Boundary regularity of a heat flow problem from Ericksen's model for nematic liquid crystals (Q6651639)

The paper is devoted to the heat flow problem posed for Ericksen's model, which describes nematic liquid crystals with variable degree of orientation. The problem consists of two equations, and the solutions consist of a pairs of functions. The authors discuss the boundary regularity properties of the solutions. To obtain the interior regularity theory, it is essential to establish a monotonicity property for an energy function and a frequency function. In this connection, special convexity assumptions are introduced on the domain. Then, if the domain is strictly convex, a monotonicity formula is established for the energy functional. As the next step, it is proved that the pair of solutions is continuous at a zero point. Moreover, it is stated that in the general case, the pair of solutions is continuous outside a singular set that has a finite \(m\)-dimensional Hausdorff measure with respect to the parabolic metric.