{"paper":{"title":"Analysis of a free boundary at contact points with Lipschitz data","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Aram Karakhanyan, Henrik Shahgholian","submitted_at":"2012-05-22T20:36:39Z","abstract_excerpt":"In this paper we consider a minimization problem for the functional $$ J(u)=\\int_{B_1^+}|\\nabla u|\\sp 2+\\lambda_{+}^2\\chi_{\\{u>0\\}}+\\lambda_{-}^2\\chi_{\\{u\\leq0\\}}, $$ in the upper half ball $B_1^+\\subset\\R^n, n\\geq 2$ subject to a Lipschitz continuous Dirichlet data on $\\partial B_1^+$. More precisely we assume that $0\\in \\partial \\{u>0\\}$ and the derivative of the boundary data has a jump discontinuity. If $0\\in \\bar{\\partial(\\{u>0\\} \\cap B_1^+)}$ then (for $n=2$ or $n>3$ and one-phase case) we prove, among other things, that the free boundary $\\partial \\{u>0\\}$ approaches the origin along on"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.5052","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"}