Ericksen's type inequalities for constrained Q-tensor models of nematic liquid crystals

This paper considers the four-elastic-constant Landau-de Gennes free-energy which characterizes nematic liquid crystal configurations in the framework of Q-tensor theory. The density for the Landau–de Gennes energy functional involves the tensor order pa- rameter Q and its spatial detivatives. The order parameter Q takes values into the set of 3 × 3 real symmetric traceless matrices, the Q-tensors. The purpose of this survey article is to give an account of the general conditions on the elastic constants which guarantee the coercivity of the free-energy density, and hence the internal consistency of the theory, in the constrained (hard) and soft Landau–de Gennes regimes. This generalizes the well-known Ericksen inequalities among the elastic constants in the classical Oseen–Frank expansion of the free-energy density. We start by recalling some background material about the Q-tensor theory of nematic liquid crystals and describing the related order parameter spaces. Next, we consider the constrained (hard) theory of uniaxial and biaxial nematic liquid crystals. We de- scribe the geometric features of the corresponding Q-tensor models, providing the Cartesian expression of the elastic invariants, and discuss coercivity conditions and existence results of the minima for the free-energy. We then address the soft theory of biaxial nematics, charac- terized by requiring the "Lyuksyutov constraint" tr(Q^2) = const. We describe the Q-tensor model for soft biaxial nematic systems and exploit the geometry of the model and the frame- indifference of the energy density to discuss the question of coercivity of the free-energy density.