Title1:Defect Curves in a Modified Ericksen Model of Nematic Liquid Crystals
Time: Tue, June 21, 2016, AM 09:30-10:30
Speaker:Prof. Robert Hardt (Rice University，USA)
Title2:Boundary Blow-up analysis for 2 dimensional approximate harmonic maps under either weak or strong anchoring boundary conditions.
Time: Tue, June 21, 2016, AM 10:30-11:30
Speaker:Prof.Changyou Wang (Purdue University)
Location:Room 4318, Building No.4, Wushan Campus
Abstract1: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 parameter 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 of points mapped to the cone point. 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 work with FH.Lin and O. Alper, we see that the singular set with this RP^2 cone model necessarily consists of Holder continuous curves.
Abstract2:In this talk, I will describe the boundary bubbling phenomena for approximate harmonic maps in dimension two under either weak or strong anchoring boundary conditions, which concludes that, among other properties, the total energy is conserved after counting the energy of bubbles. This is a joint work with Tao Huang.
