{"paper":{"title":"Solvability of the divergence equation implies John via Poincar\\'e inequality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.CA","authors_text":"Aapo Kauranen, Pekka Koskela, Renjin Jiang","submitted_at":"2013-07-04T14:16:05Z","abstract_excerpt":"Let $\\Omega \\subset \\rr^2$ be a bounded simply connected domain. We show that, for a fixed (every) $p\\in (1,\\fz),$ the divergence equation $\\mathrm{div}\\,\\mathbf{v}=f$ is solvable in $W^{1,p}_0(\\Omega)^2$ for every $f\\in L^p_0(\\Omega)$, if and only if $\\Omega$ is a John domain, if and only if the weighted Poincar\\'e inequality $$\\int_\\Omega|u(x)-u_{\\Omega}|^q\\,dx\\le C\\int_\\Omega|\\nabla u(x)|^q\\dist(x,\\partial \\Omega)^q\\,dx$$ holds for some (every) $q\\in [1,\\fz)$. In higher dimensions similar results are proved under some additional assumptions on the domain in question."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.1340","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"}