Mathematical analysis of chevron structures in liquid crystal films

This dissertation focuses on the mathematical analysis of a model for the chevron structure arising from the cooling from the Smectic-A liquid crystal to the chiral Smectic-C phase in a surface-stabilized cell with certain boundary conditions under a given electric field. This phenomenon causes severe defects in liquid crystal display devices and has attracted interest from both theoretical and practical point of view. This work consists of two parts - the investigations of static and dynamic features for chevron structure. In the static analysis, the stability of the chevron structure is established through a sequence of minimization problems converging to a reduced energy functional based on Chen-Lubensky model. This study establishes the existence of minimizers and provides a detailed analysis of the minimal energy configuration in the limiting case. When the external electric field is switched to the opposite direction, the chevron structure responds to this change due to a spontaneous polarization field induced by chirality. The dynamic analysis investigates how the system evolves over time under the switching of the electric field and establishes the existence and uniqueness of the continuous gradient flow for the Chen-Lubensky energy. A discrete in time model is constructed to study this time-dependent problem.