The defect set of minimizers of the modified Ericksen energy for nematic liquid crystals consists locally of a finite union of isolated points and Holder continuous curves with finitely many crossings.
The existence, uniqueness and regularity theory for @math -valued energy minimizing maps carry over veribatim to the case of @math -valued energy minimizing maps. However, an important difference is that simple closed geodesics, i.e. great circles of length @math , are not contractible in @math , unlike those in @math . While @math -valued maps minimizing the Dirichlet or Oseen-Frank energies do not differ from their @math -valued counterparts in terms of the size of their singular sets or their asymptotic behavior near them (cf. @cite_18 and @cite_19 ), the nontrivial topology of @math does have an effect in the context of Ericksen's variable degree of orientation model. In particular, one observes line defects for @math -valued energy minimizing maps in the case @math , @math . See Remark for an example. In particular, the Hausdorff dimension estimate @math is optimal for @math -valued energy minimizing maps, in contrast to the case of @math -valued ones.
@cite_19 @cite_18
2084946996 2035427049
We establish the existence and partial regularity for solutions of some boundary-value problems for the static theory of liquid crystals. Some related problems involving magnetic or electric fields are also discussed. Two problems concerning maps ϕ with point singularities from a domain Ω C ℝ3 toS2 are solved. The first is to determine the minimum energy of ϕ when the location and topological degree of the singularities are prescribed. In the second problem Ω is the unit ball and ϕ=g is given on ∂Ω; we show that the only cases in whichg(x |x|) minimizes the energy isg=const org(x)=±Rx withR a rotation. Extensions of these problems are also solved, e.g. points are replaced by “holes,” ℝ3,S2 is replaced by ℝ N ,SN−1 or by ℝ N , ℝPN−1, the latter being appropriate for the theory of liquid crystals.