{"paper":{"title":"Geometric Gradient Flows from Elliptic Level Sets: Normal Decomposition and Reflection Dynamics","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Mohamed El Morsalani, Mohammed Barkatou","submitted_at":"2026-07-18T18:08:13Z","abstract_excerpt":"We investigate the asymptotic geometry of shifting superlevel sets $\\Omega_t = \\{x \\in \\Omega : u(x) > t\\}$ generated by solutions to the elliptic Dirichlet problem $-\\Delta u = f$ in $\\Omega$, where the non-negative source $f \\not\\equiv 0$ is compactly supported within a strictly convex inner core $C \\subset \\Omega$. Under a quantitative radial monotonicity condition, each boundary $\\partial\\Omega_t$ is a smooth normal graph over $\\partial C$ characterized by a thickness function $d_t \\in C^{1,\\alpha}(\\partial C)$ tracking $d_0$ as $t \\to 0$.A central contribution is a rigorous decomposition "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.16912","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/2607.16912/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"}