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

TL;DR

This paper develops a method to lift Sobolev maps into Riemannian manifolds using double coverings, establishing bounds related to singularities, and applies this to analyze minimizers in a ferronematics model with mixed boundary conditions.

Contribution

It introduces a new approach to lift Sobolev maps via double coverings, providing sharp bounds on jump lengths and applying these results to a ferronematics variational problem.

Findings

Established a sharp lower bound on the jump length of the lifting.

Analyzed minimizers of a ferronematics model with mixed boundary conditions.

Connected geometric quantities like minimal connection to the analysis of singularities.

Abstract

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.