连通有限图上p$ p$ -拉普拉斯算子的特征值估计

IF 0.9 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society Pub Date : 2025-05-23 DOI:10.1112/blms.70104

Lin Feng Wang

{"title":"连通有限图上p$ p$ -拉普拉斯算子的特征值估计","authors":"Lin Feng Wang","doi":"10.1112/blms.70104","DOIUrl":null,"url":null,"abstract":"In this paper, we consider eigenvalue estimate for the <math>\n <semantics>\n <mi>p</mi>\n <annotation>$p$</annotation>\n </semantics></math>-Laplace operator <math>\n <semantics>\n <msub>\n <mi>▵</mi>\n <mi>p</mi>\n </msub>\n <annotation>$\\triangle _p$</annotation>\n </semantics></math> on a connected finite graph with the <math>\n <semantics>\n <mrow>\n <msub>\n <mi>CD</mi>\n <mi>p</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>m</mi>\n <mo>,</mo>\n <mn>0</mn>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$\\mathrm{CD}_p(m,0)$</annotation>\n </semantics></math> condition for <math>\n <semantics>\n <mrow>\n <mi>p</mi>\n <mo>></mo>\n <mn>1</mn>\n </mrow>\n <annotation>$p>1$</annotation>\n </semantics></math> and <math>\n <semantics>\n <mrow>\n <mi>m</mi>\n <mo>></mo>\n <mn>0</mn>\n </mrow>\n <annotation>$m>0$</annotation>\n </semantics></math>. We first establish elliptic gradient estimates for solutions to the eigenvalue equation. Then we establish a lower bound estimate for the first nonzero eigenvalue of <math>\n <semantics>\n <msub>\n <mi>▵</mi>\n <mi>p</mi>\n </msub>\n <annotation>$\\triangle _p$</annotation>\n </semantics></math>, which is a generalization not only of the eigenvalue estimate for the manifold setting, but also of the eigenvalue estimate for the <math>\n <semantics>\n <mi>μ</mi>\n <annotation>$\\mu$</annotation>\n </semantics></math>-Laplace operator <math>\n <semantics>\n <msub>\n <mi>▵</mi>\n <mi>μ</mi>\n </msub>\n <annotation>$\\triangle _{\\mu }$</annotation>\n </semantics></math> on the graph with the <math>\n <semantics>\n <mrow>\n <mi>CD</mi>\n <mo>(</mo>\n <mi>m</mi>\n <mo>,</mo>\n <mn>0</mn>\n <mo>)</mo>\n </mrow>\n <annotation>$\\mathrm{CD}(m,0)$</annotation>\n </semantics></math> condition.","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 8","pages":"2444-2461"},"PeriodicalIF":0.9000,"publicationDate":"2025-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Eigenvalue estimate for the \\n \\n p\\n $p$\\n -Laplace operator on a connected finite graph\",\"authors\":\"Lin Feng Wang\",\"doi\":\"10.1112/blms.70104\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In this paper, we consider eigenvalue estimate for the <math>\\n <semantics>\\n <mi>p</mi>\\n <annotation>$p$</annotation>\\n </semantics></math>-Laplace operator <math>\\n <semantics>\\n <msub>\\n <mi>▵</mi>\\n <mi>p</mi>\\n </msub>\\n <annotation>$\\\\triangle _p$</annotation>\\n </semantics></math> on a connected finite graph with the <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>CD</mi>\\n <mi>p</mi>\\n </msub>\\n <mrow>\\n <mo>(</mo>\\n <mi>m</mi>\\n <mo>,</mo>\\n <mn>0</mn>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation>$\\\\mathrm{CD}_p(m,0)$</annotation>\\n </semantics></math> condition for <math>\\n <semantics>\\n <mrow>\\n <mi>p</mi>\\n <mo>></mo>\\n <mn>1</mn>\\n </mrow>\\n <annotation>$p>1$</annotation>\\n </semantics></math> and <math>\\n <semantics>\\n <mrow>\\n <mi>m</mi>\\n <mo>></mo>\\n <mn>0</mn>\\n </mrow>\\n <annotation>$m>0$</annotation>\\n </semantics></math>. We first establish elliptic gradient estimates for solutions to the eigenvalue equation. Then we establish a lower bound estimate for the first nonzero eigenvalue of <math>\\n <semantics>\\n <msub>\\n <mi>▵</mi>\\n <mi>p</mi>\\n </msub>\\n <annotation>$\\\\triangle _p$</annotation>\\n </semantics></math>, which is a generalization not only of the eigenvalue estimate for the manifold setting, but also of the eigenvalue estimate for the <math>\\n <semantics>\\n <mi>μ</mi>\\n <annotation>$\\\\mu$</annotation>\\n </semantics></math>-Laplace operator <math>\\n <semantics>\\n <msub>\\n <mi>▵</mi>\\n <mi>μ</mi>\\n </msub>\\n <annotation>$\\\\triangle _{\\\\mu }$</annotation>\\n </semantics></math> on the graph with the <math>\\n <semantics>\\n <mrow>\\n <mi>CD</mi>\\n <mo>(</mo>\\n <mi>m</mi>\\n <mo>,</mo>\\n <mn>0</mn>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$\\\\mathrm{CD}(m,0)$</annotation>\\n </semantics></math> condition.\",\"PeriodicalId\":55298,\"journal\":{\"name\":\"Bulletin of the London Mathematical Society\",\"volume\":\"57 8\",\"pages\":\"2444-2461\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2025-05-23\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Bulletin of the London Mathematical Society\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/blms.70104\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bulletin of the London Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/blms.70104","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

摘要

本文研究了具有CD p (m)的连通有限图上p$ p$ -拉普拉斯算子p_p $\三角形_p$的特征值估计0)$ \ mathm {CD}_p(m,0)$ condition for p >；1$ and m >；0$ m>0$。我们首先建立了特征值方程解的椭圆梯度估计。然后，我们建立了第一个非零特征值的下界估计，该估计不仅是流形设定的特征值估计的推广，也讨论了μ $\mu$ -拉普拉斯算子在具有CD (m,0)$ \mathrm{CD}(m,0)$条件的图上的μ $\triangle _{\mu}$的特征值估计。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

Eigenvalue estimate for the

p
$p$
-Laplace operator on a connected finite graph

查看原文本刊更多论文

Eigenvalue estimate for the p $p$ -Laplace operator on a connected finite graph

In this paper, we consider eigenvalue estimate for the $p$ -Laplace operator $▵_{p}$ on a connected finite graph with the ${CD}_{p} (m, 0)$ condition for $p > 1$ and $m > 0$ . We first establish elliptic gradient estimates for solutions to the eigenvalue equation. Then we establish a lower bound estimate for the first nonzero eigenvalue of $▵_{p}$ , which is a generalization not only of the eigenvalue estimate for the manifold setting, but also of the eigenvalue estimate for the $μ$ -Laplace operator $▵_{μ}$ on the graph with the $CD (m, 0)$ condition.