*Accepted Paper*

**Inserted:** 17 dec 2020

**Last Updated:** 17 dec 2020

**Journal:** Rendiconti Sem. Mat. Univ. Pol. Torino

**Year:** 2020

**Abstract:**

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 parameter $Q$ and its spatial derivatives. The order parameter $Q$ takes values into the set of $3\times 3$ real symmetric traceless matrices, the $Q$-tensors. The purpose of this review paper 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 describe 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, characterized by requiring the ``Lyuksyutov constraint" trace$(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.

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