{"paper":{"title":"One phase problem for two positive harmonic function: below the codimension $1$ threshold","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.CV"],"primary_cat":"math.AP","authors_text":"Alexander Volberg","submitted_at":"2022-05-07T17:12:38Z","abstract_excerpt":"What can be said about the domain $\\Om$ in $\\bR^n$ for which its Green's function $G(z)$ satisfies $G(z)\\asymp \\dist (z, \\pd\\Om)^\\delta$? What can we say about $\\Om$ if the Boundary Harnack Principle holds in the form $u/v=\\text{real analytic}$ on the part $E$ of its boundary? Here $u, v$ are positive harmonic functions on $\\Om$ vanishing on $E$. Is this part of the boundary also nice?\n  We discuss these questions below and give answers in very special cases."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2205.03687","kind":"arxiv","version":3},"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/2205.03687/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"}