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We investigate minimizers defined on a bounded domain in $\mathbb{R}^2$ forthe Maier--Saupe Q--tensor energy used to characterize nematic liquid crystalconfigurations. The energy density is singular, as in Ball and Mujamdar'smodification of the Ginzburg--Landau Q--tensor model, so as to constrain thecompeting states to have eigenvalues in the closure of a physically realisticrange. We prove that minimizers are regular and in several model problems weare able to use this regularity to prove that minimizers have eigenvaluesstrictly within the physical range.

Regularity and the Behavior of Eigenvalues for Minimizers of a Constrained $Q$-tensor Energy for Liquid Crystals