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

Apala Majumdar (Bath, UK)

We study global minimizers of the Landau-de Gennes energy functional for nematic liquid crystals, on arbitrary three-dimensional simply connected geometries with topologically non-trivial and physically relevant Dirichlet boundary conditions. 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); (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.

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