We investigate minimizers on bounded two-dimensional domains for the Maier-Saupe energy used to characterize nematic liquid crystal configurations. The energy density is singular, as in Ball and Mujamdar's modification of the Landau-de Gennes Q-tensor model, so as to constrain the competing states to take values in the closure of a physically realistic range. We prove that minimizers are regular and in several model problems we use this regularity to prove that minimizers take on values strictly within the physical range.
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