Uniqueness of director configuration states for liquid crystals in the case of weak anchoring

In a thin layer of liquid crystal the configuration of the director field depends on the interaction between the elastic properties of the material, the thickness $d$ of the layer, the boundary conditions and the external fields that may have been applied. Suitable combinations of these factors can give rise to distorted configurations (Freedericksz transitions). In this paper we assume the Oseen-Frank model for the energy and that the director field depends only on the direction orthogonal to the layer; we assume also weak anchoring conditions at the two bounding surfaces, and we mainly study the problem of uniqueness of such distorted configurations. More precisely, we first consider the nematic case in the presence of a magnetic field $\mathbf H$, and we prove the uniqueness of the stable configuration provided the magnitude of $\mathbf H$ is between two critical thresholds, simplifying some results already known in the literature, and calculating explicitly the critical thresholds. Then we study the case of a cholesteric liquid crystal without external field. In this case the director field tends to form a right-angle helicoid around a twist axis orthogonal to the layer, and we have distorted configurations (namely oblique helicoid) for suitable value of $d$. Also in this case, with suitable restrictions on the elastic constants in the Oseen-Frank energy, we find two critical thresholds for $d$, and we prove the existence of only one stable director configuration if $d$ is between them.