On Minimizers of the Landau-de Gennes Energy Functional on Planar Domains

We study tensor-valued minimizers of the Landau-de Gennes energy functional on a simply-connected planar domain $\Omega$ with non-contractible boundary data. Here the tensorial field represents the second moment of a local orientational distribution of rod-like molecules of a nematic liquid crystal. Under the assumption that the energy depends on a single parameter---a dimensionless elastic constant $\eps>0$---we establish that, as $\eps\to0$, the minimizers converge to a projection-valued map that minimizes the Dirichlet integral away from a single point in $\Omega$. We also provide a description of the limiting map.