*Apala Majumdar*

*Professor*

Department of Mathematical Sciences

University of Bath

Nematics liquid crystals are anisotropic liquids with long-range orientational ordering, making them popular working materials for optical applications. The study of static nematic equilibria poses challenging questions in the calculus of variations and theory of partial differential equations. We study two stability problems for the prototypical radial-hedgehog solution within the Landau-de Gennes theory for nematics. The radial-hedgehog solution is an example of a uniaxial nematic equilibrium with an isotropic defect core, analogous to a degree +1-vortex solution in the Ginzburg-Landau theory of superconductivity. The first problem concerns the radial-hedgehog solution in a spherical droplet with radial boundary conditions, for low temperatures below the nematic-isotropic transition temperature.

We prove that an arbitrary sequence of Landau-de Gennes minimizers converges strongly (in W^{1,2}) to the radial-hedgehog solution in the low-temperature limit. We use the celebrated division trick for superconductivity, blow-up techniques for the singularity profile and energy estimates to show that the radial-hedgehog solution is the unique physically relevant uniaxial equilibrium in the low-temperature limit. We then compute the second variation of the Landau-de Gennes energy about the radial-hedgehog solution and demonstrate its instability with respect to higher-dimensional biaxial perturbations, for sufficiently low temperatures. We conclude that Landau-de Gennes minimizers on a spherical droplet, with radial anchoring, are always biaxial for sufficiently low temperatures.

The second problem concerns a punctured spherical droplet with radial boundary conditions. We show that the radial-hedgehog solution is locally stable for all temperatures below the nematic-isotropic transition temperature on a punctured droplet. We adapt methods from [1], [2] and use convexity-based properties of the Landau-de Gennes energy to prove that the radial-hedgehog solution is, in fact, the unique global energy minimizer in two different asymptotic limits: the vanishing elastic constant limit and the low-temperature limit, in contrast to the instability result for a spherical droplet above.

This is joint work with Duvan Henao, Adriano Pisante and Mythily Ramaswamy.

[1] A. Majumdar, A. Zarnescu, Landau-de Gennes theory of nematic liquid crystals: the Oseen-Frank limit and beyond, Arch. Rat. Mech. Anal., 196 (2010), no. 1, 227-280.

[2] F.H.Lin and C.Liu, Static and Dynamic Theories of Liquid Crystals, Journal of Partial Differential Equations, Vol 14, No. 4, 289-330 (2001).