Local minimizer and De Giorgi’s type conjecture for the isotropic–nematic interface problem

Abstract

In this paper, we investigate the structure of local minimizers for the isotropic–nematic interface based on the Landau-de Gennes energy. In the absence of the anisotropic energy, the uniaxial solution is the only local minimizer in 1-D. In 3-D, we propose a De Giorgi’s type conjecture and give an affirmative answer under a mild assumption. In the presence of the anisotropic energy with \(L_2>-\,1\) and homeotropic anchoring, the uniaxial solution is also the only local minimizer in a class of diagonal form in 1-D.