S¹-Valued Sobolev maps

Abstract

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].