On the structure of the Sobolev space H1/2 with values into the circle

Jean Bourgain; Haïm Brezis; Petru Mironescu

doi:10.1016/S0764-4442(00)00513-9

On the structure of the Sobolev space H^1/2 with values into the circle

Jean Bourgain, Haïm Brezis, Petru Mironescu

Research output: Contribution to journal › Article › peer-review

34 Scopus citations

Abstract

We are concerned with properties of H^1/2(Ω; S¹) where Ω is the boundary of a domain in ℝ³. To every u ∈ H^1/2(Ω; S¹) we associate a distribution T(u) which, in some sense, describes the location and the topological degree of singularities of u. The closure Y of C^∞(Ω; S¹) in H^1/2 coincides with the u's such that T(u) = 0. Moreover, every u ∈ Y admits a unique (mod. 2π) lifting in H^1/2 + W^1,1. We also discuss an application to the 3-d Ginzburg-Landau problem.

Original language	English (US)
Pages (from-to)	119-124
Number of pages	6
Journal	Comptes Rendus de l'Academie des Sciences - Series I: Mathematics
Volume	331
Issue number	2
DOIs	https://doi.org/10.1016/S0764-4442(00)00513-9
State	Published - Jul 15 2000
Externally published	Yes

All Science Journal Classification (ASJC) codes

Mathematics(all)

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title = "On the structure of the Sobolev space H1/2 with values into the circle",

abstract = "We are concerned with properties of H1/2(Ω; S1) where Ω is the boundary of a domain in ℝ3. To every u ∈ H1/2(Ω; S1) we associate a distribution T(u) which, in some sense, describes the location and the topological degree of singularities of u. The closure Y of C∞(Ω; S1) in H1/2 coincides with the u's such that T(u) = 0. Moreover, every u ∈ Y admits a unique (mod. 2π) lifting in H1/2 + W1,1. We also discuss an application to the 3-d Ginzburg-Landau problem.",

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AU - Brezis, Haïm

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N2 - We are concerned with properties of H1/2(Ω; S1) where Ω is the boundary of a domain in ℝ3. To every u ∈ H1/2(Ω; S1) we associate a distribution T(u) which, in some sense, describes the location and the topological degree of singularities of u. The closure Y of C∞(Ω; S1) in H1/2 coincides with the u's such that T(u) = 0. Moreover, every u ∈ Y admits a unique (mod. 2π) lifting in H1/2 + W1,1. We also discuss an application to the 3-d Ginzburg-Landau problem.

AB - We are concerned with properties of H1/2(Ω; S1) where Ω is the boundary of a domain in ℝ3. To every u ∈ H1/2(Ω; S1) we associate a distribution T(u) which, in some sense, describes the location and the topological degree of singularities of u. The closure Y of C∞(Ω; S1) in H1/2 coincides with the u's such that T(u) = 0. Moreover, every u ∈ Y admits a unique (mod. 2π) lifting in H1/2 + W1,1. We also discuss an application to the 3-d Ginzburg-Landau problem.

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