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In this article we analyze blow-ups, the free boundary $\\partial\\{u>\\psi\\}$, and the singular set close to singular points $\\mathbf{x}^{0}=(x^{0},y^{0},z^{0})$ in $\\mathbb{R}^{3}$. We show that blow-ups of the form \\[ \\lim_{j\\to\\infty}\\frac{u(r_{j}\\cdot+\\mathbf{x}^{0})}{\\|u\\|_{L^{\\infty}(B_{r_{j}}(\\mathbf{x}^{0}))}}, \\] $r"},"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":"1510.03872","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-10-13T20:16:19Z","cross_cats_sorted":[],"title_canon_sha256":"e43322e7be63213b3532820fffa46f9013785d7e913f0baabc848eb9478f8f12","abstract_canon_sha256":"4c7bf046ed9a88097b670cedcdce826741d14aec5142dbb27d8062aa67ba4649"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:30:14.687617Z","signature_b64":"MUJ1a21K8eOUiDvEEzhdCkqRzp0wy9sQZOdVBYZdK1jYcRGRZoNqLDmBe2+QJeo7N2EHsEt7DfUXgquHOzYDCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"db64a6d7a54aa201857fffbfa61694c8c031327ab8f431ea95bfbf5ee13dda64","last_reissued_at":"2026-05-18T01:30:14.686869Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:30:14.686869Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Classification of Blow-ups and Free Boundaries of Solutions to Unstable Free Boundary Problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Andreas Minne","submitted_at":"2015-10-13T20:16:19Z","abstract_excerpt":"In general, solutions $u$ to \\[ \\Delta u(\\mathbf{x})=f(\\mathbf{x})\\chi_{\\{u>\\psi\\}} \\] are not $C^{1,1}$, even for $f$ smooth and $\\psi(\\mathbf{x})\\equiv0$. 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We show that blow-ups of the form \\[ \\lim_{j\\to\\infty}\\frac{u(r_{j}\\cdot+\\mathbf{x}^{0})}{\\|u\\|_{L^{\\infty}(B_{r_{j}}(\\mathbf{x}^{0}))}}, \\] $r"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.03872","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1510.03872","created_at":"2026-05-18T01:30:14.686994+00:00"},{"alias_kind":"arxiv_version","alias_value":"1510.03872v1","created_at":"2026-05-18T01:30:14.686994+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.03872","created_at":"2026-05-18T01:30:14.686994+00:00"},{"alias_kind":"pith_short_12","alias_value":"3NSKNV5FJKRA","created_at":"2026-05-18T12:29:02.477457+00:00"},{"alias_kind":"pith_short_16","alias_value":"3NSKNV5FJKRADBL7","created_at":"2026-05-18T12:29:02.477457+00:00"},{"alias_kind":"pith_short_8","alias_value":"3NSKNV5F","created_at":"2026-05-18T12:29:02.477457+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/3NSKNV5FJKRADBL77672MFUUZD","json":"https://pith.science/pith/3NSKNV5FJKRADBL77672MFUUZD.json","graph_json":"https://pith.science/api/pith-number/3NSKNV5FJKRADBL77672MFUUZD/graph.json","events_json":"https://pith.science/api/pith-number/3NSKNV5FJKRADBL77672MFUUZD/events.json","paper":"https://pith.science/paper/3NSKNV5F"},"agent_actions":{"view_html":"https://pith.science/pith/3NSKNV5FJKRADBL77672MFUUZD","download_json":"https://pith.science/pith/3NSKNV5FJKRADBL77672MFUUZD.json","view_paper":"https://pith.science/paper/3NSKNV5F","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1510.03872&json=true","fetch_graph":"https://pith.science/api/pith-number/3NSKNV5FJKRADBL77672MFUUZD/graph.json","fetch_events":"https://pith.science/api/pith-number/3NSKNV5FJKRADBL77672MFUUZD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3NSKNV5FJKRADBL77672MFUUZD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3NSKNV5FJKRADBL77672MFUUZD/action/storage_attestation","attest_author":"https://pith.science/pith/3NSKNV5FJKRADBL77672MFUUZD/action/author_attestation","sign_citation":"https://pith.science/pith/3NSKNV5FJKRADBL77672MFUUZD/action/citation_signature","submit_replication":"https://pith.science/pith/3NSKNV5FJKRADBL77672MFUUZD/action/replication_record"}},"created_at":"2026-05-18T01:30:14.686994+00:00","updated_at":"2026-05-18T01:30:14.686994+00:00"}