{"title":"图上 $p$ 拉普拉斯算子的椭圆梯度估计","authors":"Lin Feng Wang","doi":"10.4310/ajm.2024.v28.n1.a3","DOIUrl":null,"url":null,"abstract":"Let $G(V,E)$ be a connected locally finite graph. In this paper we consider the elliptic gradient estimate for solutions to the equation\\[\\Delta_p u - \\lambda_p {\\lvert u \\rvert}^{p-2} u\\]on $G$ with the $\\mathrm{CD}^\\psi_p (m,-K)$ condition, where $p \\geq 2$, $m \\gt 0$, $K \\geq 0$, and $\\Delta_p$ denotes the $p\\textrm{-}$Laplacian. As applications, we can derive Liouville theorems and the Harnack inequality.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5000,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Elliptic gradient estimate for the $p$−Laplace operator on the graph\",\"authors\":\"Lin Feng Wang\",\"doi\":\"10.4310/ajm.2024.v28.n1.a3\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $G(V,E)$ be a connected locally finite graph. In this paper we consider the elliptic gradient estimate for solutions to the equation\\\\[\\\\Delta_p u - \\\\lambda_p {\\\\lvert u \\\\rvert}^{p-2} u\\\\]on $G$ with the $\\\\mathrm{CD}^\\\\psi_p (m,-K)$ condition, where $p \\\\geq 2$, $m \\\\gt 0$, $K \\\\geq 0$, and $\\\\Delta_p$ denotes the $p\\\\textrm{-}$Laplacian. As applications, we can derive Liouville theorems and the Harnack inequality.\",\"PeriodicalId\":55452,\"journal\":{\"name\":\"Asian Journal of Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2024-08-07\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Asian Journal of Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.4310/ajm.2024.v28.n1.a3\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Asian Journal of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4310/ajm.2024.v28.n1.a3","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
Elliptic gradient estimate for the $p$−Laplace operator on the graph
Let $G(V,E)$ be a connected locally finite graph. In this paper we consider the elliptic gradient estimate for solutions to the equation\[\Delta_p u - \lambda_p {\lvert u \rvert}^{p-2} u\]on $G$ with the $\mathrm{CD}^\psi_p (m,-K)$ condition, where $p \geq 2$, $m \gt 0$, $K \geq 0$, and $\Delta_p$ denotes the $p\textrm{-}$Laplacian. As applications, we can derive Liouville theorems and the Harnack inequality.