Journal of Mathematical Sciences

, Volume 170, Issue 3, pp 340–355 | Cite as

S1-Valued Sobolev maps



We describe the structure of the space \( {W^{s,p}}\left( {{\mathbb{S}^n};{\mathbb{S}^1}} \right) \), where 0 < s < ∞ and 1 ≤ p < ∞. According to the values of s, p, and n, maps in \( {W^{s,p}}\left( {{\mathbb{S}^n};{\mathbb{S}^1}} \right) \) can either be characterised by their phases or by a couple (singular set, phase).

Here are two examples:
$$ \begin{array}{*{20}{c}} {{W^{{{1} \left/ {{2,6}} \right.}}}\left( {{\mathbb{S}^3};{\mathbb{S}^1}} \right) = \left\{ {{e^{\iota \varphi }}:\varphi \in {W^{{{1} \left/ {{2,6}} \right.}}} + {W^{1,3}}} \right\},} \\ {{W^{{{1} \left/ {{2,3}} \right.}}}\left( {{\mathbb{S}^2};{\mathbb{S}^1}} \right) \approx D \times \left\{ {{e^{\iota \varphi }}:\varphi \in {W^{{{1} \left/ {{2,3}} \right.}}} + {W^{{{{1,3}} \left/ {2} \right.}}}} \right\}.} \\ \end{array} $$

In the second example, D is an appropriate set of infinite sums of Dirac masses. The sense of ≈ 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].


Open Problem Sobolev Space Reverse Inclusion Landau Equation Dirac Masse 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer Science+Business Media, Inc. 2010

Authors and Affiliations

  1. 1.Université de LyonLyonFrance
  2. 2.Université Lyon 1LyonFrance
  3. 3.INSA de Lyon, F-69621LyonFrance
  4. 4.École Centrale de LyonLyonFrance
  5. 5.CNRS, UMR5208, Institut Camille Jordan43 blvd du 11 novembre 1918Villeurbanne-CedexFrance

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