Planar Anchoring for a Colloid in Nematic Liquid Crystal with a Magnetic Field

Abstract

We study minimizers of the Landau-de Gennes energy in the exterior region around a smooth 2-manifold in R3 with a constant external magnetic field present. Uniaxial boundary data and a strong tangential anchoring are imposed on the surface of the manifold and we consider the large particle limit in a regime where the magnetic field is relatively weak. Before studying the general manifold, we analyze a more simple case in which the manifold is spherical. After deriving a lower bound for the energy in this limiting regime, we prove that a director field on the boundary which maximizes its vertical component yields a minimal lower bound. We then construct a recovery sequence to show that this lower bound is in fact the optimal energy bound. These steps are later repeated in more generality for a larger class of smooth manifolds.