{"paper":{"title":"$C^\\infty$ regularity in semilinear free boundary problems","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Daniel Restrepo, Xavier Ros-Oton","submitted_at":"2024-07-29T21:35:21Z","abstract_excerpt":"We study the higher regularity of solutions and free boundaries in the Alt-Phillips problem $\\Delta u=u^{\\gamma-1}$, with $\\gamma\\in(0,1)$. Our main results imply that, once free boundaries are $C^{1,\\alpha}$, then they are $C^\\infty$. In addition $u/d^{\\frac{2}{2-\\gamma}}$ and $u^{\\frac{2-\\gamma}{2}}$ are $C^\\infty$ too.\n  In order to achieve this, we need to establish fine regularity estimates for solutions of linear equations with boundary-singular Hardy potentials $-\\Delta v = \\kappa v/d^2$ in $\\Omega$, where $d$ is the distance to the boundary and $\\kappa\\leq\\frac{1}{4}$. Interestingly, w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2407.20426","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2407.20426/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"}