{"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 $C^{1,1-\\gamma}$. 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 $t$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2504.04824","kind":"arxiv","version":2},"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/2504.04824/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"}