A new representation for the Landau-de Gennes energy of nematic liquid crystals
Abstract
In the Landau-de Gennes theory on nematic liquid crystals, the well-known Landau-de Gennes energy depends on four elastic constants; L1, L2, L3, L4. For the general case of L4≠ 0, Ball-Majumdar BM found an example that the Landau-de Gennes energy functional from physics literature 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 L4, proposed by Schiele and Trimper 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 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 Chen on the Ginzburg-Landau equation does not work for the Landau-de Gennes system.