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Durham University

Research & business

Research lectures, seminars and events

The events listed in this area are research seminars, workshops and lectures hosted by Durham University departments and research institutes. If you are not a member of the University, but  wish to enquire about attending one of the events please contact the organiser or host department.


Analysis and/of PDE: Ginzburg-Landau relaxation for harmonic maps valued into manifolds

Presented by Antonin Monteil, Bristol
28 November 2019 14:00 in zoom

We will look at the classical problem of minimizing the Dirichlet 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 condition $u=\gamma$ on $\partial\Omega$. It is well known that if $\gamma$ 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 $\partial\Omega$. 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 done 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 topological decomposition of the boundary map $\gamma$ into minimizing geodesics in $N$. Moreover, we will introduce a renormalized energy whose minimizers correspond to the optimal positions of the singularities in $\Omega$.

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