On higher order variational analysis in one and three dimensions for soft boundaries

Abstract

For some problems in liquid crystal physics we need to use the Euler equation and the corresponding boundary equation in the three-dimensional case with soft boundaries. As a further complication the free energy expression, which should be minimized, might contain some second-order and third-order derivatives. These higher-order derivatives will cause the spatial derivatives of the boundary normal to appear in the boundary equation. Explicit formulae are given for the Euler equation and the corresponding surface equations for a general case. As an example, the theory is applied to nematic liquid crystals, where the general Euler equations and surface molecular fields are derived, including the effects of an imposed electric field.