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

TL;DR

This paper derives the formula for renormalized energies of harmonic maps with unit value in multiply connected domains, revealing how domain topology influences vortex behavior under different boundary conditions.

Contribution

It extends the concept of renormalized energies to multiply connected domains, accounting for topological effects on harmonic maps with boundary conditions.

Findings

Renormalized energies are modified by domain topology.

Explicit formulas for Dirichlet and Neumann boundary conditions.

Topological complexity affects vortex configurations.

Abstract

In this article we derive the expression of \textit{renormalized energies} for unit-valued harmonic maps defined on a smooth bounded domain in \(\mathbb{R}^2\) whose boundary has several connected components. The notion of renormalized energies was introduced by Bethuel-Brezis-H\'elein 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.