{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:WYZPZFK57TP6X7BEVE4FGEZ67Y","short_pith_number":"pith:WYZPZFK5","schema_version":"1.0","canonical_sha256":"b632fc955dfcdfebfc24a93853133efe03eaba53831b4916f439b30edd3e466e","source":{"kind":"arxiv","id":"1508.07447","version":3},"attestation_state":"computed","paper":{"title":"Stratification of free boundary points for a two-phase variational problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Aram L. Karakhanyan, Serena Dipierro","submitted_at":"2015-08-29T13:18:45Z","abstract_excerpt":"In this paper we study the two-phase Bernoulli type free boundary problem arising from the minimization of the functional $$ J(u):=\\int_{\\Omega}|\\nabla u|^p +\\lambda_+^p\\,\\chi_{\\{u>0\\}} +\\lambda_-^p\\,\\chi_{\\{u\\le 0\\}}, \\quad 1<p<\\infty. $$ Here $\\Omega \\subset \\R^N$ is a bounded smooth domain and $\\lambda_\\pm$ are positive constants such that $\\lambda_+^p-\\lambda^p_->0$. We prove the following dichotomy: if $x_0$ is a free boundary point then either the free boundary is smooth near $x_0$ or $u$ has linear growth at $x_0$. Furthermore, we show that for $p>1$ the free boundary has locally finite"},"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":"1508.07447","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-08-29T13:18:45Z","cross_cats_sorted":[],"title_canon_sha256":"ce0d2bd89064bd4e7d7e70fa45cb494234334190f182a215655188e908e06a1a","abstract_canon_sha256":"dc885833372f6c2c3b93d58c0099d5747185f0935606635c9307252e54b389dc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:24:38.206130Z","signature_b64":"5wpkYYrszJ5uNKicIBTpaseDqV/qB1iSK84IbSeCfBFmTX6mPkkmSL7rVvPXftKniOWu5OYEcqyL2oBzq4rwCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b632fc955dfcdfebfc24a93853133efe03eaba53831b4916f439b30edd3e466e","last_reissued_at":"2026-05-18T01:24:38.205611Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:24:38.205611Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Stratification of free boundary points for a two-phase variational problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Aram L. Karakhanyan, Serena Dipierro","submitted_at":"2015-08-29T13:18:45Z","abstract_excerpt":"In this paper we study the two-phase Bernoulli type free boundary problem arising from the minimization of the functional $$ J(u):=\\int_{\\Omega}|\\nabla u|^p +\\lambda_+^p\\,\\chi_{\\{u>0\\}} +\\lambda_-^p\\,\\chi_{\\{u\\le 0\\}}, \\quad 1<p<\\infty. $$ Here $\\Omega \\subset \\R^N$ is a bounded smooth domain and $\\lambda_\\pm$ are positive constants such that $\\lambda_+^p-\\lambda^p_->0$. We prove the following dichotomy: if $x_0$ is a free boundary point then either the free boundary is smooth near $x_0$ or $u$ has linear growth at $x_0$. Furthermore, we show that for $p>1$ the free boundary has locally finite"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.07447","kind":"arxiv","version":3},"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":"1508.07447","created_at":"2026-05-18T01:24:38.205699+00:00"},{"alias_kind":"arxiv_version","alias_value":"1508.07447v3","created_at":"2026-05-18T01:24:38.205699+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.07447","created_at":"2026-05-18T01:24:38.205699+00:00"},{"alias_kind":"pith_short_12","alias_value":"WYZPZFK57TP6","created_at":"2026-05-18T12:29:47.479230+00:00"},{"alias_kind":"pith_short_16","alias_value":"WYZPZFK57TP6X7BE","created_at":"2026-05-18T12:29:47.479230+00:00"},{"alias_kind":"pith_short_8","alias_value":"WYZPZFK5","created_at":"2026-05-18T12:29:47.479230+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/WYZPZFK57TP6X7BEVE4FGEZ67Y","json":"https://pith.science/pith/WYZPZFK57TP6X7BEVE4FGEZ67Y.json","graph_json":"https://pith.science/api/pith-number/WYZPZFK57TP6X7BEVE4FGEZ67Y/graph.json","events_json":"https://pith.science/api/pith-number/WYZPZFK57TP6X7BEVE4FGEZ67Y/events.json","paper":"https://pith.science/paper/WYZPZFK5"},"agent_actions":{"view_html":"https://pith.science/pith/WYZPZFK57TP6X7BEVE4FGEZ67Y","download_json":"https://pith.science/pith/WYZPZFK57TP6X7BEVE4FGEZ67Y.json","view_paper":"https://pith.science/paper/WYZPZFK5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1508.07447&json=true","fetch_graph":"https://pith.science/api/pith-number/WYZPZFK57TP6X7BEVE4FGEZ67Y/graph.json","fetch_events":"https://pith.science/api/pith-number/WYZPZFK57TP6X7BEVE4FGEZ67Y/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WYZPZFK57TP6X7BEVE4FGEZ67Y/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WYZPZFK57TP6X7BEVE4FGEZ67Y/action/storage_attestation","attest_author":"https://pith.science/pith/WYZPZFK57TP6X7BEVE4FGEZ67Y/action/author_attestation","sign_citation":"https://pith.science/pith/WYZPZFK57TP6X7BEVE4FGEZ67Y/action/citation_signature","submit_replication":"https://pith.science/pith/WYZPZFK57TP6X7BEVE4FGEZ67Y/action/replication_record"}},"created_at":"2026-05-18T01:24:38.205699+00:00","updated_at":"2026-05-18T01:24:38.205699+00:00"}