Uniaxial versus Biaxial Character of Landau-de Gennes Minimizers in Three Dimensions

Abstract: We study global minimizers of the Landau-de Gennes (LdG) energy functional for nematic liquid crystals, on arbitrary three-dimensional simply connected geometries with topologically non-trivial and physically relevant Dirichlet boundary conditions. The nematic configuration is described by the macroscopic LdG Q-tensor, which is a symmetric traceless three by three matrix and the nematic state is uniaxial if the Q-tensor has two degenerate eigenvalues and is maximally biaxial if the corresponding Q-tensor has a vanishing eigenvalue. Our results are specific to an asymptotic limit defined in terms of a re-scaled reduced temperature, t. We prove (i) that (re-scaled) global LdG minimizers converge uniformly to a (minimizing) limiting harmonic map, away from the singular set of the limiting map; (ii) we have points of maximal biaxiality and uniaxiality near each singular point of the limiting map (this improves recent results of Contreras and Lamy) which gives some qualitative insight into defect cores; (iii) estimates for the size of “strongly biaxial” regions in terms of the reduced temperature t. This is joint work with Duvan Henao and Adriano Pisante.