of Geometric Analysis ISSN 1050-6926