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.
In addition, using a dimension reduction argument based on the monotonicity of Almgren frequency, the following Hausdorff dimension estimates were proved in @cite_10 , @cite_21 for nontrivial minimizing maps: @math for @math and @math for @math . By @cite_26 , the first estimate is sharp, as occur in this case. On the other hand, the second estimate was improved in @cite_8 . By proving that there are no homogeneous, energy minimizing maps depending only on two variables in this case, it was shown that there cannot be any such tangent maps either. Hence, by the dimension reduction argument in @cite_10 , @cite_21 , the defect set must consist of isolated points in the case @math .
@cite_21 @cite_10 @cite_26 @cite_8
1968731434 2065357863 2060284453 2005414009
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. In this paper we represent the degree of microscopic order through a scalar-valued function, the degree of orientation, which vanishes where the fluid becomes isotropic