{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:HS3OWNEW56EOXOKRCG7PFS6XAM","short_pith_number":"pith:HS3OWNEW","schema_version":"1.0","canonical_sha256":"3cb6eb3496ef88ebb95111bef2cbd7031861438789e0c461eda0afbfae04aba7","source":{"kind":"arxiv","id":"1901.01196","version":2},"attestation_state":"computed","paper":{"title":"Regularity results for segregated configurations involving fractional Laplacian","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alessandro Zilio, Giorgio Tortone","submitted_at":"2019-01-04T16:27:29Z","abstract_excerpt":"We study the regularity of segregated profiles arising from competition - diffusion models, where the diffusion process is of nonlocal type and is driven by the fractional Laplacian of power $s \\in (0,1)$. Among others, our results apply to the regularity of the densities of an optimal partition problem involving the eigenvalues of the fractional Laplacian. More precisely, we show $C^{0,\\alpha^*}$ regularity of the density, where the exponent $\\alpha^*$ is explicit and is given by \\begin{equation*} \\alpha^* = \\begin{cases} s & \\text{for $s \\in (0,1/2]$}\\\\ 2s-1 &\\text{for $s \\in (1/2,1]$}.\\end{"},"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":"1901.01196","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-01-04T16:27:29Z","cross_cats_sorted":[],"title_canon_sha256":"a18b2ccd01fc8c40ae444120fae2be8f443c15828c5b5bace69e15e51619526a","abstract_canon_sha256":"f8c8427e1fd5400be00d85238ff086ac2d9bd677d401341a08eada7d80c49df0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:46:29.202805Z","signature_b64":"e59AgHRrV12z4PvETa0Z0Usadc8twlys72GQGsHp2fZkRoMU5QoacLCZ2ZpjC5b2CmqPzY8wVB1CP2HPKHJdAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3cb6eb3496ef88ebb95111bef2cbd7031861438789e0c461eda0afbfae04aba7","last_reissued_at":"2026-05-17T23:46:29.202184Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:46:29.202184Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Regularity results for segregated configurations involving fractional Laplacian","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alessandro Zilio, Giorgio Tortone","submitted_at":"2019-01-04T16:27:29Z","abstract_excerpt":"We study the regularity of segregated profiles arising from competition - diffusion models, where the diffusion process is of nonlocal type and is driven by the fractional Laplacian of power $s \\in (0,1)$. Among others, our results apply to the regularity of the densities of an optimal partition problem involving the eigenvalues of the fractional Laplacian. More precisely, we show $C^{0,\\alpha^*}$ regularity of the density, where the exponent $\\alpha^*$ is explicit and is given by \\begin{equation*} \\alpha^* = \\begin{cases} s & \\text{for $s \\in (0,1/2]$}\\\\ 2s-1 &\\text{for $s \\in (1/2,1]$}.\\end{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.01196","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":"1901.01196","created_at":"2026-05-17T23:46:29.202268+00:00"},{"alias_kind":"arxiv_version","alias_value":"1901.01196v2","created_at":"2026-05-17T23:46:29.202268+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1901.01196","created_at":"2026-05-17T23:46:29.202268+00:00"},{"alias_kind":"pith_short_12","alias_value":"HS3OWNEW56EO","created_at":"2026-05-18T12:33:18.533446+00:00"},{"alias_kind":"pith_short_16","alias_value":"HS3OWNEW56EOXOKR","created_at":"2026-05-18T12:33:18.533446+00:00"},{"alias_kind":"pith_short_8","alias_value":"HS3OWNEW","created_at":"2026-05-18T12:33:18.533446+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/HS3OWNEW56EOXOKRCG7PFS6XAM","json":"https://pith.science/pith/HS3OWNEW56EOXOKRCG7PFS6XAM.json","graph_json":"https://pith.science/api/pith-number/HS3OWNEW56EOXOKRCG7PFS6XAM/graph.json","events_json":"https://pith.science/api/pith-number/HS3OWNEW56EOXOKRCG7PFS6XAM/events.json","paper":"https://pith.science/paper/HS3OWNEW"},"agent_actions":{"view_html":"https://pith.science/pith/HS3OWNEW56EOXOKRCG7PFS6XAM","download_json":"https://pith.science/pith/HS3OWNEW56EOXOKRCG7PFS6XAM.json","view_paper":"https://pith.science/paper/HS3OWNEW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1901.01196&json=true","fetch_graph":"https://pith.science/api/pith-number/HS3OWNEW56EOXOKRCG7PFS6XAM/graph.json","fetch_events":"https://pith.science/api/pith-number/HS3OWNEW56EOXOKRCG7PFS6XAM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HS3OWNEW56EOXOKRCG7PFS6XAM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HS3OWNEW56EOXOKRCG7PFS6XAM/action/storage_attestation","attest_author":"https://pith.science/pith/HS3OWNEW56EOXOKRCG7PFS6XAM/action/author_attestation","sign_citation":"https://pith.science/pith/HS3OWNEW56EOXOKRCG7PFS6XAM/action/citation_signature","submit_replication":"https://pith.science/pith/HS3OWNEW56EOXOKRCG7PFS6XAM/action/replication_record"}},"created_at":"2026-05-17T23:46:29.202268+00:00","updated_at":"2026-05-17T23:46:29.202268+00:00"}