On Minimizers of a 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 epsilon > 0-we establish that, as epsilon -> 0, 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.