A new representation for the Landau-de Gennes energy of nematic liquid crystals

In the Landau-de Gennes theory on nematic liquid crystals, the well-known Landau-de Gennes energy depends on four elastic constants; $L_1$, $L_2$, $L_3$, $L_4$. For the general case of $L_4\neq 0$, Ball-Majumdar \cite {BM} found an example that the Landau-de Gennes energy functional from physics literature \cite{MN} does not satisfy a coercivity condition, which causes a problem in mathematics to establish existence of energy minimizers. In order to solve this problem, we observe that the original third order term on $L_4$, proposed by Schiele and Trimper \cite{ST} in physics, is a linear combination of a fourth order term and a second order term. Therefore, we can propose a new Landau-de Gennes energy, which is equal to the original for uniaxial nematic $Q$-tensors. The new Landau-de Gennes energy with general elastic constants satisfies the coercivity condition for all $Q$-tensors, which establishes a new link between mathematical and physical theory. Similarly to the work of Majumdar-Zarnescu \cite{MZ}, we prove existence and convergence of minimizers of the new Landau-de Gennes energy. Moreover, we find a new way to study the limiting problem of the Landau-de Gennes system since the cross product method \cite{Chen} on the Ginzburg-Landau equation does not work for the Landau-de Gennes system.