关于平面凸域中诺伊曼问题的波利亚猜想

IF 3.1 1区数学 Q1 MATHEMATICS Communications on Pure and Applied Mathematics Pub Date : 2024-10-10 DOI:10.1002/cpa.22231

N. Filonov

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Pólya conjectured that <math>\n <semantics>\n <mrow>\n <msub>\n <mi>N</mi>\n <mi>N</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>Ω</mi>\n <mo>,</mo>\n <mi>λ</mi>\n <mo>)</mo>\n </mrow>\n <mo>⩾</mo>\n <msup>\n <mrow>\n <mo>(</mo>\n <mn>4</mn>\n <mi>π</mi>\n <mo>)</mo>\n </mrow>\n <mrow>\n <mo>−</mo>\n <mn>1</mn>\n </mrow>\n </msup>\n <mrow>\n <mo>|</mo>\n <mi>Ω</mi>\n <mo>|</mo>\n </mrow>\n <mi>λ</mi>\n </mrow>\n <annotation>$N_{\\cal N} (\\Omega,\\lambda) \\geqslant (4\\pi)^{-1} |\\Omega | \\lambda$</annotation>\n </semantics></math>. 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Here <math>\n <semantics>\n <msub>\n <mi>j</mi>\n <mn>0</mn>\n </msub>\n <annotation>$j_0$</annotation>\n </semantics></math> is the first zero of the Bessel function <math>\n <semantics>\n <msub>\n <mi>J</mi>\n <mn>0</mn>\n </msub>\n <annotation>$J_0$</annotation>\n </semantics></math>.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"78 3","pages":"537-544"},"PeriodicalIF":3.1000,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the Pólya conjecture for the Neumann problem in planar convex domains\",\"authors\":\"N. Filonov\",\"doi\":\"10.1002/cpa.22231\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Denote by <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>N</mi>\\n <mi>N</mi>\\n </msub>\\n <mrow>\\n <mo>(</mo>\\n <mi>Ω</mi>\\n <mo>,</mo>\\n <mi>λ</mi>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation>$N_{\\\\cal N} (\\\\Omega,\\\\lambda)$</annotation>\\n </semantics></math> the counting function of the spectrum of the Neumann problem in the domain <math>\\n <semantics>\\n <mi>Ω</mi>\\n <annotation>$\\\\Omega$</annotation>\\n </semantics></math> on the plane. G. 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引用次数: 0

摘要

用平面域中诺伊曼问题谱的计数函数表示.波利亚猜想 .我们证明，对于凸域 .这里是贝塞尔函数的第一个零点 .

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On the Pólya conjecture for the Neumann problem in planar convex domains

Denote by $N_{N} (Ω, λ)$ the counting function of the spectrum of the Neumann problem in the domain $Ω$ on the plane. G. Pólya conjectured that $N_{N} (Ω, λ) ⩾ {(4 π)}^{- 1} | Ω | λ$ . We prove that for convex domains $N_{N} (Ω, λ) ⩾ {(2 \sqrt{3} j_{0}^{2})}^{- 1} | Ω | λ$ . Here $j_{0}$ is the first zero of the Bessel function $J_{0}$ .

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来源期刊

Communications on Pure and Applied Mathematics 数学-数学

CiteScore

6.70

自引率

3.30%

发文量

审稿时长

>12 weeks

期刊介绍： Communications on Pure and Applied Mathematics (ISSN 0010-3640) is published monthly, one volume per year, by John Wiley & Sons, Inc. © 2019. The journal primarily publishes papers originating at or solicited by the Courant Institute of Mathematical Sciences. It features recent developments in applied mathematics, mathematical physics, and mathematical analysis. The topics include partial differential equations, computer science, and applied mathematics. CPAM is devoted to mathematical contributions to the sciences; both theoretical and applied papers, of original or expository type, are included.

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