Renormalized energies for unit-valued harmonic maps in multiply connected domains

In this article we derive the expression of renormalized energies for unit-valued harmonic maps defined on a smooth bounded domain in R 2 whose boundary has several connected components. The notion was introduced by Bethuel–Brezis–Hélein order to describe position limiting Ginzburg–Landau vortices simply domains. We show here, how non-trivial topology modifies energies. treat case Dirichlet conditions and Neumann as well.