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Line Defects in a Modified Ericksen Model of Nematic Liquid Crystals

Saturday, October 24, 2015 - 3:35pm - 4:15pm
Robert Hardt (Rice University)
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 director vectorfield on a region of R^3 but also a scalar order parmeter quantifying 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 defect set $^{-1}{0}$. The optimal partial regularity result of R.Hardt-FH.Lin in 1993, for this model, improved this to regularity away from isolated points. This result 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 involving RP^2 . In recent ongoing work with FH.Lin and O. Alper, we see that the singular set with the RP^2 cone model necessarily consists of Holder continuous curves.
MSC Code: 
76A15