{"title":"多连通域中单位调和映射的重正化能量","authors":"Rémy Rodiac, Pa'ul Ubill'us","doi":"10.3233/ASY-211712","DOIUrl":null,"url":null,"abstract":"In this article we derive the expression of \\textit{renormalized energies} for unit-valued harmonic maps defined on a smooth bounded domain in \\(\\mathbb{R}^2\\) whose boundary has several connected components. The notion of renormalized energies was introduced by Bethuel-Brezis-Helein in order to describe the position of limiting Ginzburg-Landau vortices in simply connected domains. We show here, how a non-trivial topology of the domain modifies the expression of the renormalized energies. We treat the case of Dirichlet boundary conditions and Neumann boundary conditions as well.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"40 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2020-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Renormalized energies for unit-valued harmonic maps in multiply connected domains\",\"authors\":\"Rémy Rodiac, Pa'ul Ubill'us\",\"doi\":\"10.3233/ASY-211712\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In this article we derive the expression of \\\\textit{renormalized energies} for unit-valued harmonic maps defined on a smooth bounded domain in \\\\(\\\\mathbb{R}^2\\\\) whose boundary has several connected components. The notion of renormalized energies was introduced by Bethuel-Brezis-Helein in order to describe the position of limiting Ginzburg-Landau vortices in simply connected domains. We show here, how a non-trivial topology of the domain modifies the expression of the renormalized energies. We treat the case of Dirichlet boundary conditions and Neumann boundary conditions as well.\",\"PeriodicalId\":8445,\"journal\":{\"name\":\"arXiv: Analysis of PDEs\",\"volume\":\"40 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2020-11-05\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv: Analysis of PDEs\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.3233/ASY-211712\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv: Analysis of PDEs","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3233/ASY-211712","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Renormalized energies for unit-valued harmonic maps in multiply connected domains
In this article we derive the expression of \textit{renormalized energies} for unit-valued harmonic maps defined on a smooth bounded domain in \(\mathbb{R}^2\) whose boundary has several connected components. The notion of renormalized energies was introduced by Bethuel-Brezis-Helein in order to describe the position of limiting Ginzburg-Landau vortices in simply connected domains. We show here, how a non-trivial topology of the domain modifies the expression of the renormalized energies. We treat the case of Dirichlet boundary conditions and Neumann boundary conditions as well.