ON HIGHER-ORDER VARIATIONAL ANALYSIS IN ONE-DIMENSIONS AND 3-DIMENSIONS FOR SOFT BOUNDARIES

For some problems in liquid crystal physics we need to use the Euler e quation and the corresponding boundary equation in the three-dimension al case with soft boundaries. As a further complication the free energ y expression, which should be minimized, might contain some second-ord er and third-order derivatives. These higher-order derivatives will ca use the spatial derivatives of the boundary normal to appear in the bo undary equation. Explicit formulae are given for the Euler equation an d the corresponding surface equations for a general case. As an exampl e, the theory is applied to nematic liquid crystals, where the general Euler equations and surface molecular fields are derived, including t he effects of an imposed electric field.