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.
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.