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.
See also @cite_24 for an extensive discussion on the use of @math in modeling uniaxial nematics, relations with the Landau-de Gennes theory, issues of suitable function spaces, orientability of line fields, and boundary conditions. For another result on line defects in liquid crystals, see @cite_23 , which considers the vanishing elastic constant limit in a Landau-de Gennes model, in the spirit of the Ginzburg-Landau theory.
@cite_24 @cite_23
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Uniaxial nematic liquid crystals are modelled in the Oseen–Frank theory through a unit vector field n. This theory has the apparent drawback that it does not respect the head-to-tail symmetry in which n should be equivalent to −n. This symmetry is preserved in the constrained Landau–de Gennes theory that works with the tensor ( Q=s (n n- 1 3 Id ) ). We study the differences and the overlaps between the two theories. These depend on the regularity class used as well as on the topology of the underlying domain. We show that for simply-connected domains and in the natural energy class W1,2 the two theories coincide, but otherwise there can be differences between the two theories, which we identify. In the case of planar domains with holes and various boundary conditions, for the simplest form of the energy functional, we completely characterise the instances in which the predictions of the constrained Landau–de Gennes theory differ from those of the Oseen–Frank theory. We consider the Landau-de Gennes variational model for nematic liquid crystals, in three-dimensional domains. More precisely, we study the asymptotic behaviour of minimizers as the elastic constant tends to zero, under the assumption that minimizers are uniformly bounded and their energy blows up as the logarithm of the elastic constant. We show that there exists a closed set ( S _ line ) of finite length, such that minimizers converge to a locally harmonic map away from ( S _ line ). Moreover, ( S _ line ) restricted to the interior of the domain is a locally finite union of straight line segments. We provide sufficient conditions, depending on the domain and the boundary data, under which our main results apply. We also discuss some examples.