Remarks on uniaxial solutions in the Landau–de Gennes theory

Majumdar, Apala and Wang, Yiwei (2018) Remarks on uniaxial solutions in the Landau–de Gennes theory. Journal of Mathematical Analysis and Applications, 464 (1). pp. 328-353. ISSN 0022-247X (https://doi.org/10.1016/j.jmaa.2018.04.003)

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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 ez 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.

ORCID iDs

Majumdar, Apala ORCID: https://orcid.org/0000-0003-4802-6720

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  Item type:	Article
  ID code:	70831
  Dates:	Date Event 1 August 2018 Published 6 April 2018 Published Online 1 April 2018 Accepted
  Subjects:	Science > Mathematics
  Department:	Faculty of Science > Mathematics and Statistics
  Depositing user:	Pure Administrator
  Date deposited:	12 Dec 2019 10:12
  Last modified:	11 Nov 2024 12:32
  URI:	https://strathprints.strath.ac.uk/id/eprint/70831