Uniform estimates of Landau-de Gennes minimizers in the vanishing elasticity limit with line defects

TL;DR

This paper establishes uniform regularity and compactness results for Landau-de Gennes minimizers modeling nematic liquid crystals, especially near line defects, extending classical results to a more complex setting with improved energy bounds.

Contribution

It extends the classical compactness theorem to the Landau-de Gennes model with line defects, providing uniform bounds and compactness in the vanishing elasticity limit.

Findings

Relatively compact minimizers in $W^{1,p}$ for $1<p<2$

Uniform local bounds on bulk energy potential

Extension of Bourgain-Brézis-Mironescu theorem to Landau-de Gennes setting

Abstract

For the Landau-de Gennes functional modeling nematic liquid crystals in dimension three, we prove that, if the energy is bounded by $C(\log\frac{1}{\varepsilon}+1)$, then the sequence of minimizers $\{\mathbf{Q}_{\varepsilon}\}_{\varepsilon\in (0,1)}$ is relatively compact in $W_{\operatorname{loc}}^{1,p}$ for every $1<p<2$. This extends the classical compactness theorem of Bourgain-Br\'{e}zis-Mironescu [Publ. Math., IH\'{E}S, 99:1-115, 2004] for complex Ginzburg-Landau minimizers to the $\mathbb R\mathbf P^2$-valued Landau-de Gennes setting. Moreover, We obtain local bounds on the integral of the bulk energy potential that are uniform in $ \varepsilon $, improving the estimate that follows directly from the assumption.