{
  "id": "1910.05740",
  "version": "v1",
  "published": "2019-10-13T12:35:59.000Z",
  "updated": "2019-10-13T12:35:59.000Z",
  "title": "A Reduced Study for Nematic Equilibria on Two-Dimensional Polygons",
  "authors": [
    "Yucen Han",
    "Apala Majumdar",
    "Lei Zhang"
  ],
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "We study reduced nematic equilibria on regular two-dimensional polygons with Dirichlet tangent boundary conditions, in a reduced two-dimensional 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 centre of the polygon for $K \\neq 4$. The ring solution is unique. For sufficiently large $\\lambda$, we deduce the existence of at least $\\left[K/2 \\right]$ 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-13T12:35:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "reduced study",
      "dirichlet tangent boundary conditions",
      "study reduced nematic equilibria",
      "regular two-dimensional polygons",
      "full three-dimensional framework"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}