{"title":"全维椭球的调和自映射无穷族","authors":"Volker Branding , Anna Siffert","doi":"10.1016/j.na.2025.113874","DOIUrl":null,"url":null,"abstract":"<div><div>We prove that for given <span><math><mrow><mi>k</mi><mo>∈</mo><mi>N</mi></mrow></math></span>, <span><math><mrow><mi>k</mi><mo>≥</mo><mn>3</mn></mrow></math></span>, <span><math><mrow><mi>d</mi><mo>∈</mo><mi>N</mi></mrow></math></span> and each <span><math><mrow><mi>a</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mo>∗</mo></mrow></msup></mrow></math></span> with <span><span><span><math><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup><mo><</mo><mn>4</mn><mi>d</mi><mrow><mo>(</mo><mi>d</mi><mo>+</mo><mi>k</mi><mo>−</mo><mn>2</mn><mo>)</mo></mrow><msup><mrow><mrow><mo>(</mo><mi>k</mi><mo>−</mo><mn>2</mn><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow></math></span></span></span>the ellipsoid <span><math><mrow><msub><mrow><mi>E</mi></mrow><mrow><mi>a</mi></mrow></msub><mo>≔</mo><mrow><mo>{</mo><mi>x</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>k</mi></mrow></msup><mspace></mspace><mo>|</mo><mspace></mspace><msup><mrow><mi>a</mi></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msup><msubsup><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><mo>…</mo><mo>+</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>=</mo><mn>1</mn><mo>}</mo></mrow></mrow></math></span> admits infinitely many harmonic self-maps.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"261 ","pages":"Article 113874"},"PeriodicalIF":1.3000,"publicationDate":"2025-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Infinite families of harmonic self-maps of ellipsoids in all dimensions\",\"authors\":\"Volker Branding , Anna Siffert\",\"doi\":\"10.1016/j.na.2025.113874\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>We prove that for given <span><math><mrow><mi>k</mi><mo>∈</mo><mi>N</mi></mrow></math></span>, <span><math><mrow><mi>k</mi><mo>≥</mo><mn>3</mn></mrow></math></span>, <span><math><mrow><mi>d</mi><mo>∈</mo><mi>N</mi></mrow></math></span> and each <span><math><mrow><mi>a</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mo>∗</mo></mrow></msup></mrow></math></span> with <span><span><span><math><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup><mo><</mo><mn>4</mn><mi>d</mi><mrow><mo>(</mo><mi>d</mi><mo>+</mo><mi>k</mi><mo>−</mo><mn>2</mn><mo>)</mo></mrow><msup><mrow><mrow><mo>(</mo><mi>k</mi><mo>−</mo><mn>2</mn><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow></math></span></span></span>the ellipsoid <span><math><mrow><msub><mrow><mi>E</mi></mrow><mrow><mi>a</mi></mrow></msub><mo>≔</mo><mrow><mo>{</mo><mi>x</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>k</mi></mrow></msup><mspace></mspace><mo>|</mo><mspace></mspace><msup><mrow><mi>a</mi></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msup><msubsup><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><mo>…</mo><mo>+</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>=</mo><mn>1</mn><mo>}</mo></mrow></mrow></math></span> admits infinitely many harmonic self-maps.</div></div>\",\"PeriodicalId\":49749,\"journal\":{\"name\":\"Nonlinear Analysis-Theory Methods & Applications\",\"volume\":\"261 \",\"pages\":\"Article 113874\"},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2025-07-02\",\"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/S0362546X25001282\",\"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/S0362546X25001282","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
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