{"paper":{"title":"A note on the uniformity of the constant in the Poincar\\'e inequality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"David Ruiz","submitted_at":"2012-08-29T22:38:24Z","abstract_excerpt":"The classical Poincar\\'e inequality establishes that for any bounded regular domain $\\Omega\\subset \\R^N$ there exists a constant $C=C(\\Omega)>0$ such that $$ \\int_{\\Omega} |u|^2\\, dx \\leq C \\int_{\\Omega} |\\nabla u|^2\\, dx \\ \\ \\forall u \\in H^1(\\Omega),\\ \\int_{\\Omega} u(x) \\, dx=0.$$ In this note we show that $C$ can be taken independently of $\\Omega$ when $\\Omega$ is in a certain class of domains. Our result generalizes previous results in this direction."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.6045","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"}