{"paper":{"title":"Properties of Mean Value Sets: Angle Conditions, Blowup Solutions, and Nonconvexity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Niles Armstrong","submitted_at":"2018-01-20T05:42:01Z","abstract_excerpt":"We study the mean values sets of the second order divergence form elliptic operator with principal coefficients defined as $$a^{ij}_k(x):= \\begin{cases} \\alpha_k \\delta^{ij}(x) &x_n>0 \\beta_k \\delta^{ij}(x) &x_n<0. \\end{cases}$$ In particular, we will show that the mean value sets associated to such an operator need not be convex as $\\alpha_k$ and $\\beta_k$ converge to 1. This example then leads to an example of nonconvex mean value sets for smooth $a^{ij}(x)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.06627","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"}