Liftings of Sobolev maps into closed Riemannian manifolds via double coverings and minimal connections relative to planar sets, with an application to ferronematics

We consider Sobolev maps from a planar domain into a closed Riemannian manifold and their BV liftings via a double covering of the target. We establish a sharp lower bound on the jump length of the lifting, expressed in terms of a geometric quantity: the minimal connection, relative to the domain, of the non-orientable singularities. As an application, we analyse minimisers of a two-dimensional model of ferronematics under ``mixed’’ boundary conditions – that is, Dirichlet conditions for the liquid crystal order parameter and Neumann conditions for the magnetisation vector.