{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:BTRUETJNI43TEG7WMMJ523GKNP","short_pith_number":"pith:BTRUETJN","schema_version":"1.0","canonical_sha256":"0ce3424d2d4737321bf66313dd6cca6bca61f8378850da531578c5c8e0005719","source":{"kind":"arxiv","id":"1904.06270","version":3},"attestation_state":"computed","paper":{"title":"A nonlocal free boundary problem with Wasserstein distance","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Aram Karakhanyan","submitted_at":"2019-04-12T15:21:11Z","abstract_excerpt":"We study the probability measures $\\rho\\in \\mathcal M(\\mathbb R^2)$ minimizing the functional \\[ J[\\rho]=\\iint \\log\\frac1{|x-y|}d\\rho(x)d\\rho(y)+d^2(\\rho, \\rho_0), \\] where $\\rho_0$ is a given probability measure and $d(\\rho, \\rho_0)$ is the 2-Wasserstein distance of $\\rho$ and $\\rho_0$. %\n  We prove the existence of minimizers $\\rho$ and show that the potential $U^\\rho=-\\log|x|\\ast \\rho$ solves a degenerate obstacle problem, the obstacle being the transport potential. Every minimizer $\\rho$ is absolutely continuous with respect to the Lebesgue measure. The singular set of the free boundary of"},"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":"1904.06270","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-04-12T15:21:11Z","cross_cats_sorted":[],"title_canon_sha256":"6972a40e7fda684585e20b43d1d12fe79557df894cc83d2b9acf27e0775d57e4","abstract_canon_sha256":"c4a5e4b00664b5b91f2f1ef6a60b1f07815e448020d9a1c6f3b8eb0d89a8dac4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:45:44.122929Z","signature_b64":"+llyYXXIHIXfvCf/mAFlfEZ+vKeKVAaoffbxat0Aq9q40JclBJ6kJQqAGE0ZMB586AvApPYwkGUksocrYj1sDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0ce3424d2d4737321bf66313dd6cca6bca61f8378850da531578c5c8e0005719","last_reissued_at":"2026-05-17T23:45:44.122141Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:45:44.122141Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A nonlocal free boundary problem with Wasserstein distance","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Aram Karakhanyan","submitted_at":"2019-04-12T15:21:11Z","abstract_excerpt":"We study the probability measures $\\rho\\in \\mathcal M(\\mathbb R^2)$ minimizing the functional \\[ J[\\rho]=\\iint \\log\\frac1{|x-y|}d\\rho(x)d\\rho(y)+d^2(\\rho, \\rho_0), \\] where $\\rho_0$ is a given probability measure and $d(\\rho, \\rho_0)$ is the 2-Wasserstein distance of $\\rho$ and $\\rho_0$. %\n  We prove the existence of minimizers $\\rho$ and show that the potential $U^\\rho=-\\log|x|\\ast \\rho$ solves a degenerate obstacle problem, the obstacle being the transport potential. Every minimizer $\\rho$ is absolutely continuous with respect to the Lebesgue measure. The singular set of the free boundary of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.06270","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":"1904.06270","created_at":"2026-05-17T23:45:44.122289+00:00"},{"alias_kind":"arxiv_version","alias_value":"1904.06270v3","created_at":"2026-05-17T23:45:44.122289+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1904.06270","created_at":"2026-05-17T23:45:44.122289+00:00"},{"alias_kind":"pith_short_12","alias_value":"BTRUETJNI43T","created_at":"2026-05-18T12:33:12.712433+00:00"},{"alias_kind":"pith_short_16","alias_value":"BTRUETJNI43TEG7W","created_at":"2026-05-18T12:33:12.712433+00:00"},{"alias_kind":"pith_short_8","alias_value":"BTRUETJN","created_at":"2026-05-18T12:33:12.712433+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/BTRUETJNI43TEG7WMMJ523GKNP","json":"https://pith.science/pith/BTRUETJNI43TEG7WMMJ523GKNP.json","graph_json":"https://pith.science/api/pith-number/BTRUETJNI43TEG7WMMJ523GKNP/graph.json","events_json":"https://pith.science/api/pith-number/BTRUETJNI43TEG7WMMJ523GKNP/events.json","paper":"https://pith.science/paper/BTRUETJN"},"agent_actions":{"view_html":"https://pith.science/pith/BTRUETJNI43TEG7WMMJ523GKNP","download_json":"https://pith.science/pith/BTRUETJNI43TEG7WMMJ523GKNP.json","view_paper":"https://pith.science/paper/BTRUETJN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1904.06270&json=true","fetch_graph":"https://pith.science/api/pith-number/BTRUETJNI43TEG7WMMJ523GKNP/graph.json","fetch_events":"https://pith.science/api/pith-number/BTRUETJNI43TEG7WMMJ523GKNP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BTRUETJNI43TEG7WMMJ523GKNP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BTRUETJNI43TEG7WMMJ523GKNP/action/storage_attestation","attest_author":"https://pith.science/pith/BTRUETJNI43TEG7WMMJ523GKNP/action/author_attestation","sign_citation":"https://pith.science/pith/BTRUETJNI43TEG7WMMJ523GKNP/action/citation_signature","submit_replication":"https://pith.science/pith/BTRUETJNI43TEG7WMMJ523GKNP/action/replication_record"}},"created_at":"2026-05-17T23:45:44.122289+00:00","updated_at":"2026-05-17T23:45:44.122289+00:00"}