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We also obtain $C^{2,\\alpha}$ regularity of Lipschitz "},"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":"1205.1755","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-05-08T17:33:59Z","cross_cats_sorted":[],"title_canon_sha256":"7248c47c71d5d511d29e91b42ef03836965f028811253eadc10e16965dc2be82","abstract_canon_sha256":"d0a8626977d35983ec014ea4565f31f91dacb83ef8289d3b45ad87c8b2f37dbb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:56:09.040847Z","signature_b64":"Ldb9IKRoPlR7YMVKQI1MnVnyP8CcRrjIF7Gk5vYAYAbdxgwfPuUtIaePJt2R1ywotvpDMjrEUN/D2+p8yVsWDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"44d2519a159d84fad07b800457f597614ebb8b6c3e088f3ed6e219b49d7e4dd0","last_reissued_at":"2026-05-18T03:56:09.036550Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:56:09.036550Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Regularity of Lipschitz free boundaries for the thin one-phase problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Daniela De Silva, Ovidiu Savin","submitted_at":"2012-05-08T17:33:59Z","abstract_excerpt":"We study regularity properties of the free boundary for the thin one-phase problem which consists of minimizing the energy functional $$\\label{E} E(u,\\Omega) = \\int_\\Omega |\\nabla u|^2 dX + \\mathcal{H}^n(\\{u>0\\} \\cap \\{x_{n+1} = 0\\}), \\quad \\Omega \\subset \\R^{n+1},$$ among all functions $u\\ge 0$ which are fixed on $\\p \\Omega$.\n  We prove that the free boundary $F(u)=\\p_{\\R^n}\\{u>0\\}$ of a minimizer $u$ has locally finite $\\mathcal{H}^{n-1}$ measure and is a $C^{2,\\alpha}$ surface except on a small singular set of Hausdorff dimension $n-3$. 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