{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:4AXRQUTIDGIKCJ5F5EWCD77MMY","short_pith_number":"pith:4AXRQUTI","schema_version":"1.0","canonical_sha256":"e02f1852681990a127a5e92c21ffec6607a0896d293c4a7c8c0f59de9971b8f0","source":{"kind":"arxiv","id":"1212.1492","version":2},"attestation_state":"computed","paper":{"title":"The two-phase fractional obstacle problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Arshak Petrosyan, Erik Lindgren, Mark Allen","submitted_at":"2012-12-06T22:44:51Z","abstract_excerpt":"We study minimizers of the functional $$ \\int_{B_1^+}|\\nabla u|^2 x_n^a\\,d x +2\\int_{B_1'} (\\lambda_+ u^++\\lambda_- u^-)\\,d x', $$ for $a\\in(-1,1)$. The problem arises in connection with heat flow with control on the boundary. It can also be seen as a non-local analogue of the, by now well studied, two-phase obstacle problem. Moreover, when $u$ does not change signs this is equivalent to the fractional obstacle problem. Our main results are the optimal regularity of the minimizer and the separation of the two free boundaries $\\Gamma^+=\\partial'\\{u(\\cdot,0)>0\\}$ and $\\Gamma^-=\\partial'\\{u(\\cdot"},"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":"1212.1492","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-12-06T22:44:51Z","cross_cats_sorted":[],"title_canon_sha256":"4d067f000d6e52af74cb34480f771220a0854887dfdaa5e36ade33cf863998a0","abstract_canon_sha256":"15b88bc9115dd1cecd0b51f3604b316947434aa0a83b403a8ce97d12a8ff56b1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:49:18.492378Z","signature_b64":"+2XQGSLG9nmlobL+KlrlCTyseN1bsdNissnyWmuZYhGQ3WXzWOI9kZeMNs0QLHMXBaXxkljUwkvWU18q3hVBAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e02f1852681990a127a5e92c21ffec6607a0896d293c4a7c8c0f59de9971b8f0","last_reissued_at":"2026-05-18T02:49:18.491838Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:49:18.491838Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The two-phase fractional obstacle problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Arshak Petrosyan, Erik Lindgren, Mark Allen","submitted_at":"2012-12-06T22:44:51Z","abstract_excerpt":"We study minimizers of the functional $$ \\int_{B_1^+}|\\nabla u|^2 x_n^a\\,d x +2\\int_{B_1'} (\\lambda_+ u^++\\lambda_- u^-)\\,d x', $$ for $a\\in(-1,1)$. The problem arises in connection with heat flow with control on the boundary. It can also be seen as a non-local analogue of the, by now well studied, two-phase obstacle problem. Moreover, when $u$ does not change signs this is equivalent to the fractional obstacle problem. Our main results are the optimal regularity of the minimizer and the separation of the two free boundaries $\\Gamma^+=\\partial'\\{u(\\cdot,0)>0\\}$ and $\\Gamma^-=\\partial'\\{u(\\cdot"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.1492","kind":"arxiv","version":2},"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":"1212.1492","created_at":"2026-05-18T02:49:18.491917+00:00"},{"alias_kind":"arxiv_version","alias_value":"1212.1492v2","created_at":"2026-05-18T02:49:18.491917+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.1492","created_at":"2026-05-18T02:49:18.491917+00:00"},{"alias_kind":"pith_short_12","alias_value":"4AXRQUTIDGIK","created_at":"2026-05-18T12:26:53.410803+00:00"},{"alias_kind":"pith_short_16","alias_value":"4AXRQUTIDGIKCJ5F","created_at":"2026-05-18T12:26:53.410803+00:00"},{"alias_kind":"pith_short_8","alias_value":"4AXRQUTI","created_at":"2026-05-18T12:26:53.410803+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/4AXRQUTIDGIKCJ5F5EWCD77MMY","json":"https://pith.science/pith/4AXRQUTIDGIKCJ5F5EWCD77MMY.json","graph_json":"https://pith.science/api/pith-number/4AXRQUTIDGIKCJ5F5EWCD77MMY/graph.json","events_json":"https://pith.science/api/pith-number/4AXRQUTIDGIKCJ5F5EWCD77MMY/events.json","paper":"https://pith.science/paper/4AXRQUTI"},"agent_actions":{"view_html":"https://pith.science/pith/4AXRQUTIDGIKCJ5F5EWCD77MMY","download_json":"https://pith.science/pith/4AXRQUTIDGIKCJ5F5EWCD77MMY.json","view_paper":"https://pith.science/paper/4AXRQUTI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1212.1492&json=true","fetch_graph":"https://pith.science/api/pith-number/4AXRQUTIDGIKCJ5F5EWCD77MMY/graph.json","fetch_events":"https://pith.science/api/pith-number/4AXRQUTIDGIKCJ5F5EWCD77MMY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4AXRQUTIDGIKCJ5F5EWCD77MMY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4AXRQUTIDGIKCJ5F5EWCD77MMY/action/storage_attestation","attest_author":"https://pith.science/pith/4AXRQUTIDGIKCJ5F5EWCD77MMY/action/author_attestation","sign_citation":"https://pith.science/pith/4AXRQUTIDGIKCJ5F5EWCD77MMY/action/citation_signature","submit_replication":"https://pith.science/pith/4AXRQUTIDGIKCJ5F5EWCD77MMY/action/replication_record"}},"created_at":"2026-05-18T02:49:18.491917+00:00","updated_at":"2026-05-18T02:49:18.491917+00:00"}