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Specifically, we prove that under the zero Dirichlet boundary condition on $\\{x_n=0\\}$, the optimal regularity of $u$ up to the flat boundary $\\{x_n=0\\}$ is $C^{1,1-\\gamma}$. Moreover, for the homogeneous equations, we establish that the optimal regularity of $u$ is $C^{2,1-\\gamma}$ in the spatial variables, and that $x_n^{-\\gamma}u$ is smooth in the variables $x'$ and $t$."},"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":"2504.04824","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-04-07T08:25:02Z","cross_cats_sorted":[],"title_canon_sha256":"47754b214f388f46367f3f10e01990f5272cacd6884f46bb6aa046b57931024a","abstract_canon_sha256":"df773646ef3108532c4c034ba3f8c128498bd654b59d8c2a3e81630e012efe2f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:46:04.790359Z","signature_b64":"5W9L6PZTe81oIlW3dyG1fhRUdgunsO+spVFb89f1cZMsT77HAXjr5/8h8GRSynsyjK7UfwrFg5KzHSKcy4vOBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c825c7c4b39fa9ac47ebf4fc960ed888ce58edced34a4e0358a7ed32adbe5749","last_reissued_at":"2026-07-05T10:46:04.789871Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:46:04.789871Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Optimal regularity for degenerate parabolic equations on a flat boundary","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Hyungsung Yun","submitted_at":"2025-04-07T08:25:02Z","abstract_excerpt":"We establish the optimal regularity of viscosity solutions to\n  \\begin{equation*}\n  u_t - x_n^\\gamma \\Delta u = f,\n  \\end{equation*} which arises in the regularity theory for the porous medium equation. Specifically, we prove that under the zero Dirichlet boundary condition on $\\{x_n=0\\}$, the optimal regularity of $u$ up to the flat boundary $\\{x_n=0\\}$ is $C^{1,1-\\gamma}$. 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