Geodesic
$\mathbb S^1$-Valued Sobolev maps
Contemporary Mathematics. Fundamental Directions, Proceedings of the Fifth International Conference on Differential and Functional-Differential Equations (Moscow, August 17–24, 2008). Part 1, Tome 35 (2010), pp. 86-100.

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We describe the structure of the space $W^{s,p}(\mathbb S^n;\mathbb S^1)$, where $0$, $1\le p\infty$. According to the values of $s$, $p$ and $n$, maps in $W^{s,p}(\mathbb S^n;\mathbb S^1)$ can either be characterised by their phases or by a couple (singular set, phase). Here are two examples: $W^{1/2,6}(\mathbb S^3;\mathbb S^1)=\{e^{\imath\varphi}\colon\varphi\in W^{1/2,6}+W^{1,3}\}$, $W^{1/2,3}(\mathbb S^2;\mathbb S^1)\approx D\times\{e^{\imath\varphi}\colon\varphi\in W^{1/2,3}+W^{1,3/2}\}$. In the second example, $D$ is an appropriate set of infinite sums of Dirac masses. The sense of $\approx$ will be explained in the paper. The presentation is based on the papers of H.-M. Nguyen [22], of the author [20], and on a joint forthcoming paper of H. Brezis, H.-M. Nguyen, and the author [15]. 
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     author = {P. Mironescu},
     title = {$\mathbb S^1${-Valued} {Sobolev} maps},
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P. Mironescu. $\mathbb S^1$-Valued Sobolev maps. Contemporary Mathematics. Fundamental Directions, Proceedings of the Fifth International Conference on Differential and Functional-Differential Equations (Moscow, August 17–24, 2008). Part 1, Tome 35 (2010), pp. 86-100. http://geodesic.mathdoc.fr/item/CMFD_2010_35_a6/

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