{"title":"论紧凑集补集中具有罗宾边界条件的 $p$ 拉普拉斯算子的第一个特征值","authors":"Lukas Bundrock, Tiziana Giorgi, Robert Smits","doi":"arxiv-2408.06236","DOIUrl":null,"url":null,"abstract":"We consider the first eigenvalue $\\lambda_1$ of the $p$-Laplace operator\nsubject to Robin boundary conditions in the exterior of a compact set. We\ndiscuss the conditions for the existence of a variational $\\lambda_1$,\ndepending on the boundary parameter, the space dimension, and $p$. Our analysis\ninvolves the first $p$-harmonic Steklov eigenvalue in exterior domains. We\nestablish properties of $\\lambda_1$ for the exterior of a ball, including\ngeneral inequalities, the asymptotic behavior as the boundary parameter\napproaches zero, and a monotonicity result with respect to a special type of\ndomain inclusion. In two dimensions, we generalized to $p\\neq 2$ some known\nshape optimization results.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"7 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the First Eigenvalue of the $p$-Laplace Operator with Robin Boundary Conditions in the Complement of a Compact Set\",\"authors\":\"Lukas Bundrock, Tiziana Giorgi, Robert Smits\",\"doi\":\"arxiv-2408.06236\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We consider the first eigenvalue $\\\\lambda_1$ of the $p$-Laplace operator\\nsubject to Robin boundary conditions in the exterior of a compact set. We\\ndiscuss the conditions for the existence of a variational $\\\\lambda_1$,\\ndepending on the boundary parameter, the space dimension, and $p$. Our analysis\\ninvolves the first $p$-harmonic Steklov eigenvalue in exterior domains. We\\nestablish properties of $\\\\lambda_1$ for the exterior of a ball, including\\ngeneral inequalities, the asymptotic behavior as the boundary parameter\\napproaches zero, and a monotonicity result with respect to a special type of\\ndomain inclusion. In two dimensions, we generalized to $p\\\\neq 2$ some known\\nshape optimization results.\",\"PeriodicalId\":501373,\"journal\":{\"name\":\"arXiv - MATH - Spectral Theory\",\"volume\":\"7 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-08-12\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Spectral Theory\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2408.06236\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Spectral Theory","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.06236","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
On the First Eigenvalue of the $p$-Laplace Operator with Robin Boundary Conditions in the Complement of a Compact Set
We consider the first eigenvalue $\lambda_1$ of the $p$-Laplace operator
subject to Robin boundary conditions in the exterior of a compact set. We
discuss the conditions for the existence of a variational $\lambda_1$,
depending on the boundary parameter, the space dimension, and $p$. Our analysis
involves the first $p$-harmonic Steklov eigenvalue in exterior domains. We
establish properties of $\lambda_1$ for the exterior of a ball, including
general inequalities, the asymptotic behavior as the boundary parameter
approaches zero, and a monotonicity result with respect to a special type of
domain inclusion. In two dimensions, we generalized to $p\neq 2$ some known
shape optimization results.