{"paper":{"title":"Volume Estimates for Singular sets and Critical Sets of Elliptic Equations with H\\\"older Coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.AP","authors_text":"Wenshuai Jiang, Yiqi Huang","submitted_at":"2023-09-15T01:14:58Z","abstract_excerpt":"Consider the solutions $u$ to the elliptic equation $\\mathcal{L}(u) = \\partial_i(a^{ij}(x) \\partial_j u) + b^i(x) \\partial_i u + c(x) u= 0$ with $a^{ij}$ assumed only to be H\\\"older continuous. In this paper we prove an explicit bound for $(n-2)$-dimensional Minkowski estimates of singular set $\\mathcal{S}(u) = \\{ x \\in B_1 : u(x) = |\\nabla u(x)| = 0\\}$ and critical set $\\mathcal{C}(u) \\equiv \\{ x\\in B_{1} : |\\nabla u(x)| = 0 \\}$ in terms of the bound on doubling index, depending on $c \\equiv 0$ or not. Here the H\\\"older assumption is sharp as it is the weakest condition in order to define the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2309.08089","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/2309.08089/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"}