Peculiar behavior of the principal Laplacian eigenvalue for large negative Robin parameters

IF 1.2 2区数学 Q1 MATHEMATICS

Journal of the London Mathematical Society-Second Series Pub Date : 2025-07-22 DOI:10.1112/jlms.70242

Charlotte Dietze, Konstantin Pankrashkin

{"title":"Peculiar behavior of the principal Laplacian eigenvalue for large negative Robin parameters","authors":"Charlotte Dietze, Konstantin Pankrashkin","doi":"10.1112/jlms.70242","DOIUrl":null,"url":null,"abstract":"Let <math>\n <semantics>\n <mrow>\n <mi>Ω</mi>\n <mo>⊂</mo>\n <msup>\n <mi>R</mi>\n <mi>n</mi>\n </msup>\n </mrow>\n <annotation>$\\Omega \\subset \\mathbb {R}^n$</annotation>\n </semantics></math> with <math>\n <semantics>\n <mrow>\n <mi>n</mi>\n <mo>⩾</mo>\n <mn>2</mn>\n </mrow>\n <annotation>$n\\geqslant 2$</annotation>\n </semantics></math> be a bounded Lipschitz domain with outer unit normal <math>\n <semantics>\n <mi>ν</mi>\n <annotation>$\\nu$</annotation>\n </semantics></math>. For <math>\n <semantics>\n <mrow>\n <mi>α</mi>\n <mo>∈</mo>\n <mi>R</mi>\n </mrow>\n <annotation>$\\alpha \\in \\mathbb {R}$</annotation>\n </semantics></math>, let <math>\n <semantics>\n <msubsup>\n <mi>R</mi>\n <mi>Ω</mi>\n <mi>α</mi>\n </msubsup>\n <annotation>$R_\\Omega ^\\alpha$</annotation>\n </semantics></math> be the Laplacian in <math>\n <semantics>\n <mi>Ω</mi>\n <annotation>$\\Omega$</annotation>\n </semantics></math> with the Robin boundary condition <math>\n <semantics>\n <mrow>\n <msub>\n <mi>∂</mi>\n <mi>ν</mi>\n </msub>\n <mi>u</mi>\n <mo>+</mo>\n <mi>α</mi>\n <mi>u</mi>\n <mo>=</mo>\n <mn>0</mn>\n </mrow>\n <annotation>$\\partial _\\nu u+\\alpha u=0$</annotation>\n </semantics></math>, and denote by <math>\n <semantics>\n <mrow>\n <mi>E</mi>\n <mo>(</mo>\n <msubsup>\n <mi>R</mi>\n <mi>Ω</mi>\n <mi>α</mi>\n </msubsup>\n <mo>)</mo>\n </mrow>\n <annotation>$E(R^\\alpha _\\Omega)$</annotation>\n </semantics></math> its principal eigenvalue. In 2017, Bucur, Freitas, and Kennedy stated the following open question: Does the limit of the ratio <math>\n <semantics>\n <mrow>\n <mi>E</mi>\n <mrow>\n <mo>(</mo>\n <msubsup>\n <mi>R</mi>\n <mi>Ω</mi>\n <mi>α</mi>\n </msubsup>\n <mo>)</mo>\n </mrow>\n <mo>/</mo>\n <msup>\n <mi>α</mi>\n <mn>2</mn>\n </msup>\n </mrow>\n <annotation>$E(R_\\Omega ^\\alpha)/ \\alpha ^2$</annotation>\n </semantics></math> for <math>\n <semantics>\n <mrow>\n <mi>α</mi>\n <mo>→</mo>\n <mo>−</mo>\n <mi>∞</mi>\n </mrow>\n <annotation>$\\alpha \\rightarrow -\\infty$</annotation>\n </semantics></math> always exist? We give a negative answer.","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"112 1","pages":""},"PeriodicalIF":1.2000,"publicationDate":"2025-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70242","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the London Mathematical Society-Second Series","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.70242","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

Abstract

Let $Ω \subset R^{n}$ with $n ⩾ 2$ be a bounded Lipschitz domain with outer unit normal $ν$ . For $α \in R$ , let $R_{Ω}^{α}$ be the Laplacian in $Ω$ with the Robin boundary condition $\partial_{ν} u + α u = 0$ , and denote by $E (R_{Ω}^{α})$ its principal eigenvalue. In 2017, Bucur, Freitas, and Kennedy stated the following open question: Does the limit of the ratio $E (R_{Ω}^{α}) / α^{2}$ for $α \to - \infty$ always exist? We give a negative answer.

Abstract Image

查看原文本刊更多论文

大负Robin参数下主拉普拉斯特征值的特殊行为

设Ω∧R n $\Omega \subset \mathbb {R}^n$， n小于或等于$n\geqslant 2$为外单位为法向ν的有界李普希茨域$\nu$。对于α∈R $\alpha \in \mathbb {R}$，设R Ω α $R_\Omega ^\alpha$为Ω $\Omega$中的拉普拉斯函数，具有Robin边界条件∂ν u +α u = 0 $\partial _\nu u+\alpha u=0$，用E (R Ω α) $E(R^\alpha _\Omega)$表示其主特征值。2017年，Bucur、Freitas和Kennedy提出了以下开放性问题：比值E (R Ω α) / α 2 $E(R_\Omega ^\alpha)/ \alpha ^2$的极限为α→−∞$\alpha \rightarrow -\infty$永远存在？我们给出否定的答案。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Journal of the London Mathematical Society-Second Series 数学-数学

CiteScore

1.90

自引率

0.00%

发文量

186

审稿时长

6-12 weeks

期刊介绍： The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.