{"title":"三维第一个卷曲特征值的最佳凸域","authors":"A. Enciso, Wadim Gerner, D. Peralta-Salas","doi":"10.1090/tran/8914","DOIUrl":null,"url":null,"abstract":"<p>We prove that there exists a bounded convex domain <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Omega subset-of double-struck upper R cubed\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi mathvariant=\"normal\">Ω</mml:mi>\n <mml:mo>⊂</mml:mo>\n <mml:msup>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">R</mml:mi>\n </mml:mrow>\n <mml:mn>3</mml:mn>\n </mml:msup>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\Omega \\subset \\mathbb {R}^3</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> of fixed volume that minimizes the first positive curl eigenvalue among all other bounded convex domains of the same volume. We show that this optimal domain cannot be analytic, and that it cannot be stably convex if it is sufficiently smooth (e.g., of class <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C Superscript 1 comma 1\">\n <mml:semantics>\n <mml:msup>\n <mml:mi>C</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mn>1</mml:mn>\n <mml:mo>,</mml:mo>\n <mml:mn>1</mml:mn>\n </mml:mrow>\n </mml:msup>\n <mml:annotation encoding=\"application/x-tex\">C^{1,1}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>). Existence results for uniformly Hölder optimal domains in a box (that is, contained in a fixed bounded domain <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D subset-of double-struck upper R cubed\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>D</mml:mi>\n <mml:mo>⊂</mml:mo>\n <mml:msup>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">R</mml:mi>\n </mml:mrow>\n <mml:mn>3</mml:mn>\n </mml:msup>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">D\\subset \\mathbb {R}^3</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>) are also presented.</p>","PeriodicalId":23209,"journal":{"name":"Transactions of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Optimal convex domains for the first curl eigenvalue in dimension three\",\"authors\":\"A. Enciso, Wadim Gerner, D. Peralta-Salas\",\"doi\":\"10.1090/tran/8914\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We prove that there exists a bounded convex domain <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"normal upper Omega subset-of double-struck upper R cubed\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi mathvariant=\\\"normal\\\">Ω</mml:mi>\\n <mml:mo>⊂</mml:mo>\\n <mml:msup>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"double-struck\\\">R</mml:mi>\\n </mml:mrow>\\n <mml:mn>3</mml:mn>\\n </mml:msup>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\Omega \\\\subset \\\\mathbb {R}^3</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> of fixed volume that minimizes the first positive curl eigenvalue among all other bounded convex domains of the same volume. We show that this optimal domain cannot be analytic, and that it cannot be stably convex if it is sufficiently smooth (e.g., of class <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper C Superscript 1 comma 1\\\">\\n <mml:semantics>\\n <mml:msup>\\n <mml:mi>C</mml:mi>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mn>1</mml:mn>\\n <mml:mo>,</mml:mo>\\n <mml:mn>1</mml:mn>\\n </mml:mrow>\\n </mml:msup>\\n <mml:annotation encoding=\\\"application/x-tex\\\">C^{1,1}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>). Existence results for uniformly Hölder optimal domains in a box (that is, contained in a fixed bounded domain <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper D subset-of double-struck upper R cubed\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>D</mml:mi>\\n <mml:mo>⊂</mml:mo>\\n <mml:msup>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"double-struck\\\">R</mml:mi>\\n </mml:mrow>\\n <mml:mn>3</mml:mn>\\n </mml:msup>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">D\\\\subset \\\\mathbb {R}^3</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>) are also presented.</p>\",\"PeriodicalId\":23209,\"journal\":{\"name\":\"Transactions of the American Mathematical Society\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2024-04-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Transactions of the American Mathematical Society\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/tran/8914\",\"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":"Transactions of the American Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/tran/8914","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
我们证明存在一个固定体积的有界凸域 Ω ⊂ R 3 \Omega \subset \mathbb {R}^3，它在所有其他相同体积的有界凸域中最小化第一个正卷积特征值。我们证明了这个最优域不可能是解析的，而且如果它足够光滑（例如，类 C 1 , 1 C^{1,1} ），它就不可能是稳定凸的。我们还给出了盒中均匀霍尔德最优域（即包含在一个固定有界域 D ⊂ R 3 D\subset \mathbb {R}^3 中）的存在性结果。
Optimal convex domains for the first curl eigenvalue in dimension three
We prove that there exists a bounded convex domain Ω⊂R3\Omega \subset \mathbb {R}^3 of fixed volume that minimizes the first positive curl eigenvalue among all other bounded convex domains of the same volume. We show that this optimal domain cannot be analytic, and that it cannot be stably convex if it is sufficiently smooth (e.g., of class C1,1C^{1,1}). Existence results for uniformly Hölder optimal domains in a box (that is, contained in a fixed bounded domain D⊂R3D\subset \mathbb {R}^3) are also presented.
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