A REDUCED STUDY FOR NEMATIC EQUILIBRIA ON TWO-DIMENSIONAL POLYGONS

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 -> 0 limit, with a unique point defect at the center of the polygon for K not equal 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.