| Title | Remarks on uniaxial solutions in the Landau-de Gennes theory |
| Authors | Majumdar, Apala Wang, Yiwei |
| Affiliation | Univ Bath, Dept Math Sci, Bath BA2 7AY, Avon, England. Peking Univ, Sch Math Sci, Beijing 100871, Peoples R China. |
| Keywords | Landau-de Gennes Uniaxial solutions Symmetric solutions NEMATIC LIQUID-CRYSTALS RADIAL-HEDGEHOG MINIMIZATION SYMMETRY MODELS |
| Issue Date | 2018 |
| Publisher | JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS |
| Citation | JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS. 2018, 464(1), 328-353. |
| Abstract | We study uniaxial solutions of the Euler-Lagrange equations for a Landau de Gennes free energy for nematic liquid crystals, with a fourth order bulk potential, with and without elastic anisotropy. These uniaxial solutions are characterised by a director and a scalar order parameter. In the elastic isotropic case, we show that (i) all uniaxial solutions, with a director field of a certain specified symmetry, necessarily have the radial-hedgehog structure modulo an orthogonal transformation, (ii) the "escape into third dimension" director cannot correspond to a purely uniaxial solution of the Landau-de Gennes Euler-Lagrange equations and we do not use artificial assumptions on the scalar order parameter and (iii) we use the structure of the Euler-Lagrange equations to exclude non-trivial uniaxial solutions with e(z) as a fixed eigenvector i.e. such uniaxial solutions necessarily have a constant eigenframe. In the elastic anisotropic case, we prove that all uniaxial solutions of the corresponding "anisotropic" Euler-Lagrange equations, with a certain specified symmetry, are strictly of the radial-hedgehog type, i.e. the elastic anisotropic case enforces the radial-hedgehog structure (or the degree +1-vortex structure) more strongly than the elastic isotropic case and the associated partial differential equations are technically far more difficult than in the elastic isotropic case. (C) 2018 Elsevier Inc. All rights reserved. |
| URI | http://hdl.handle.net/20.500.11897/518023 |
| ISSN | 0022-247X |
| DOI | 10.1016/j.jmaa.2018.04.003 |
| Indexed | SCI(E) |
| Appears in Collections: | 数学科学学院 |
