Remarks on Uniaxial Solutions in the Landau-de Gennes Theory

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. In the elastic isotropic case, we show that (i) all uniaxial solutions of the Euler-Lagrange equations, with a director field of certain 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) there are no non-trivial uniaxial solutions that have $\mathbf{e}_z$ as an eigenvector. In the elastic anisotropic case, we prove that all uniaxial solutions of the corresponding Euler-Lagrange equations, with a certain 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.