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 prove regularity of minimizers of the functional (F(s, u) = ( | s |^2 + s^2 | u |^2 + (s) ) dx ) recently suggested by Ericksen [10] for the statics of nematic liquid crystals. We show that, given locally minimizing pairs (s, u),s has a continuous representative, ands, u are smooth outside the set s=0 . The proof relies upon higher integrability estimates, monotonicity, and decay lemmas. We consider a new mathematical model for nematic liquid crystal proposed by J. Ericksen using Landau’s order parameter approach. In this model, the static configuration of liquid crystal can be described by a map minimizing certain degenerate variational integral. Here we prove that minimizers exist and are Holder continuous. Energy minimizing static configurations of nematic liquid crystals with variable degree of orientation can be described by a map which minimizes certain degenerate variational integrals. This mathematical model was recently posed by J. L. Erickson. In this paper, we shall prove the existence, local uniqueness, and global regularity of minimizers. We shall also estimate the size of possible defects.  This is a summary of some recent mathematical advances in the theory of defects in nematic liquid crystals. It also includes some new results concerning disclinations (line defects) which have not been published elsewhere. We prove existence of minimizers of the functional ( ( | s |^2 + s^2 | u |^2 + (s) ) dx ) recently suggested by Ericksen [8] for the statics of nematic liquid crystals. A set of necessary conditions for the minimizers and a monotonicity formula are also found.