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UID:7107@i2m.univ-amu.fr
DTSTART;TZID=Europe/Paris:20190305T110000
DTEND;TZID=Europe/Paris:20190305T120000
DTSTAMP:20241120T203422Z
URL:https://www.i2m.univ-amu.fr/evenements/ginzburg-landau-relaxation-for-
 harmonic-maps-valued-into-manifolds/
SUMMARY: (...): Ginzburg-Landau relaxation for harmonic maps valued into ma
 nifolds
DESCRIPTION:: We will look at the classical problem of minimizing the Diric
 hlet energy of a map $u :\\Omega\\subset\\mathbb{R}^2\\to N$ valued into a
  compact Riemannian manifold $N$ and subjected to a Dirichlet boundary con
 dition $u=\\gamma$ on $\\partial\\Omega$. It is well known that if $\\gamm
 a$ has a non-trivial homotopy class in $N$\, then there are no maps in the
  critical Sobolev space $H^1(\\Omega\,N)$ such that $u=\\gamma$ on $\\part
 ial\\Omega$. To overcome this obstruction\, a way is to rather consider a 
 relaxed version of the Dirichlet energy leading to singular harmonic maps 
 with a finite number of topological singularities in $\\Omega$. This was d
 one in the 90's in a pioneering work by Bethuel-Brezis-Helein in the case 
 $N=\\mathbb{S}^1$\, related to the Ginzburg-Landau theory. In general\, we
  will see that minimizing the energy leads at main order to a non-trivial 
 combinatorial problem which consists in finding the energetically best top
 ological decomposition of the boundary map $\\gamma$ into minimizing geode
 sics in $N$. Moreover\, we will introduce a renormalized energy whose mini
 mizers correspond to the optimal positions of the singularities in $\\Omeg
 a$.http://perso.uclouvain.be/antonin.monteil/
CATEGORIES:Séminaire,Analyse Appliquée
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