A reduced study for nematic equilibria on two-dimensional polygons

Han, Yucen and Majumdar, Apala and Zhang, Lei (2020) A reduced study for nematic equilibria on two-dimensional polygons. SIAM Journal of Applied Mathematics, 80 (4). pp. 1678-1703.

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    Abstract

    We study reduced nematic equilibria on regular two-dimensional polygons with Dirichlet tangent boundary conditions in a reduced two-dimensional Landau--de Gennes framework, discussing their relevance in the full three-dimensional framework too. We work at a fixed temperature and study the reduced stable equilibria in terms of the edge length, $\lambda$, of the regular polygon, $E_K$, with $K$ edges. We analytically compute a novel “ring solution” in the $\lambda \to 0$ limit, with a unique point defect at the center of the polygon for $K \neq 4$. The ring solution is unique. For sufficiently large $\lambda$, we deduce the existence of at least $[K/2 ]$ classes of stable equilibria and numerically compute bifurcation diagrams for reduced equilibria on a pentagon and hexagon, as a function of $\lambda^2$, thus illustrating the effects of geometry on the structure, locations, and dimensionality of defects in this framework.