Variational methods in the Landau-de Gennes theory of nematic liquid crystals

We study the so-called "biaxial torus solutions" in the Landau-de Gennes (LdG) model for nematic liquid crystals. These are smooth minimizers, so far only known through numerical experiments, of a widely-studied LdG energy functional w.r.t. a radial Dirichlet boundary datum, in conditions of "low temperature", with a very rich structure and a peculiar S^1-equivariant profile. We analytically prove that minimizing the energy in the class of S^1-equivariant Q-tensors of constant pointwise norm (which is, surprisingly, physically reasonable) agreeing the assigned boundary condition on the boundary gives back either biaxial torus solutions or the analogue of a special kind of singular solution, also expected from simulations. We then investigate the behavior of minimizers w.r.t. deformations of the boundary datum and, exploiting the very precise classification we got of the possible singularities, we also establish counterparts of some well-known theorems in harmonic maps. Joint work with Vincent Millot (Paris Diderot) and Adriano Pisante (Sapienza).