Andrea Colesanti, Elisa Francini, Galyna Livshyts, Paolo Salani
{"title":"The Brunn-Minkowski inequality for the first eigenvalue of the Ornstein-Uhlenbeck operator and log-concavity of the relevant eigenfunction","authors":"Andrea Colesanti, Elisa Francini, Galyna Livshyts, Paolo Salani","doi":"arxiv-2407.21354","DOIUrl":null,"url":null,"abstract":"We prove that the first (nontrivial) Dirichlet eigenvalue of the\nOrnstein-Uhlenbeck operator $$ L(u)=\\Delta u-\\langle\\nabla u,x\\rangle\\,, $$ as\na function of the domain, is convex with respect to the Minkowski addition, and\nwe characterize the equality cases in some classes of convex sets. We also\nprove that the corresponding (positive) eigenfunction is log-concave if the\ndomain is convex.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Metric Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2407.21354","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We prove that the first (nontrivial) Dirichlet eigenvalue of the
Ornstein-Uhlenbeck operator $$ L(u)=\Delta u-\langle\nabla u,x\rangle\,, $$ as
a function of the domain, is convex with respect to the Minkowski addition, and
we characterize the equality cases in some classes of convex sets. We also
prove that the corresponding (positive) eigenfunction is log-concave if the
domain is convex.