Alexander Logunov, Lakshmi Priya M. E., Andrea Sartori
{"title":"谐函数节点体积的近似尖锐下界","authors":"Alexander Logunov, Lakshmi Priya M. E., Andrea Sartori","doi":"10.1002/cpa.22207","DOIUrl":null,"url":null,"abstract":"<p>This paper focuses on a relation between the growth of harmonic functions and the Hausdorff measure of their zero sets. Let <span></span><math>\n <semantics>\n <mi>u</mi>\n <annotation>$u$</annotation>\n </semantics></math> be a real-valued harmonic function in <span></span><math>\n <semantics>\n <msup>\n <mi>R</mi>\n <mi>n</mi>\n </msup>\n <annotation>$\\mathbb {R}^n$</annotation>\n </semantics></math> with <span></span><math>\n <semantics>\n <mrow>\n <mi>u</mi>\n <mo>(</mo>\n <mn>0</mn>\n <mo>)</mo>\n <mo>=</mo>\n <mn>0</mn>\n </mrow>\n <annotation>$u(0)=0$</annotation>\n </semantics></math> and <span></span><math>\n <semantics>\n <mrow>\n <mi>n</mi>\n <mo>≥</mo>\n <mn>3</mn>\n </mrow>\n <annotation>$n\\ge 3$</annotation>\n </semantics></math>. We prove\n\n </p>","PeriodicalId":3,"journal":{"name":"ACS Applied Electronic Materials","volume":null,"pages":null},"PeriodicalIF":4.3000,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.22207","citationCount":"0","resultStr":"{\"title\":\"Almost sharp lower bound for the nodal volume of harmonic functions\",\"authors\":\"Alexander Logunov, Lakshmi Priya M. E., Andrea Sartori\",\"doi\":\"10.1002/cpa.22207\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>This paper focuses on a relation between the growth of harmonic functions and the Hausdorff measure of their zero sets. Let <span></span><math>\\n <semantics>\\n <mi>u</mi>\\n <annotation>$u$</annotation>\\n </semantics></math> be a real-valued harmonic function in <span></span><math>\\n <semantics>\\n <msup>\\n <mi>R</mi>\\n <mi>n</mi>\\n </msup>\\n <annotation>$\\\\mathbb {R}^n$</annotation>\\n </semantics></math> with <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>u</mi>\\n <mo>(</mo>\\n <mn>0</mn>\\n <mo>)</mo>\\n <mo>=</mo>\\n <mn>0</mn>\\n </mrow>\\n <annotation>$u(0)=0$</annotation>\\n </semantics></math> and <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>n</mi>\\n <mo>≥</mo>\\n <mn>3</mn>\\n </mrow>\\n <annotation>$n\\\\ge 3$</annotation>\\n </semantics></math>. We prove\\n\\n </p>\",\"PeriodicalId\":3,\"journal\":{\"name\":\"ACS Applied Electronic Materials\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":4.3000,\"publicationDate\":\"2024-05-29\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.22207\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"ACS Applied Electronic Materials\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/cpa.22207\",\"RegionNum\":3,\"RegionCategory\":\"材料科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"ENGINEERING, ELECTRICAL & ELECTRONIC\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Electronic Materials","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/cpa.22207","RegionNum":3,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"ENGINEERING, ELECTRICAL & ELECTRONIC","Score":null,"Total":0}
Almost sharp lower bound for the nodal volume of harmonic functions
This paper focuses on a relation between the growth of harmonic functions and the Hausdorff measure of their zero sets. Let be a real-valued harmonic function in with and . We prove
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