\(H^{1/2}\) maps with values into the circle: minimal connections, lifting, and the Ginzburg-Landau equation
type: Item
instance of: scholarly article
title: \(H^{1/2}\) maps with values into the circle: minimal connections, lifting, and the Ginzburg-Landau equation
zbMATH Open document ID: 1051.49030
author: Petru Mironescu
author: Jean Bourgain
author: Haïm Brezis
published in: Publications Mathématiques
publication date: 2004-09-27T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/104206
review text: Let \(G\) be a smooth bounded domain in \({\mathbb R}^3\) with \(\Omega=\partial G\) simply connected. This paper deals with the study of the manifold \(H^{1/2}(\Omega ;S^1)=\{g\in H^{1/2}(\Omega ;{\mathbb R}^2)\); \(| g| =1\) a.e. on \(\Omega\}\). Using a density result of \textit{T. Rivière}, [Ann. Global Anal. Geom. 18, No. 5, 517--528 (2000; Zbl 0960.35022)], the authors prove that for any \(g\in H^{1/2}(\Omega;S^1)\), there exist two sequences \((P_i)\) and \((N_i)\) in \(\Omega\) such that \(\sum_i| P_i-N_i| <\infty\) and \(\langle T(g),\varphi\rangle =2\pi\sum_i(\varphi (P_i)-\varphi(N_i))\) for all \(\varphi\in Lip\,(\Omega;{\mathbb R})\), where \(T(g)\in{\mathcal D}'(\Omega;{\mathbb R})\) is the distribution associated to \(g\in H^{1/2}(\Omega;S^1)\) [and which describes the location and the topological degree of the singularities of \(g\)]. Moreover, if the distribution \(T(g)\) is a measure (of finite total mass), then \(T(g)=2\pi\sum_{\text{finite}}d_j\delta_{a_j}\), with \(d_j\in{\mathbb Z}\) and \(a_j\in\Omega\). Next, the authors establish a representation formula for functions belonging to \(Y:=\overline{C^\infty(\Omega;S^1)}^{H^{1/2}}\). More precisely, it is shown that for every \(g\in Y\) there exists \(\varphi\in H^{1/2}(\Omega;{\mathbb R})+W^{1,1}(\Omega;{\mathbb R})\), which is unique (modulo \(2\pi\)), such that \(g=e^{i\varphi}\). Conversely, if \(g\in H^{1/2}(\Omega;S^1)\) can be written as \(g=e^{i\varphi}\) with \(\varphi\in H^{1/2}(\Omega;{\mathbb R})+W^{1,1}(\Omega;{\mathbb R})\), then \(g\in Y\). The proof of this result relies on the notion of paraproduct, in the sense of J.-M. Bony and Y. Meyer. The heart of the matter is the estimate \[ \| \varphi\| _{H^{1/2}+W^{1,1}}\leq C_\Omega\| e^{i\varphi}\| _{H^{1/2}}(1+\| e^{i\varphi}\| _{H^{1/2}}), \] which holds for any smooth real-valued function \(\varphi\). The same estimate enables the authors to prove that for all \(g\in H^{1/2}(\Omega;S^1)\) there exists \(\varphi\in H^{1/2}(\Omega;{\…
reviewed by: Vicenţiu D. Rădulescu
Mathematics Subject Classification ID: 49Q20
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 46T05
Mathematics Subject Classification ID: 47H11
Mathematics Subject Classification ID: 49J35
Mathematics Subject Classification ID: 82D50
Mathematics Subject Classification ID: 82D55
Mathematics Subject Classification ID: 35J20
zbMATH DE Number: 2103608
zbMATH Keywords: Ginzburg-Landau equation
zbMATH Keywords: minimization problem
zbMATH Keywords: minimal connection
zbMATH Keywords: topological degree
zbMATH Keywords: lifting
MaRDI profile type: Publication
OpenAlex ID: W2066518895
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DOI: 10.1007/S10240-004-0019-5
description: scientific article; zbMATH DE number 2103608