{"paper":{"title":"Differential equations defined by Kre\\u{\\i}n-Feller operators on Riemannian manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Lei Ouyang, Sze-Man Ngai","submitted_at":"2024-08-09T04:29:34Z","abstract_excerpt":"We study linear and semi-linear wave, heat, and Schr\\\"odinger equations defined by Kre\\u{\\i}n-Feller operator $-\\Delta_\\mu$ on a complete Riemannian $n$-manifolds $M$, where $\\mu$ is a finite positive Borel measure on a bounded open subset $\\Omega$ of $M$ with support contained in $\\overline{\\Omega}$. Under the assumption that $\\underline{\\operatorname{dim}}_{\\infty}(\\mu)>n-2$, we prove that for a linear or semi-linear equation of each of the above three types, there exists a unique weak solution. We study the crucial condition $\\dim_(\\mu)>n-2$ and provide examples of measures on $\\mathbb{S}^2"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2408.04858","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/2408.04858/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"}