scholarly journals Ginzburg–Landau Relaxation for Harmonic Maps on Planar Domains into a General Compact Vacuum Manifold

2021 ◽  
Author(s):  
Antonin Monteil ◽  
Rémy Rodiac ◽  
Jean Van Schaftingen
Keyword(s):  
Harmonic Maps ◽  
Ginzburg Landau ◽  
Planar Domains

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

2021 ◽  
pp. 1-32
Author(s):  
Rémy Rodiac ◽  
Paúl Ubillús
Keyword(s):  
Boundary Conditions ◽  
Harmonic Maps ◽  
Neumann Boundary ◽  
Dirichlet Boundary Conditions ◽  
Connected Components ◽  
Multiply Connected Domains ◽  
Smooth Bounded Domain ◽  
Ginzburg Landau ◽  
Simply Connected ◽  
Simply 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 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.

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