{"paper":{"title":"A Liouville theorem for convex functions with periodic Monge-Amp\\`ere measure","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Hung V. Tran, Tianling Jin, Xushan Tu, Yanyan Li","submitted_at":"2025-11-19T01:38:59Z","abstract_excerpt":"We study global convex solutions of the Monge-Amp\\`ere equation \\[ \\det D^2 u = \\mu \\quad \\text{in } \\mathbb{R}^n, \\] where $\\mu \\not\\equiv 0$ is a nonnegative locally finite periodic Borel measure on $\\mathbb{R}^n$. We prove a Liouville-type theorem showing that every such solution admits a unique decomposition, up to an additive constant, as the sum of a quadratic polynomial and a periodic function. This extends earlier results of Caffarelli-Li and Li-Lu, which required $\\mu$ to have a density with regular or bounded logarithm, to the full generality of periodic measures, allowing degeneracy"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2511.15021","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/2511.15021/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"}