  We study nematic liquid crystalline films within the framework of the
Landau-de Gennes theory in the limit when both the thickness of the film and
the nematic correlation length are vanishingly small compared to the lateral
extent of the film. We prove $\Gamma$-convergence for a sequence of singularly
perturbed functionals with a potential vanishing on a high-dimensional set and
a Dirichlet condition imposed on admissible functions. This then allows us to
prove the existence of local minimizers of the Landau-de Gennes energy in the
spirit of a theorem due to Kohn and Sternberg despite the lack of compactness
arising from the high-dimensional structure of the wells. The limiting energy
consists of leading order perimeter terms, similar to Allen-Cahn models, and
lower order terms arising from vortex structures reminiscent of Ginzburg-Landau
models.
