Distances between classes in $W^{1,1}(\Omega;{\mathbb S}^1)$
Résumé
In the space $W^{1,1}(\Omega;{\mathbb S}^1)$, we introduce the equivalence relation $u\sim v$ iff $v=e^{\imath\varphi}\, u$ for some $\varphi\in W^{1,1}(\Omega ; {\mathbb R})$. This is a natural analog of the equivalence relation $f\sim g$ iff deg $f$ = deg $g$ for continuous maps $f, g : {\mathbb S}^N \to {\mathbb S}^N$. We determine the metric and Hausdorff distances between the equivalence classes. We also investigate the distances between the equivalence classes in $W^{1,p}(\Omega;{\mathbb S}^1)$ with $p>1$.
Origine | Fichiers produits par l'(les) auteur(s) |
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