Stability of non-constant equilibrium states in a finite sample of ferroelectric liquid crystals

The non-constant equilibrium solutions to a dynamic equation governing the director reorientation within a finite sample of ferroelectric liquid crystal are obtained in terms of Jacobian elliptic integrals of the first kind. The stability results for these solutions, for various values of the measurable parameters of the problem, depend upon proving the positivity of the eigenvalues of a regular Sturm-Liouville problem which arises from applying a perturbation analysis to such equilibrium solutions in a quite general setting. A Hardy-type integral inequality is further employed to prove the positivity of eigenvalues in a more restricted class of perturbations which allows improved results on the ranges of parameters for which stability holds.