{"paper":{"title":"Volume estimates on the critical sets of solutions to elliptic PDEs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.AP","authors_text":"Aaron Naber, Daniele Valtorta","submitted_at":"2014-03-17T17:21:26Z","abstract_excerpt":"In this paper we study solutions to elliptic linear equations $L(u)=\\partial_i(a^{ij}(x)\\partial_j u) + b^i(x) \\partial_i u + c(x) u=0$, either on $R^n$ or a Riemannian manifold, under the assumption of Lipschitz control on the coefficients $a^{ij}$. We focus our attention on the critical set $Cr(u)\\equiv\\{x:|\\nabla u|=0\\}$ and the singular set $S(u)\\equiv\\{x:u=|\\nabla u|=0\\}$, and more importantly on effective versions of these. Currently, under the coefficient control we have assumed, the strongest results in the literature say that the singular set is n-2-dimensional, however at this point "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.4176","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":""},"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"}