In this talk, we make quantitative comparisons between two widely-used liquid crystal modelling approaches - the continuum Landau-de Gennes theory and mesoscopic mean-field theories, such as the Maier-Saupe and Onsager theories. We use maximum principle arguments for elliptic partial differential equations to compute explicit bounds for the norm of static equilibria within the Landau-de Gennes framework. These bounds yield an explicit prescription of the temperature regime within which the LdG and the mean-field predictions are consistent, for both spatially homogeneous and inhomogeneous systems. We find that the Landau-de Gennes theory can make physically unrealistic predictions in the low-temperature regime. In my joint work with John Ball, we formulate a new theory that interpolates between mean-field and continuum approaches and remedies the deficiencies of the Landau-de Gennes theory in the low-temperature regime. In particular, we define a new thermotropic potential that blows up whenever the mean-field constraints are violated. The main novelty of this work is the incorporation of spatial inhomogeneities (outside the scope of mean-field theory) along with retention of mean-field level information.
