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We prove that u\\in C^{1,1}(\\Omega) under the A3 condition and A3w^+ condition respectively. In the former case, we construct a suitable auxiliary function to obtain uniform {\\it a priori} estimates directly. In the latter case, the main argument is to establish the Pogorelov type estimates, which are interesting independently."},"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":"1806.01720","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-06-05T14:39:21Z","cross_cats_sorted":[],"title_canon_sha256":"151fb7516e582f6d473e974229a068e9025b67440264bde94f918fada738f7f4","abstract_canon_sha256":"41ba8e8653a9265b66716de267049896a093b396b1f02788a89969a8da2b2390"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:14:11.966821Z","signature_b64":"1yYz+XUIEE1a3DmFu+nqxm6wARhcBbJ7aFIgc+7evgSjesBlIqLYeF74dP1fpRxBC1WeCvUVZXfSNVdVQbtlBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"311b30dad95732f2be528b3776e6e4623186d24165a9d235c444ce71c165281e","last_reissued_at":"2026-05-18T00:14:11.966237Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:14:11.966237Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Interior C^{1,1} regularity of solutions to degenerate Monge-Amp\\`{e}re type equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Feida Jiang, Juhua Shi, Xiaoping Yang","submitted_at":"2018-06-05T14:39:21Z","abstract_excerpt":"In this paper, we study the interior C^{1,1} regularity of viscosity solutions for a degenerate Monge-Amp\\`{e}re type equation \\det[D^{2}u-A(x, u, Du)]=B(x, u, Du) when B \\geq 0 and B^{\\frac{1}{n-1}}\\in C^{1,1}(\\bar{\\Omega}\\times\\mathbb{R}\\times \\mathbb{R}^n). We prove that u\\in C^{1,1}(\\Omega) under the A3 condition and A3w^+ condition respectively. In the former case, we construct a suitable auxiliary function to obtain uniform {\\it a priori} estimates directly. In the latter case, the main argument is to establish the Pogorelov type estimates, which are interesting independently."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.01720","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":""},"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":"1806.01720","created_at":"2026-05-18T00:14:11.966330+00:00"},{"alias_kind":"arxiv_version","alias_value":"1806.01720v1","created_at":"2026-05-18T00:14:11.966330+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.01720","created_at":"2026-05-18T00:14:11.966330+00:00"},{"alias_kind":"pith_short_12","alias_value":"GENTBWWZK4ZP","created_at":"2026-05-18T12:32:25.280505+00:00"},{"alias_kind":"pith_short_16","alias_value":"GENTBWWZK4ZPFPSS","created_at":"2026-05-18T12:32:25.280505+00:00"},{"alias_kind":"pith_short_8","alias_value":"GENTBWWZ","created_at":"2026-05-18T12:32:25.280505+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/GENTBWWZK4ZPFPSSRM3XNZXEMI","json":"https://pith.science/pith/GENTBWWZK4ZPFPSSRM3XNZXEMI.json","graph_json":"https://pith.science/api/pith-number/GENTBWWZK4ZPFPSSRM3XNZXEMI/graph.json","events_json":"https://pith.science/api/pith-number/GENTBWWZK4ZPFPSSRM3XNZXEMI/events.json","paper":"https://pith.science/paper/GENTBWWZ"},"agent_actions":{"view_html":"https://pith.science/pith/GENTBWWZK4ZPFPSSRM3XNZXEMI","download_json":"https://pith.science/pith/GENTBWWZK4ZPFPSSRM3XNZXEMI.json","view_paper":"https://pith.science/paper/GENTBWWZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1806.01720&json=true","fetch_graph":"https://pith.science/api/pith-number/GENTBWWZK4ZPFPSSRM3XNZXEMI/graph.json","fetch_events":"https://pith.science/api/pith-number/GENTBWWZK4ZPFPSSRM3XNZXEMI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GENTBWWZK4ZPFPSSRM3XNZXEMI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GENTBWWZK4ZPFPSSRM3XNZXEMI/action/storage_attestation","attest_author":"https://pith.science/pith/GENTBWWZK4ZPFPSSRM3XNZXEMI/action/author_attestation","sign_citation":"https://pith.science/pith/GENTBWWZK4ZPFPSSRM3XNZXEMI/action/citation_signature","submit_replication":"https://pith.science/pith/GENTBWWZK4ZPFPSSRM3XNZXEMI/action/replication_record"}},"created_at":"2026-05-18T00:14:11.966330+00:00","updated_at":"2026-05-18T00:14:11.966330+00:00"}