{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:A4VFHHN77LELXMHGCLVSGVDCVS","short_pith_number":"pith:A4VFHHN7","schema_version":"1.0","canonical_sha256":"072a539dbffac8bbb0e612eb235462acbaeb581481e421cd632cc3abcded0ad1","source":{"kind":"arxiv","id":"1409.7721","version":3},"attestation_state":"computed","paper":{"title":"Fractional elliptic equations, Caccioppoli estimates and regularity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.AP","authors_text":"L. A. Caffarelli, P. R. Stinga","submitted_at":"2014-09-26T20:44:41Z","abstract_excerpt":"Let $L=-\\operatorname{div}_x(A(x)\\nabla_x)$ be a uniformly elliptic operator in divergence form in a bounded domain $\\Omega$. We consider the fractional nonlocal equations $$\\begin{cases} L^su=f,&\\hbox{in}~\\Omega,\\\\ u=0,&\\hbox{on}~\\partial\\Omega, \\end{cases}\\quad \\hbox{and}\\quad \\begin{cases} L^su=f,&\\hbox{in}~\\Omega,\\\\ \\partial_Au=0,&\\hbox{on}~\\partial\\Omega. \\end{cases}$$ Here $L^s$, $0<s<1$, is the fractional power of $L$ and $\\partial_Au$ is the conormal derivative of $u$ with respect to the coefficients $A(x)$. We reproduce Caccioppoli type estimates that allow us to develop the regularit"},"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":"1409.7721","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-09-26T20:44:41Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"84c76a8a25ce556559ace3ab8eb4f3f7faf0677ed634e610e7a94538625d3083","abstract_canon_sha256":"c607fbe0dc550839e9065620dd76afb04331b14a81ccb12f082cb1ec0e128977"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:36:41.756105Z","signature_b64":"9ynRYgbY9+Vyv3Ms2UWlxz3DjVPhAkYucp81VygovGLN+uxYa4eXxgSeEUbAVgtXGqj8pUpLLru48gWtunP3Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"072a539dbffac8bbb0e612eb235462acbaeb581481e421cd632cc3abcded0ad1","last_reissued_at":"2026-05-18T00:36:41.755594Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:36:41.755594Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Fractional elliptic equations, Caccioppoli estimates and regularity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.AP","authors_text":"L. A. Caffarelli, P. R. Stinga","submitted_at":"2014-09-26T20:44:41Z","abstract_excerpt":"Let $L=-\\operatorname{div}_x(A(x)\\nabla_x)$ be a uniformly elliptic operator in divergence form in a bounded domain $\\Omega$. We consider the fractional nonlocal equations $$\\begin{cases} L^su=f,&\\hbox{in}~\\Omega,\\\\ u=0,&\\hbox{on}~\\partial\\Omega, \\end{cases}\\quad \\hbox{and}\\quad \\begin{cases} L^su=f,&\\hbox{in}~\\Omega,\\\\ \\partial_Au=0,&\\hbox{on}~\\partial\\Omega. \\end{cases}$$ Here $L^s$, $0<s<1$, is the fractional power of $L$ and $\\partial_Au$ is the conormal derivative of $u$ with respect to the coefficients $A(x)$. We reproduce Caccioppoli type estimates that allow us to develop the regularit"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.7721","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":"1409.7721","created_at":"2026-05-18T00:36:41.755692+00:00"},{"alias_kind":"arxiv_version","alias_value":"1409.7721v3","created_at":"2026-05-18T00:36:41.755692+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.7721","created_at":"2026-05-18T00:36:41.755692+00:00"},{"alias_kind":"pith_short_12","alias_value":"A4VFHHN77LEL","created_at":"2026-05-18T12:28:19.803747+00:00"},{"alias_kind":"pith_short_16","alias_value":"A4VFHHN77LELXMHG","created_at":"2026-05-18T12:28:19.803747+00:00"},{"alias_kind":"pith_short_8","alias_value":"A4VFHHN7","created_at":"2026-05-18T12:28:19.803747+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/A4VFHHN77LELXMHGCLVSGVDCVS","json":"https://pith.science/pith/A4VFHHN77LELXMHGCLVSGVDCVS.json","graph_json":"https://pith.science/api/pith-number/A4VFHHN77LELXMHGCLVSGVDCVS/graph.json","events_json":"https://pith.science/api/pith-number/A4VFHHN77LELXMHGCLVSGVDCVS/events.json","paper":"https://pith.science/paper/A4VFHHN7"},"agent_actions":{"view_html":"https://pith.science/pith/A4VFHHN77LELXMHGCLVSGVDCVS","download_json":"https://pith.science/pith/A4VFHHN77LELXMHGCLVSGVDCVS.json","view_paper":"https://pith.science/paper/A4VFHHN7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1409.7721&json=true","fetch_graph":"https://pith.science/api/pith-number/A4VFHHN77LELXMHGCLVSGVDCVS/graph.json","fetch_events":"https://pith.science/api/pith-number/A4VFHHN77LELXMHGCLVSGVDCVS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/A4VFHHN77LELXMHGCLVSGVDCVS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/A4VFHHN77LELXMHGCLVSGVDCVS/action/storage_attestation","attest_author":"https://pith.science/pith/A4VFHHN77LELXMHGCLVSGVDCVS/action/author_attestation","sign_citation":"https://pith.science/pith/A4VFHHN77LELXMHGCLVSGVDCVS/action/citation_signature","submit_replication":"https://pith.science/pith/A4VFHHN77LELXMHGCLVSGVDCVS/action/replication_record"}},"created_at":"2026-05-18T00:36:41.755692+00:00","updated_at":"2026-05-18T00:36:41.755692+00:00"}