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

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