Existence and uniqueness of minimizers for axisymmetric nematic films

Nematic surfaces are thin liquid films endowed with in-plane orientational order. We study a variational model in which the nematic director is constrained to lie in the tangent space of an axisymmetric surface, and the associated surface energy accounts for both surface tension and elastic nematic contributions. Here we adopt the surface gradient as the differential operator on the surface, we restrict our analysis to revolution surfaces spanning two coaxial rings, and we assume that the nematic director is aligned along parallels. In this setting, the energy functional reduces to a one-dimensional variational problem. We rigorously prove the existence and uniqueness of minimizers and we provide their complete geometric characterization. Finally, we run some numerical simulations.