Analysis of a variational model for nematic shells

Abstract

We analyze an elastic surface energy which was recently introduced by G. Napoli and L. Vergori to model thin films of nematic liquid crystals. We show how a novel approach in modeling the surface’s extrinsic geometry leads to considerable differences with respect to the classical intrinsic energy. Our results concern three connected aspects: (i) using methods of the calculus of variations, we establish a relation between the existence of minimizers and the topology of the surface; (ii) we prove, by a Ginzburg–Landau approximation, the well-posedness of the gradient flow of the energy; (iii) in the case of a parametrized torus we obtain a stronger characterization of global and local minimizers, which we supplement with numerical experiments.