{"paper":{"title":"On the asymptotics of the principal eigenvalue for a Robin problem with a large parameter in planar domains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"Konstantin Pankrashkin","submitted_at":"2013-05-14T20:29:24Z","abstract_excerpt":"Let $\\Omega\\subset \\RR^2$ be a domain having a compact boundary $\\Sigma$ which is Lipschitz and piecewise $C^4$ smooth, and let $\\nu$ denote the inward unit normal vector on $\\Sigma$. We study the principal eigenvalue $E(\\beta)$ of the Laplacian in $\\Omega$ with the Robin boundary conditions $\\partial f/\\partial\\nu +\\beta f=0$ on $\\Sigma$, where $\\beta$ is a positive number. Assuming that $\\Sigma$ has no convex corners we show the estimate $E(\\beta)=-\\beta^2- \\gamma_\\mx\\beta + O\\big(\\beta^\\{2}{3}\\big)$ as $\\beta\\to+\\infty$, where $\\gamma_\\mx$ is the maximal curvature of the boundary."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.3293","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}