Radial symmetry and biaxiality in nematic liquid crystals
We study the model problem of a nematic liquid crystal confined to a spherical droplet subject to radial anchoring conditions, in the context of the Landau-de Gennes continuum theory. Based on the recent radial symmetry result by Millot & Pisante (J. Eur. Math. Soc. 2010) and Pisante (J. Funct. Anal. 2011) for the vector-valued Ginzburg-Landau equations in three-dimensional superconductivity theory, we prove that global Landau-de Gennes minimizers in the class of uniaxial Q-tensors converge, in the low-temperature limit, to the radial-hegdehog solution of the tensor-valued Ginzburg-Landau equations. Combining this with the result by Majumdar (Eur. J. App. Math. 2012) and by Gartland & Mkaddem (Phys. Rev. E. 1999) that the radial-hedgehog equilibrium is unstable under biaxial perturbations, we obtain the non-purely uniaxial character of global minimizers for sufficiently low temperatures.