Remarks on uniaxial solutions in the Landau–de Gennes theory

Apala Majumdar, Yiwei Wang

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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.

Original languageEnglish
Pages (from-to)328-353
Number of pages26
JournalJournal of Mathematical Analysis and Applications
Issue number1
Early online date6 Apr 2018
Publication statusPublished - 1 Aug 2018


  • Landau–de Gennes
  • Symmetric solutions
  • Uniaxial solutions

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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