{"paper":{"title":"Nodal sets and continuity of eigenfunctions of Krein-Feller operators on Riemannian manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Sze-Man Ngai, Wen-Quan Zhao","submitted_at":"2024-12-13T09:46:13Z","abstract_excerpt":"Let $d\\geq1$, $\\Omega$ be a bounded domain of a smooth complete Riemannian d-manifold M, and $\\mu$ be a positive finite Borel measure with compact support in $\\overline{\\Omega}$. We prove the Courant nodal domain theorem for the eigenfunctions of Kre\\u{i}n-Feller operator $\\Delta_{\\mu}$ under the assumption that such eigenfunctions are continuous on $\\overline{\\Omega}$. For $d\\geq2$, We prove that on a bounded domain $\\Omega\\subset M$ with smooth boundary and on which the Green's function of the Laplace-Beltrami operator exists, the eigenfunctions of $\\Delta_{\\mu}$ are continuous on $\\Omega$. "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.10007","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/2412.10007/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"}