Renormalised energies and renormalisable singular harmonic maps into a compact manifold on planar domains

We define renormalised energies for maps that describe the first-order asymptotics of harmonic maps outside of singularities arising due to obstructions generated by the boundary data and the mutliple connectedness of the target manifold. The constructions generalise the definition by Bethuel, Brezis and H\'elein for the circle (Ginzburg-Landau vortices, 1994). In general, the singularities are geometrical objects and the dependence on homotopic singularities can be studied through a new notion of synharmony. The renormalised energies are showed to be coercive and Lipschitz-continuous. The renormalised energies are associated to minimising renormalisable singular harmonic maps and minimising configurations of points can be characterised by the flux of the stress-energy tensor at the singularities. We compute the singular energy and the renormalised energy in several particular cases.