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

Abstract

AbstractThe Landau-de Gennes energy in nematic liquid crystals depends on four elastic constants $$L_1$$ L 1 , $$L_2$$ L 2 , $$L_3$$ L 3 , $$L_4$$ L 4 . In the case of $$L_4\ne 0$$ L 4 ≠ 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 $$L_4\ne 0$$ L 4 ≠ 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 2 , $$L_3$$ L 3 , $$L_4$$ L 4 .