Robert M Hardt


Rice University


Monday, October 12, 2015 - 3:00pm to 4:00pm




In 1985, J. Ericksen derived a model for uniaxial liquid crystals to allow for disclinations (i.e. line defects or curve singularities). It involved not only a unit orientation vectorfield on a region of R^3  but also a scalar order parmeter quantify- ing the expected inner product between this vector and the molecular orientation. FH.Lin, in several papers, related this model, for certain material constants, to harmonic maps to a metric cone over S^2. He showed that a minimizer would be continuous everywhere but would have higher regularity fail on the singular de- fect set s^{-1}(0). The optimal partial regularity result of R.Hardt-FH.Lin in 1993, for this model, led to regularity away from isolated points, which unfortunately still excluded line singularities. This paper accordingly also introduced a modified model involving maps to a metric cone over RP^2, the real projective plane. Here the nontrivial homotopy leads to the optimal estimate of the singular set being 1 dimensional. In 2010, J. Ball and A.Zarnescu discussed a derivation from the de Gennes Q tensor and interesting orientability questions using RP2. In recent ongo- ing work with FH.Lin and O. Alper, we see that the singular set with this model necessarily consists of Holder continuous curves. We will also survey some of the many more elaborate liquid crystal PDE’s involving a general director functional, the full Q tensor model, and possible coupling with fluid velocity.