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

Article type: Research Article

Authors: Rodiac, Rémy^{a; *} | Ubillús, Paúl^b

Affiliations: [a] CNRS, Laboratoire de mathématiques d’Orsay, Université Paris-Saclay, 91405, Orsay, France | [b] Institut de Recherche en Mathématique et Physique (IRMP), Universite catholique de Louvain, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium

Correspondence: [*] Corresponding author. E-mail: [email protected].

Abstract: In this article we derive the expression of renormalized energies for unit-valued harmonic maps defined on a smooth bounded domain in R2 whose boundary has several connected components. The notion of renormalized energies was introduced by Bethuel–Brezis–Hélein in order to describe the position of limiting Ginzburg–Landau vortices in simply connected domains. We show here, how a non-trivial topology of the domain modifies the expression of the renormalized energies. We treat the case of Dirichlet boundary conditions and Neumann boundary conditions as well.

Keywords: Renormalized energies, Harmonic maps, Ginzburg–Landau vortices

DOI: 10.3233/ASY-211712

Journal: Asymptotic Analysis, vol. 128, no. 3, pp. 413-444, 2022