Uniform Estimates of Landau–de Gennes Minimizers in the Vanishing Elasticity Limit with Line Defects

Abstract.

For the Landau–de Gennes functional modeling nematic liquid crystals in dimension three, we prove that if the energy is bounded by [Formula: see text], then the sequence of minimizers [Formula: see text] is relatively compact in [Formula: see text] for every [Formula: see text]. This extends the classical compactness theorem of Bourgain, Brézis, and Mironescu [ Publ. Math., Inst. Hautes Étud. Sci., 99 (2004), pp. 1–115] for complex Ginzburg–Landau minimizers to the [Formula: see text]-valued Landau–de Gennes setting. Moreover, we obtain local bounds on the integral of the bulk energy potential that are uniform in [Formula: see text], improving the estimate that follows directly from the assumption.