Existence of minimizers and convergence of critical points for a new Landau-de Gennes energy functional in nematic liquid crystals

TL;DR

This paper introduces a new Landau-de Gennes energy functional for nematic liquid crystals that ensures coercivity, proves convergence of solutions to the Oseen-Frank system, and generalizes existing convergence results to cases with non-zero elastic constants.

Contribution

It proposes a modified energy density satisfying coercivity for all Q-tensors and extends convergence analysis to more general elastic constants.

Findings

New energy density is equivalent to the original for uniaxial tensors.

Solutions of the Landau-de Gennes system can approach the Q-tensor Oseen-Frank system.

Generalized convergence results to cases with non-zero elastic constants.

Abstract

The Landau-de Gennes energy in nematic liquid crystals depends on four elastic constants $L_1$, $L_2$, $L_3$, $L_4$. In the case of $L_4\neq 0$, Ball and Majumdar (Mol. Cryst. Liq. Cryst., 2010) found an example that the original Landau-de Gennes energy functional in physics does not satisfy a coercivity condition, which causes a problem in mathematics to establish existence of energy minimizers. At first, we introduce a new Landau-de Gennes energy density with $L_4\neq 0$, which is equivalent to the original Landau-de Gennes density for uniaxial tensors and satisfies the coercivity condition for all $Q$-tensors. Secondly, we prove that solutions of the Landau-de Gennes system can approach a solution of the $Q$-tensor Oseen-Frank system without using energy minimizers. Thirdly, we develop a new approach to generalize the Nguyen and Zarnescu (Calc. Var. PDEs, 2013) convergence result to the case of non-zero elastic constants $L_2$, $L_3$, $L_4$.