A reduced study for nematic equilibria on two-dimensional polygons

Yucen Han, Apala Majumdar, Lei Zhang

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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.
Original languageEnglish
Pages (from-to)1678-1703
Number of pages26
JournalSIAM Journal of Applied Mathematics
Volume80
Issue number4
Early online date16 Jul 2020
DOIs
Publication statusE-pub ahead of print - 16 Jul 2020

Keywords

  • reduced nematic equilibria
  • two-dimensional polygons
  • Dirichlet tangent boundary conditions

Cite this

Han, Y., Majumdar, A., & Zhang, L. (2020). A reduced study for nematic equilibria on two-dimensional polygons. SIAM Journal of Applied Mathematics, 80(4), 1678-1703. https://doi.org/10.1137/19M1293156