Stability of point defects of degree \(\pm \frac{1}{2}\) in a two-dimensional nematic liquid crystal model. (English) Zbl 1353.82076
Summary: We study \(k\)-radially symmetric solutions corresponding to topological defects of charge \(\frac{k}{2}\) for integer \(k\neq 0\) in the Landau-de Gennes model describing liquid crystals in two-dimensional domains. We show that the solutions whose radial profiles satisfy a natural sign invariance are stable when \(|k| = 1\) (unlike the case \(|k| > 1\) which we treated before). The proof crucially uses the monotonicity of the suitable components, obtained by making use of the cooperative character of the system. A uniqueness result for the radial profiles is also established.
MSC:
| 82D30 | Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses) |
| 35A15 | Variational methods applied to PDEs |
| 35B38 | Critical points of functionals in context of PDEs (e.g., energy functionals) |
| 49J10 | Existence theories for free problems in two or more independent variables |
| 49J30 | Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.) |
| 76A15 | Liquid crystals |
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