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Profile decomposition of unimodular maps Mironescu, Petru


We present the structure of a sequence $u_n: {\mathbb S}^1\to {\mathbb S}^1$ bounded in a critical space $W^{1/p,p}$, $1<p<\infty$. Up to a subsequence and to factors of the form $e^{i \varphi_n}$, with bounded $\varphi_n$, such sequences exhibit “profiles”, which are Moebius maps. We give applications to nonlocal critical variational problems, in particular the Ginzburg-Landau equations with semi-stiff boundary conditions. We also present other types of techniques useful in this setting, based e.g. on Wente type estimates or Dirichlet-to-Neumann operators. Part of the results we present were obtained with L.V. Berlyand, A. Farina, X. Lamy, V. Rybalko and É. Sandier.

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