Daniel Phillips

We will describe work on two problems. First, we examine the transition between nematic and smectic phases in liquid crystals. These are characterized as minimizers for an elliptic energy consisting of the Frank energy for the nematic director, coupled with a Ginzburg-Landau energy for an order parameter characterizing smectic structure. Temperature appears as a parameter in the energy. We determine thresholds, such that if the temperature is too large then minimizers are nematic (the order parameter vanishes identically) and if the temperature is too low then minimizers must have smectic structure.

Second, we describe the analysis of a model for polymer stabilized liquid crystals. Here a parallel array of polymer fibers is placed in a liquid crystal matrix. The l.c. molecules tend to align with the fibers, and this pattern is switched off by applying a magnetic field perpendicular to the fibers. The problem is modeled by a nonlinear elliptic energy on a perforated domain. We investigate equilibrium configurations and the effective limit as the polymer array becomes finer and finer.