{"paper":{"title":"Quantitative propagation of smallness for solutions of elliptic equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alexander Logunov, Eugenia Malinnikova","submitted_at":"2017-11-28T01:12:38Z","abstract_excerpt":"Let $u$ be a solution to an elliptic equation $\\text{div}(A\\nabla u)=0$ with Lipschitz coefficients in $\\mathbb{R}^n$.\n  Assume $|u|$ is bounded by $1$ in the ball $B=\\{|x|\\leq 1\\}$. We show that if $|u| < \\varepsilon$ on a set $ E \\subset \\frac{1}{2} B$ with positive $n$-dimensional Hausdorf measure, then $$|u|\\leq C\\varepsilon^\\gamma \\text{ on } \\frac{1}{2}B,$$ where $C>0, \\gamma \\in (0,1)$ do not depend on $u$ and depend only on $A$ and the measure of $E$.\n  We specify the dependence on the measure of $E$ in the form of the Remez type inequality. Similar estimate holds for sets $E$ with Hau"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.10076","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":""},"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"}