{"paper":{"title":"Regularity and classification of the free boundary for a Monge-Amp\\`ere obstacle problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jingang Xiong, Tianling Jin, Xushan Tu","submitted_at":"2025-04-30T01:42:12Z","abstract_excerpt":"We study convex solutions to the Monge-Amp\\`ere obstacle problem \\[ \\operatorname{det} D^2 v=g v^q\\chi_{\\{v>0\\}}, \\quad v \\geq 0, \\] where $q \\in [0,n)$ is a constant and $g$ is a bounded positive function. This problem emerges from the $L_p$ Minkowski problem. We establish $C^{1, \\alpha}$ regularity for the strictly convex part of the free boundary $\\partial\\{v=0\\}$. Furthermore, when $g \\in C^{\\alpha}$, we prove a Schauder-type estimate. As a consequence, when $g\\equiv 1$, we obtain a Liouville theorem for entire solutions with unbounded coincidence sets $\\{v=0\\}$. Combined with existing res"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2504.21253","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/2504.21253/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"}