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$\mathbb S^1$-Valued Sobolev maps
P. Mironescu Université de Lyon, France
Abstract:
We describe the structure of the space
$W^{s,p}(\mathbb S^n;\mathbb S^1)$, where
$0<s<\infty$,
$1\le p<\infty$. 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].
UDC:
517.9