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Journal Articles Journal of Mathematical Sciences Year : 2010

${\mathbb S}^1$-valued Sobolev maps

Abstract

We describe the structure of the space $W^{s,p}({\mathbb S}^n ; {\mathbb S}^1)$, where $0$<$s$<$\infty$ and $1\le p$<$\infty$. 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: a) $W^{1/2,6}({\mathbb S}^3 ; {\mathbb S}^1) = \{e^{\imath\varphi};\, \varphi\in W^{1/2,6} + W^{1,3}\}$; b) $W^{1/2,3}( {\mathbb S}^2 ; {\mathbb S}^1) \approx D \times \{e^{\imath \varphi} ;\, \varphi\in W^{1/2,3} + W^{1,3/2}\}$. 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 a paper of H.-M. Nguyen (C. R. Acad. Sci. Paris 2008), and on a forthcoming paper of the author.
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Dates and versions

hal-00356333 , version 1 (27-01-2009)

Identifiers

  • HAL Id : hal-00356333 , version 1

Cite

Petru Mironescu. ${\mathbb S}^1$-valued Sobolev maps. Journal of Mathematical Sciences, 2010, 170 (3), pp.340-355. ⟨hal-00356333⟩
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