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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 radialhedgehog 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 nontrivial 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 radialhedgehog type, i.e. the elastic anisotropic case enforces the radialhedgehog structure (or the degree +1vortex 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 language  English 

Pages (fromto)  328353 
Number of pages  26 
Journal  Journal of Mathematical Analysis and Applications 
Volume  464 
Issue number  1 
Early online date  6 Apr 2018 
DOIs  
Publication status  Published  1 Aug 2018 
Keywords
 Landau–de Gennes
 Symmetric solutions
 Uniaxial solutions
ASJC Scopus subject areas
 Analysis
 Applied Mathematics
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Dive into the research topics of 'Remarks on uniaxial solutions in the Landau–de Gennes theory'. Together they form a unique fingerprint.Projects
 1 Finished

Fellowship  The Mathematics of Liquid Crystals: Analysis, Computation and Applications
Majumdar, A.
Engineering and Physical Sciences Research Council
1/08/12 → 30/09/16
Project: Research council