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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1111.0079","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2011-10-31T23:43:15Z","cross_cats_sorted":[],"title_canon_sha256":"8f532abc9fd8bc2e0acf8f0a81f71cb42ac073f86e73e2988beca171404f256f","abstract_canon_sha256":"6f8a3374a8b7ba3519a24e9b378e5e585d8c8f06c18381e7afe5032595080383"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:09:46.455226Z","signature_b64":"TH1pqSCS7tWmScPUYCAbUIDJ81sqS0dld20YjLgjIS51TYgXqWzmYyxxdDlQ3QhBMX63wqc2nwBlnue6iP20Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8b3c7e838378471153f7a07370c0be93dbd1bb552af133bd0447f9d37c737fe2","last_reissued_at":"2026-05-18T04:09:46.454814Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:09:46.454814Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $\\bar\\Omega\\times I$. 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