{"paper":{"title":"The Integrability of Negative Powers of the Solution of the Saint Venant Problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.AP","authors_text":"Anthony Carbery, David J. Rule, Marius Mitrea, Vladimir Maz'ya","submitted_at":"2012-04-20T11:46:05Z","abstract_excerpt":"We initiate the study of the finiteness condition $\\int_{\\Omega}u(x)^{-\\beta}\\,dx\\leq C(\\Omega,\\beta)<+\\infty$ where $\\Omega\\subseteq{\\mathbb{R}}^n$ is an open set and $u$ is the solution of the Saint Venant problem $\\Delta u=-1$ in $\\Omega$, $u=0$ on $\\partial\\Omega$. The central issue which we address is that of determining the range of values of the parameter $\\beta>0$ for which the aforementioned condition holds under various hypotheses on the smoothness of $\\Omega$ and demands on the nature of the constant $C(\\Omega,\\beta)$. Classes of domains for which our analysis applies include bounde"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.4590","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"}