{"paper":{"title":"Spectral bounds for the torsion function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.SP","authors_text":"Michiel van den Berg","submitted_at":"2017-01-09T13:40:39Z","abstract_excerpt":"Let $\\Omega$ be an open set in Euclidean space $\\R^m,\\, m=2,3,...$, and let $v_{\\Omega}$ denote the torsion function for $\\Omega$. It is known that $v_{\\Omega}$ is bounded if and only if the bottom of the spectrum of the Dirichlet Laplacian acting in $\\Leb^2(\\Omega)$, denoted by $\\lambda(\\Omega)$, is bounded away from $0$. It is shown that the previously obtained bound $\\|v_{\\Omega}\\|_{\\Leb^{\\infty}(\\Omega)}\\lambda(\\Omega)\\ge 1$ is sharp: for $m\\in\\{2,3,...\\}$, and any $\\epsilon>0$ we construct an open, bounded and connected set $\\Omega_{\\epsilon}\\subset \\R^m$ such that $\\|v_{\\Omega_{\\epsilon}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.02172","kind":"arxiv","version":2},"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"}