{"paper":{"title":"On the Boundary Behavior of Positive Solutions of Elliptic Differential Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.CA","authors_text":"A.A.Logunov","submitted_at":"2013-06-15T12:35:34Z","abstract_excerpt":"Let $u$ be a positive harmonic function in the unit ball $B_1 \\subset \\mathbb{R}^n$ and let $\\mu$ be the boundary measure of $u$. Consider a point $x\\in \\partial B_1$ and let $n(x)$ denote the unit normal vector at $x$. Let $\\alpha$ be a number in $(-1,n-1]$ and $A \\in [0,+\\infty) $. We prove that $u(x+n(x)t)t^{\\alpha} \\to A$ as $t \\to +0$ if and only if $\\frac{\\mu({B_r(x)})}{r^{n-1}} r^{\\alpha} \\to C_\\alpha A$ as $r\\to+0$, where ${C_\\alpha= \\frac{\\pi^{n/2}}{\\Gamma(\\frac{n-\\alpha+1}{2})\\Gamma(\\frac{\\alpha+1}{2})}}$. For $\\alpha=0$ it follows from the theorems by Rudin and Loomis which claim th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.3571","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"}