{"title":"通过严格Faber-Krahn型不等式的第一特征值的定义域变分","authors":"T. Anoop, K. Kumar","doi":"10.57262/ade028-0708-537","DOIUrl":null,"url":null,"abstract":"For $d\\geq 2$ and $\\frac{2d+2}{d+2}<p<\\infty $, we prove a strict Faber-Krahn type inequality for the first eigenvalue $\\lambda _1(\\Omega )$ of the $p$-Laplace operator on a bounded Lipschitz domain $\\Omega \\subset \\mathbb{R}^d$ (with mixed boundary conditions) under the polarizations. We apply this inequality to the obstacle problems on the domains of the form $\\Omega \\setminus \\mathscr{O}$, where $\\mathscr{O}\\subset \\subset \\Omega $ is an obstacle. Under some geometric assumptions on $\\Omega $ and $\\mathscr{O}$, we prove the strict monotonicity of $\\lambda _1 (\\Omega \\setminus \\mathscr{O})$ with respect to certain translations and rotations of $\\mathscr{O}$ in $\\Omega $.","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":" ","pages":""},"PeriodicalIF":1.5000,"publicationDate":"2022-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"5","resultStr":"{\"title\":\"Domain variations of the first eigenvalue via a strict Faber-Krahn type inequality\",\"authors\":\"T. Anoop, K. Kumar\",\"doi\":\"10.57262/ade028-0708-537\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"For $d\\\\geq 2$ and $\\\\frac{2d+2}{d+2}<p<\\\\infty $, we prove a strict Faber-Krahn type inequality for the first eigenvalue $\\\\lambda _1(\\\\Omega )$ of the $p$-Laplace operator on a bounded Lipschitz domain $\\\\Omega \\\\subset \\\\mathbb{R}^d$ (with mixed boundary conditions) under the polarizations. We apply this inequality to the obstacle problems on the domains of the form $\\\\Omega \\\\setminus \\\\mathscr{O}$, where $\\\\mathscr{O}\\\\subset \\\\subset \\\\Omega $ is an obstacle. Under some geometric assumptions on $\\\\Omega $ and $\\\\mathscr{O}$, we prove the strict monotonicity of $\\\\lambda _1 (\\\\Omega \\\\setminus \\\\mathscr{O})$ with respect to certain translations and rotations of $\\\\mathscr{O}$ in $\\\\Omega $.\",\"PeriodicalId\":53312,\"journal\":{\"name\":\"Advances in Differential Equations\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":1.5000,\"publicationDate\":\"2022-02-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"5\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Advances in Differential Equations\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.57262/ade028-0708-537\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Differential Equations","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.57262/ade028-0708-537","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
期刊介绍：
Advances in Differential Equations will publish carefully selected, longer research papers on mathematical aspects of differential equations and on applications of the mathematical theory to issues arising in the sciences and in engineering. Papers submitted to this journal should be correct, new and non-trivial. Emphasis will be placed on papers that are judged to be specially timely, and of interest to a substantial number of mathematicians working in this area.
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