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We establish the estimate under the sharp conditions that the inhomogeneous term $f\\in C^{\\alpha}$ and the domains are convex and $C^{1,\\alpha}$ smooth. When $f\\in C^0$ (resp. $1/C<f<C$ for some positive constant $C$), we also obtain the global $W^{2,p}$ (resp. $W^{2,1+\\epsilon}$) regularity."},"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":"1806.09482","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-06-25T14:23:03Z","cross_cats_sorted":[],"title_canon_sha256":"82e3563d5e5f67236d3bc0ab6e54feb7edcd28b21359f1096e796ac8ebf5d401","abstract_canon_sha256":"cd97c9a0f27fcc0fe2573a36e472a1547e4fec18fd39bbe9326d603904e5e36f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:12:27.155381Z","signature_b64":"G6qb7HaMAk4yQRMCQU5GqOaTaAORNowu7I/jhoXM5ZD73BLWwDxaWY+3KkSmyESvAfucPBc4dF85wu7DoFkLAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d9c52731be67fb6b3aca8d6c696129781ebd5b89ae2faf1632aab6a042ec1d9a","last_reissued_at":"2026-05-18T00:12:27.154760Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:12:27.154760Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Boundary regularity for the second boundary-value problem of Monge-Amp\\`ere equations in dimension two","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jiakun Liu, Shibing Chen, Xu-Jia Wang","submitted_at":"2018-06-25T14:23:03Z","abstract_excerpt":"In this paper, we introduce an iteration argument to prove that a convex solution to the Monge-Amp\\`ere equation $\\mbox{det } D^2 u =f $ in dimension two subject to the natural boundary condition $Du(\\Omega) = \\Omega^*$ is $C^{2,\\alpha}$ smooth up to the boundary. 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