Rectifiability of line defects in liquid crystals with variable degree of orientation

Abstract

In [AHL] Hardt, Lin and the author proved that the defect set of minimizers of the modified Ericksen energy for nematic liquid crystals consists locally of a finite union of isolated points and Hölder continuous curves with finitely many crossings. In this article, we show that each Hölder continuous curve in the defect set is of finite length. Hence, the defect set is locally rectifiable. For the most part, the proof follows the work of De Lellis, Marchese, Spadaro and Valtorta [DLMSV] on harmonic $\mathcal{Q}$-valued maps closely. The blow-up analysis in [AHL] allows us to simplify the covering arguments in [DLMSV] and locally estimate the length of line defects in a geometric fashion.

Minor revisions; added a new section restating the results (by R. Hardt, F.H. Lin and the author) used in the article; 28 pages