Convexity criteria for expansions of the Landau-de Gennes elastic free energy

Abstract

The Landau-de Gennes elastic free energy of liquid crystals is a functional expressed in terms of a symmetric traceless tensor order parameter. We derive convexity criteria for SO(3)-invariant expansions of the elastic energy that are quadratic in the gradient and up to fourth-order overall in the tensor order parameter. To do so we write the integrand as the sum of squares of linearly independent spherical tensors from which definiteness criteria can be ascertained. In particular, transformations between Cartesian and spherical tensors representations are computed based on irreducible tensor representations of the rotation group.