Constrained Energy Minimization in Liquid Crystal Models
Liquid crystals are a distinct phase of matter existing between the chaos of isotropic liquids and the order of crystalline solids. In addition to being partially ordered, they manifest sensitivity to changes in temperature, concentration, or electric and magnetic fields. These properties combined render liquid crystals useful in various optical and biological applications. We present two models describing each of these types of applications. One is formed by bent-core molecules, a shape that endows the material with electric responsiveness. The second is composed of disc-like molecules that form rings when added to a solution, and these in turn aggregate into interesting geometrical shapes. An important question in both setups is how the dominant mechanism - switching in the first model and shape formation in the second - is affected by specific system parameters. We formulate both models as energy minimization problems allowing us to use several variational tools. We emphasize how we can deal with challenges that arise from constraints and nonlinearities. Our results address existence, uniqueness, and computation of solutions to the ensuing partial differential equations, which in tun shed light on the physical mechanisms observed.