{"paper":{"title":"Dirichlet and Neumann problems for planar domains with parameter","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Florian Bertrand, Xianghong Gong","submitted_at":"2011-10-31T23:43:15Z","abstract_excerpt":"Let $\\Gamma(\\cdot,\\lambda)$ be smooth, i.e.\\, $\\mathcal C^\\infty$, embeddings from $\\bar{\\Omega}$ onto $\\bar{\\Omega^{\\lambda}}$, where $\\Omega$ and $\\Omega^\\lambda$ are bounded domains with smooth boundary in the complex plane and $\\lambda$ varies in $I=[0,1]$. Suppose that $\\Gamma$ is smooth on $\\bar\\Omega\\times I$ and $f$ is a smooth function on $\\partial\\Omega\\times I$. Let $u(\\cdot,\\lambda)$ be the harmonic functions on $\\Omega^\\lambda$ with boundary values $f(\\cdot,\\lambda)$. We show that $u(\\Gamma(z,\\lambda),\\lambda)$ is smooth on $\\bar\\Omega\\times I$. Our main result is proved for suita"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.0079","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"}