{"paper":{"title":"Critical sets of elliptic equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Aaron Naber, Daniele Valtorta, Jeff Cheeger","submitted_at":"2012-07-17T23:54:28Z","abstract_excerpt":"Given a solution $u$ to a linear homogeneous second order elliptic equation with Lipschitz coefficients, we introduce techniques for giving improved estimates of the critical set $\\Cr(u)\\equiv \\{x:|\\nabla u|(x)=0\\}$. The results are new even for harmonic functions on $\\dR^n$. Given such a $u$, the standard {\\it first order} stratification $\\{\\cS^k\\}$ of $u$ separates points $x$ based on the degrees of symmetry of the leading order polynomial of $u-u(x)$. In this paper we give a quantitative stratification $\\{\\cS^k_{\\eta,r}\\}$ of $u$, which separates points based on the number of {\\it almost} s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.4236","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}