We report on some recent progresses in the study of global minimisers of a continuum Landau-de Gennes energy for nematic liquid crystals in three dimensional domains. First, we show the absence of singularities under pointwise norm constraint for minimising configurations. Then, we will see that, under suitable assumptions on the topology of the domain and on the Dirichlet boundary conditions, the combination of smoothness and absence of isotropic phase yields the emergence of non-trivial topological structures in the biaxiality level sets. Next, we discuss the above properties under the additional constraint of axial symmetry. In this case, only a partial regularity result is available and minimisers are smooth except for a finite (possibly empty) singular set located on the symmetry axis. Nevertheless, the asymptotic behaviour around possible singularities is so precisely characterised that we can take advantage of axial symmetry to obtain a very refined description of the topology of biaxiality level sets for both smooth and singular minimisers. We remark that singularities may really appear in minimisers for reasons of energy efficiency (and they can be absent for the same reason). Finally, still assuming the norm and symmetry constraints and starting from the case of a nematic droplet (i.e., a ball) with the radial hedgehog boundary condition, we will discuss how singular or smooth minimisers or even both do appear as energy minimisers when deforming either the domain or the boundary data. As a consequence, we derive symmetry breaking results for the minimisation among all competitors.
Joint work with Vincent Millot (Paris Creteil) and Adriano Pisante (Sapienza).