On equilibrium configurations of nematic liquid crystals droplet with anisotropic elastic energy

We investigate the effect of anisotropic elastic energy on defect patterns of liquid crystals confined in a three-dimensional spherical domain within the framework of Landau–de Gennes model. Two typical strong anchoring boundary conditions, namely homeotropic and mirror-homeotropic anchoring conditions, are considered. For the homeotropic anchoring, we find three different configurations: uniaxial hedgehog, ring and split-core, in both cases with or without the anisotropic energy. For the mirror-homeotropic anchoring, there are also three analogue solutions: the uniaxial hyperbolic hedgehog, ring and split-core for the isotropic energy case. However, when the anisotropic energy is taken into account, the numerical results and rigorous analysis reveal that the uniaxial hyperbolic hedgehog is no longer a solution. Indeed, we find ring solution only for negative L2 (the elastic coefficient of the anisotropic energy), while both split-core and ring solutions can be stable minimizers for positive L2. More precisely, the uniaxial hyperbolic hedgehog for L2 = 0 bifurcates to a split-core solution when L2 increases and to a ring solution when L2 decreases. This example shows that the anisotropic energy may significantly affect the symmetry of point defects with degree−1 whenever it is introduced.