{"title":"环面间 j 度映射的 Dirichlet 能量最小化","authors":"","doi":"10.1016/j.na.2024.113671","DOIUrl":null,"url":null,"abstract":"<div><div>Let <span><math><mi>A</mi></math></span> and <span><math><msub><mrow><mi>A</mi></mrow><mrow><mo>∗</mo></mrow></msub></math></span> be circular annuli in the complex plane, and consider the Dirichlet energy integral of <span><math><mi>j</mi></math></span>-degree mappings between <span><math><mi>A</mi></math></span> and <span><math><msub><mrow><mi>A</mi></mrow><mrow><mo>∗</mo></mrow></msub></math></span>. We aim to minimize this energy integral. The minimizer is a <span><math><mi>j</mi></math></span>-degree harmonic mapping between the annuli <span><math><mi>A</mi></math></span> and <span><math><msub><mrow><mi>A</mi></mrow><mrow><mo>∗</mo></mrow></msub></math></span>, provided it exists. If such a harmonic mapping does not exist, then the minimizer is still a <span><math><mi>j</mi></math></span>-degree mapping which is harmonic in <span><math><mrow><msup><mrow><mi>A</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>⊂</mo><mi>A</mi></mrow></math></span>, and it is a squeezing mapping in its complementary annulus <span><math><mrow><msup><mrow><mi>A</mi></mrow><mrow><mo>′</mo><mo>′</mo></mrow></msup><mo>=</mo><mi>A</mi><mo>∖</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></math></span>. This result is an extension of a certain result by Astala et al. (2010).</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Minimization of Dirichlet energy of j−degree mappings between annuli\",\"authors\":\"\",\"doi\":\"10.1016/j.na.2024.113671\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>Let <span><math><mi>A</mi></math></span> and <span><math><msub><mrow><mi>A</mi></mrow><mrow><mo>∗</mo></mrow></msub></math></span> be circular annuli in the complex plane, and consider the Dirichlet energy integral of <span><math><mi>j</mi></math></span>-degree mappings between <span><math><mi>A</mi></math></span> and <span><math><msub><mrow><mi>A</mi></mrow><mrow><mo>∗</mo></mrow></msub></math></span>. We aim to minimize this energy integral. The minimizer is a <span><math><mi>j</mi></math></span>-degree harmonic mapping between the annuli <span><math><mi>A</mi></math></span> and <span><math><msub><mrow><mi>A</mi></mrow><mrow><mo>∗</mo></mrow></msub></math></span>, provided it exists. If such a harmonic mapping does not exist, then the minimizer is still a <span><math><mi>j</mi></math></span>-degree mapping which is harmonic in <span><math><mrow><msup><mrow><mi>A</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>⊂</mo><mi>A</mi></mrow></math></span>, and it is a squeezing mapping in its complementary annulus <span><math><mrow><msup><mrow><mi>A</mi></mrow><mrow><mo>′</mo><mo>′</mo></mrow></msup><mo>=</mo><mi>A</mi><mo>∖</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></math></span>. This result is an extension of a certain result by Astala et al. (2010).</div></div>\",\"PeriodicalId\":49749,\"journal\":{\"name\":\"Nonlinear Analysis-Theory Methods & Applications\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2024-10-09\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Nonlinear Analysis-Theory Methods & Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0362546X24001901\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Nonlinear Analysis-Theory Methods & Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0362546X24001901","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
假设 A 和 A∗ 是复平面上的圆形环面,并考虑 A 和 A∗ 之间 j 阶映射的 Dirichlet 能量积分。我们的目标是最小化这个能量积分。最小值是环面 A 和 A∗ 之间的 j 度谐波映射,前提是它存在。如果不存在这样的调和映射,那么最小化映射仍然是一个在 A′⊂A 中调和的 j 度映射,并且是其互补环面 A′′=A∖A′ 中的挤压映射。这一结果是对阿斯塔拉等人(2010)的某个结果的扩展。
Minimization of Dirichlet energy of j−degree mappings between annuli
Let and be circular annuli in the complex plane, and consider the Dirichlet energy integral of -degree mappings between and . We aim to minimize this energy integral. The minimizer is a -degree harmonic mapping between the annuli and , provided it exists. If such a harmonic mapping does not exist, then the minimizer is still a -degree mapping which is harmonic in , and it is a squeezing mapping in its complementary annulus . This result is an extension of a certain result by Astala et al. (2010).
期刊介绍:
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