关于柱面上能量最小化调和型映射的对称性

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2021-10-17 DOI:10.3934/mine.2023056

G. Fratta, A. Fiorenza, V. Slastikov

{"title":"关于柱面上能量最小化调和型映射的对称性","authors":"G. Fratta, A. Fiorenza, V. Slastikov","doi":"10.3934/mine.2023056","DOIUrl":null,"url":null,"abstract":"The paper concerns the analysis of global minimizers of a Dirichlet-type energy functional in the class of $ \\mathbb{S}^2 $-valued maps defined in cylindrical surfaces. The model naturally arises as a curved thin-film limit in the theories of nematic liquid crystals and micromagnetics. We show that minimal configurations are $ z $-invariant and that energy minimizers in the class of weakly axially symmetric competitors are, in fact, axially symmetric. Our main result is a family of sharp Poincaré-type inequality on the circular cylinder, which allows for establishing a nearly complete picture of the energy landscape. The presence of symmetry-breaking phenomena is highlighted and discussed. Finally, we provide a complete characterization of in-plane minimizers, which typically appear in numerical simulations for reasons we explain.","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2021-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"On symmetry of energy minimizing harmonic-type maps on cylindrical surfaces\",\"authors\":\"G. Fratta, A. Fiorenza, V. Slastikov\",\"doi\":\"10.3934/mine.2023056\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The paper concerns the analysis of global minimizers of a Dirichlet-type energy functional in the class of $ \\\\mathbb{S}^2 $-valued maps defined in cylindrical surfaces. The model naturally arises as a curved thin-film limit in the theories of nematic liquid crystals and micromagnetics. We show that minimal configurations are $ z $-invariant and that energy minimizers in the class of weakly axially symmetric competitors are, in fact, axially symmetric. Our main result is a family of sharp Poincaré-type inequality on the circular cylinder, which allows for establishing a nearly complete picture of the energy landscape. The presence of symmetry-breaking phenomena is highlighted and discussed. Finally, we provide a complete characterization of in-plane minimizers, which typically appear in numerical simulations for reasons we explain.\",\"PeriodicalId\":1,\"journal\":{\"name\":\"Accounts of Chemical Research\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":16.4000,\"publicationDate\":\"2021-10-17\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Accounts of Chemical Research\",\"FirstCategoryId\":\"5\",\"ListUrlMain\":\"https://doi.org/10.3934/mine.2023056\",\"RegionNum\":1,\"RegionCategory\":\"化学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"CHEMISTRY, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"5","ListUrlMain":"https://doi.org/10.3934/mine.2023056","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}

引用次数: 2

摘要

本文讨论了在柱面上定义的$ \mathbb{S}^2 $值映射类中的dirichlet型能量泛函的全局极小值分析。该模型作为向列液晶和微磁学理论中的弯曲薄膜极限而自然出现。我们证明了最小构型是$ z $不变的，并且弱轴对称竞争类中的能量最小值实际上是轴对称的。我们的主要结果是圆柱体上的一个尖锐的庞加莱姆齐式不等式族，它允许建立一个几乎完整的能源图景。强调并讨论了对称破缺现象的存在。最后，我们提供了平面内最小化器的完整表征，它通常出现在数值模拟中，原因我们解释了。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

On symmetry of energy minimizing harmonic-type maps on cylindrical surfaces

The paper concerns the analysis of global minimizers of a Dirichlet-type energy functional in the class of $ \mathbb{S}^2 $-valued maps defined in cylindrical surfaces. The model naturally arises as a curved thin-film limit in the theories of nematic liquid crystals and micromagnetics. We show that minimal configurations are $ z $-invariant and that energy minimizers in the class of weakly axially symmetric competitors are, in fact, axially symmetric. Our main result is a family of sharp Poincaré-type inequality on the circular cylinder, which allows for establishing a nearly complete picture of the energy landscape. The presence of symmetry-breaking phenomena is highlighted and discussed. Finally, we provide a complete characterization of in-plane minimizers, which typically appear in numerical simulations for reasons we explain.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.