{"paper":{"title":"Quantitative stratification and sharp regularity estimates for supercritical semilinear elliptic equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Haotong Fu, Wei Wang, Zhifei Zhang","submitted_at":"2024-08-13T08:37:12Z","abstract_excerpt":"In this paper, we investigate the interior regularity theory for stationary solutions of the supercritical nonlinear elliptic equation $$ -\\Delta u=|u|^{p-1}u\\quad\\text{in }\\Omega,\\quad p>\\frac{n+2}{n-2}, $$ where $ \\Omega\\subset\\mathbb{R}^n $ is a bounded domain with $ n\\geq 3 $. Our primary focus is on the structure of stratification for the singular sets. We define the $ k $-th stratification $ S^k(u) $ of $ u $ based on the tangent functions and measures. We show that the Hausdorff dimension of $ S^k(u) $ is at most $ k $ and $ S^k(u) $ is $ k $-rectifiable, and establish estimates for vol"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2408.06726","kind":"arxiv","version":2},"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/2408.06726/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"}