Distances between classes in W 1,1 (Ω; S 1 ) Haim Brezis 1,3 , Petru Mironescu 2 and Itai Shafrir 3 1 Department of Mathematics, Rutgers University, USA 2 Universit´e de Lyon, Universit´e Lyon 1, CNRS UMR 5208, Institut Camille Jordan, 69622 Villeurbanne, France 3 Department of Mathematics, Technion - I.I.T., 32 000 Haifa, Israel February 5, 2022 Abstract We introduce an equivalence relation on the space W 1,1 (Ω; S 1 ) which classifies maps according to their “topological singularities”. We establish sharp bounds for the dis- tances (in the usual sense and in the Hausdorff sense) between the equivalence classes. Similar questions are examined for the space W 1,p (Ω; S 1 ) when p> 1. 1 Introduction Let Ω be a smooth bounded domain in R N , N ≥ 2. (Many of the results in this paper remain valid if Ω is replaced by a manifold M, with or without boundary, and the case M = S 1 is already of interest (see [13, 14]).) In some places we will assume in addition that Ω is simply connected (and this will be mentioned explicitly). Our basic setting is W 1,1 (Ω; S 1 )= {u ∈ W 1,1 (Ω; R 2 ) ’ W 1,1 (Ω; C); |u| = 1 a.e.}. It is clear that if u, v ∈ W 1,1 (Ω; S 1 ) then uv ∈ W 1,1 (Ω; S 1 ); moreover if u n → u and v n → v in W 1,1 (Ω; S 1 ) then u n v n → uv in W 1,1 (Ω; S 1 ). (1.1) In particular, W 1,1 (Ω; S 1 ) is a topological group. We call the attention of the reader that maps u of the form u = e ıϕ with ϕ ∈ W 1,1 (Ω; R) belong to W 1,1 (Ω; S 1 ). However they do not exhaust W 1,1 (Ω; S 1 ): there exist maps in W 1,1 (Ω; S 1 ) which cannot be written as u = e ıϕ for some ϕ ∈ W 1,1 (Ω; R). A typical example is the map u(x)= x/|x| in Ω =unit disc in R 2 ; This was originally observed in [4] (with roots in [30]) and is based on degree theory; see also [8, 11]. Set E = {u ∈ W 1,1 (Ω; S 1 ); u = e ıϕ for some ϕ ∈ W 1,1 (Ω; R)}. (1.2) We claim that E is closed in W 1,1 (Ω; S 1 ). Indeed, let u n = e ıϕn with u n → u in W 1,1 . Then ∇ϕ n = -ı u n ∇u n converges in L 1 to -ı u∇u. By adding an integer multiple of 2π to ϕ n we may assume that   ´ Ω ϕ n   ≤ 2π|Ω|. Thus, a subsequence of {ϕ n } converges in W 1,1 to some ϕ and u = e ıϕ . 2010 Mathematics Subject Classification. Primary 58D15; Secondary 46E35. 1 arXiv:1606.01526v3 [math.FA] 2 Oct 2017