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More precisely we assume that $0\\in \\partial \\{u>0\\}$ and the derivative of the boundary data has a jump discontinuity. If $0\\in \\bar{\\partial(\\{u>0\\} \\cap B_1^+)}$ then (for $n=2$ or $n>3$ and one-phase case) we prove, among other things, that the free boundary $\\partial \\{u>0\\}$ approaches the origin along on"},"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":"1205.5052","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-05-22T20:36:39Z","cross_cats_sorted":[],"title_canon_sha256":"f536416b7eae0a50ca97127d019d23bc43cf1ad3c796e29d41a8f27da3bf2535","abstract_canon_sha256":"a3245a31c2ed07a077b9315183d90a888429fa230cd62632bb55fff40ba3334d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:12:35.176745Z","signature_b64":"uiXzoiWQXYHtHRQaDPc79YKfDJFmhvRFMFU4zZrgTNY7ZpDD9S6e1US0KO1ivX7hSmFet768WaOPDT3c/ewFCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"928dbdc44679cfbc3caa920f47c5b3c005af754045f576e8aa990f15467614e4","last_reissued_at":"2026-05-18T03:12:35.175876Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:12:35.175876Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Analysis of a free boundary at contact points with Lipschitz data","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Aram Karakhanyan, Henrik Shahgholian","submitted_at":"2012-05-22T20:36:39Z","abstract_excerpt":"In this paper we consider a minimization problem for the functional $$ J(u)=\\int_{B_1^+}|\\nabla u|\\sp 2+\\lambda_{+}^2\\chi_{\\{u>0\\}}+\\lambda_{-}^2\\chi_{\\{u\\leq0\\}}, $$ in the upper half ball $B_1^+\\subset\\R^n, n\\geq 2$ subject to a Lipschitz continuous Dirichlet data on $\\partial B_1^+$. 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