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Journal Articles Calculus of Variations and Partial Differential Equations Year : 2018

Distances between classes in $W^{1,1}(\Omega;{\mathbb S}^1)$

Abstract

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 $p>1$.
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Dates and versions

hal-01326786 , version 1 (05-06-2016)
hal-01326786 , version 2 (08-12-2017)

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Cite

Haim Brezis, Petru Mironescu, Itai Shafrir. Distances between classes in $W^{1,1}(\Omega;{\mathbb S}^1)$. Calculus of Variations and Partial Differential Equations, 2018, 57 (1), pp.57:14. ⟨10.1007/s00526-017-1280-z⟩. ⟨hal-01326786v2⟩
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