{"paper":{"title":"Sharp H\\\"older regularity of weak solutions of the Neumann problem and applications to nonlocal PDE in metric measure spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.AP","authors_text":"Luca Capogna, Nageswari Shanmugalingam, Riikka Korte, Ryan Gibara","submitted_at":"2025-05-20T22:23:02Z","abstract_excerpt":"We prove global H\\\"older regularity result for weak solutions $u\\in N^{1,p}(\\Omega, \\mu)$ to a PDE of $p$-Laplacian type with a measure as non-homogeneous term: \\[ -\\text{div}\\!\\left( |\\nabla u|^{p-2}\\nabla u \\right)=\\overline\\nu, \\] \nwhere $1<p<\\infty$ and $\\overline\\nu \\in (N^{1,p}(\\Omega,\\mu))^*$ is a signed Radon measure supported in $\\overline \\Omega$. Here, $\\Omega$ is a John domain in a metric measure space satisfying a doubling condition and a $p$-Poincar\\'e inequality, and $\\nabla u$ is the Cheeger gradient. The regularity results obtained in this paper improve on earlier estimates pr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.14950","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/2505.14950/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"}