{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:XZ6TSLL46PM4PEO3R43RXPW36C","short_pith_number":"pith:XZ6TSLL4","schema_version":"1.0","canonical_sha256":"be7d392d7cf3d9c791db8f371bbedbf0a5abe0e31e2937b189f102506ea31340","source":{"kind":"arxiv","id":"2501.09487","version":1},"attestation_state":"computed","paper":{"title":"Higher Sobolev regularity on the mixed local and nonlocal p-Laplace equations","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Chao Zhang, Dingding Li, Yuzhou Fang","submitted_at":"2025-01-16T11:49:50Z","abstract_excerpt":"We develop a systematic study of the interior Sobolev regularity of weak solutions to the mixed local and nonlocal $p$-Laplace equations. To be precise, we show that the weak solution $u$ belongs to $W^{2, p}_\\mathrm{loc}$ and even $W^{2, 2}_{\\rm loc}$ Sobolev spaces in the subquadratic case, while $|\\nabla u|^{\\frac{p-2}{2}}\\nabla u$ is of the class $W^{1, 2}_\\mathrm{loc}$ in the superquadratic scenario, both of which coincide with that of the classical $p$-Laplace equations. Moreover, an improved higher fractional differentiability and integrability result $u\\in W^{1+\\beta, q}_\\mathrm{loc}$ "},"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":"2501.09487","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-01-16T11:49:50Z","cross_cats_sorted":[],"title_canon_sha256":"d4a41bb3fe0241832a8fd29edcc7c7e2cafbc0b9852761354be1173c1c326e75","abstract_canon_sha256":"dcc00930262e0cefc1e8bee16a7f75c6eb80f4d3ffa7bff833408d6620177eed"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:01:47.848676Z","signature_b64":"XErx/hO+hO8R6nhmJso7Qz1hL+ZjZ4MkZTbTHuFfqs6GAUnOyOKFSGZDg4ZLo2QZACOBacVyowS0K5OHuwYWBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"be7d392d7cf3d9c791db8f371bbedbf0a5abe0e31e2937b189f102506ea31340","last_reissued_at":"2026-07-05T10:01:47.848114Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:01:47.848114Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Higher Sobolev regularity on the mixed local and nonlocal p-Laplace equations","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Chao Zhang, Dingding Li, Yuzhou Fang","submitted_at":"2025-01-16T11:49:50Z","abstract_excerpt":"We develop a systematic study of the interior Sobolev regularity of weak solutions to the mixed local and nonlocal $p$-Laplace equations. To be precise, we show that the weak solution $u$ belongs to $W^{2, p}_\\mathrm{loc}$ and even $W^{2, 2}_{\\rm loc}$ Sobolev spaces in the subquadratic case, while $|\\nabla u|^{\\frac{p-2}{2}}\\nabla u$ is of the class $W^{1, 2}_\\mathrm{loc}$ in the superquadratic scenario, both of which coincide with that of the classical $p$-Laplace equations. Moreover, an improved higher fractional differentiability and integrability result $u\\in W^{1+\\beta, q}_\\mathrm{loc}$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2501.09487","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2501.09487/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2501.09487","created_at":"2026-07-05T10:01:47.848173+00:00"},{"alias_kind":"arxiv_version","alias_value":"2501.09487v1","created_at":"2026-07-05T10:01:47.848173+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2501.09487","created_at":"2026-07-05T10:01:47.848173+00:00"},{"alias_kind":"pith_short_12","alias_value":"XZ6TSLL46PM4","created_at":"2026-07-05T10:01:47.848173+00:00"},{"alias_kind":"pith_short_16","alias_value":"XZ6TSLL46PM4PEO3","created_at":"2026-07-05T10:01:47.848173+00:00"},{"alias_kind":"pith_short_8","alias_value":"XZ6TSLL4","created_at":"2026-07-05T10:01:47.848173+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2505.23288","citing_title":"Sobolev regularity for the nonlocal $(1, p)$-Laplace equations in the superquadratic case","ref_index":31,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/XZ6TSLL46PM4PEO3R43RXPW36C","json":"https://pith.science/pith/XZ6TSLL46PM4PEO3R43RXPW36C.json","graph_json":"https://pith.science/api/pith-number/XZ6TSLL46PM4PEO3R43RXPW36C/graph.json","events_json":"https://pith.science/api/pith-number/XZ6TSLL46PM4PEO3R43RXPW36C/events.json","paper":"https://pith.science/paper/XZ6TSLL4"},"agent_actions":{"view_html":"https://pith.science/pith/XZ6TSLL46PM4PEO3R43RXPW36C","download_json":"https://pith.science/pith/XZ6TSLL46PM4PEO3R43RXPW36C.json","view_paper":"https://pith.science/paper/XZ6TSLL4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2501.09487&json=true","fetch_graph":"https://pith.science/api/pith-number/XZ6TSLL46PM4PEO3R43RXPW36C/graph.json","fetch_events":"https://pith.science/api/pith-number/XZ6TSLL46PM4PEO3R43RXPW36C/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XZ6TSLL46PM4PEO3R43RXPW36C/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XZ6TSLL46PM4PEO3R43RXPW36C/action/storage_attestation","attest_author":"https://pith.science/pith/XZ6TSLL46PM4PEO3R43RXPW36C/action/author_attestation","sign_citation":"https://pith.science/pith/XZ6TSLL46PM4PEO3R43RXPW36C/action/citation_signature","submit_replication":"https://pith.science/pith/XZ6TSLL46PM4PEO3R43RXPW36C/action/replication_record"}},"created_at":"2026-07-05T10:01:47.848173+00:00","updated_at":"2026-07-05T10:01:47.848173+00:00"}