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Local minimizer and de Giorgi’s type conjecture for the isotropic-nematic interface problem. (English) Zbl 1400.82287

Summary: 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.

MSC:

82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
35J47 Second-order elliptic systems
35J61 Semilinear elliptic equations
82B24 Interface problems; diffusion-limited aggregation arising in equilibrium statistical mechanics
82B26 Phase transitions (general) in equilibrium statistical mechanics
82B27 Critical phenomena in equilibrium statistical mechanics
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