{"title":"Maximizing the Second Robin Eigenvalue of Simply Connected Curved Membranes","authors":"Jeffrey J. Langford, Richard S. Laugesen","doi":"10.1007/s40315-023-00516-1","DOIUrl":null,"url":null,"abstract":"<p>The second eigenvalue of the Robin Laplacian is shown to be maximal for a spherical cap among simply connected Jordan domains on the 2-sphere, for substantial intervals of positive and negative Robin parameters and areas. Geodesic disks in the hyperbolic plane similarly maximize the eigenvalue on a natural interval of negative Robin parameters. These theorems extend work of Freitas and Laugesen from the Euclidean case (zero curvature) and the authors’ hyperbolic and spherical results for Neumann eigenvalues (zero Robin parameter). Complicating the picture is the numerically observed fact that the second Robin eigenfunction on a large spherical cap is purely radial, with no angular dependence, when the Robin parameter lies in a certain negative interval depending on the cap aperture.</p>","PeriodicalId":49088,"journal":{"name":"Computational Methods and Function Theory","volume":"71 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2023-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Computational Methods and Function Theory","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s40315-023-00516-1","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
The second eigenvalue of the Robin Laplacian is shown to be maximal for a spherical cap among simply connected Jordan domains on the 2-sphere, for substantial intervals of positive and negative Robin parameters and areas. Geodesic disks in the hyperbolic plane similarly maximize the eigenvalue on a natural interval of negative Robin parameters. These theorems extend work of Freitas and Laugesen from the Euclidean case (zero curvature) and the authors’ hyperbolic and spherical results for Neumann eigenvalues (zero Robin parameter). Complicating the picture is the numerically observed fact that the second Robin eigenfunction on a large spherical cap is purely radial, with no angular dependence, when the Robin parameter lies in a certain negative interval depending on the cap aperture.
期刊介绍:
CMFT is an international mathematics journal which publishes carefully selected original research papers in complex analysis (in a broad sense), and on applications or computational methods related to complex analysis. Survey articles of high standard and current interest can be considered for publication as well.