The Radial-Hedgehog Solution in Landau–de Gennes’ theory

We study the radial-hedgehog solution on a unit ball in three dimensions, with homeotropic boundary conditions, within the Landau-de Gennes theory for nematic liquid crystals. The radial-hedgehog solution is a candidate for a globally stable configuration in this model framework and is also a prototype configuration for studying isolated point defects in condensed matter physics. We use a combination of Ginzburg-Landau techniques, perturbation methods and stability analyses to study the qualitative properties of the radial-hedgehog solution, the structure of its defect core, its stability and instability with respect to biaxial perturbations. Our results complement previous work in the field, are rigorous in nature, give information about the role of geometry, elastic constants and temperature on the properties of the radial-hedgehog solution and the associated biaxial instabilities.