The investigator studies free energy functionals associated with liquid-crystalline systems and gradient flow evolutions (and more general dynamics) associated with these functionals. The study of critical points of the free energy functionals provides information about the equilibrium states of the corresponding systems, whereas the gradient flows provide insights into the out-of-equilibrium phenomena. In particular, the investigator applies the methods recently developed for the Onsager model of spatially homogeneous nematic liquid crystals to a more general setting. Employing a novel approach based on the combination of Oseen-Frank, Landau-de Gennes, and Onsager's theories, the investigator obtains an analytically tractable theory explaining the spatial variations of orientational ordering. Further on, the investigator exploits the fact that a particular case of two-dimensional nematics is closely related to complex Ginzburg-Landau-type models and the problem of harmonic mapping of a domain into a unit circle. This connection allows him to establish rigorous asymptotics for various patterns (twisters, vortices, domain walls, etc.,) arising in this particular setting using the tools from Ginzburg-Landau theory. The dynamical part of this project concentrates on the extent to which the kinetic equations for evolution of orientation probability densities and correlation functions (described via gradient flows in Wasserstein spaces of probability measures or more general mass-transportation dynamics) are reducible to the commonly used Ginsburg-Landau-type dynamics of the order parameters. A systematic analysis of such reductions is requisite for a complete understanding of dynamics of liquid-crystalline systems.
Liquid crystals are substances that share characteristics with conventional liquids and crystalline solids. Such an exotic combination of material properties allows for a high degree of dynamic control, which in turn allows for a wide range of useful applications. Liquid Crystal Display (LCD) technology is, undoubtedly, the prime example. Modern applications employ the fact that optical properties of liquid crystals change under external influences such as applied electric or magnetic fields, stresses, etc. By manipulating the latter in an appropriate manner one can achieve the desired effects. The theory of liquid crystals has been developing since the middle of twentieth century. Its success has been recognized by a Nobel Prize awarded in 1991 to P.-G. de Gennes for fundamental contributions to our understanding of order phenomena in complex systems. Still, due to enormous growth in applicability of liquid crystals and rapid development of the associated technologies, it has become even more important to go beyond the general principles and to further expand our understanding. Although computer simulations provide valuable insights, for precise control over liquid-crystalline materials it is essential to support the experimental findings and computer simulations with a rigorous mathematical theory. Development of such a theory is the primary goal of the investigator's work. His particular interest is mathematical analysis of patterns, their formation and evolution, and dynamics of phase transitions in liquid crystals.
